{"version":2,"work":"monographs-physics","title":"Physics","source_revision":"4a90f329bcdcd73cca3a269ccc18fd8ba132ec08","sources":[{"repository_path":"monographs/physics/README.md","file_url":"documents/files/04b72724bf5d4c550c3cf22bf2293d28dbff92a8/README.md","url":"read/monographs-physics/readme-f6065a976c8a.html","sha256":"11404507e4c1db9fe11e25f4745cf5f31e8834f6dd73eade986e8a2960752525","git_blob_sha1":"04b72724bf5d4c550c3cf22bf2293d28dbff92a8","size":6336,"source_url":"read/monographs-physics/readme-f6065a976c8a-source.html","passages":0,"macros":{}},{"repository_path":"monographs/physics/formal/continuum/README.md","file_url":"documents/files/6f2f391e5728f9dcd4a2bd7ecd0498b01502cec6/README.md","url":"read/monographs-physics/readme-64e53a93e35f.html","sha256":"ee8540b9bad13a8912475b43c67804fdc3a37c1d153c1102f7bc57e75bc76d29","git_blob_sha1":"6f2f391e5728f9dcd4a2bd7ecd0498b01502cec6","size":4578,"source_url":"read/monographs-physics/readme-64e53a93e35f-source.html","passages":0,"macros":{}},{"repository_path":"monographs/physics/formal/continuum/docs/BLOCK_51_PROOF_GUIDE.md","file_url":"documents/files/317db1b2267bcd7db2942c96362ff562062b00ea/BLOCK_51_PROOF_GUIDE.md","url":"read/monographs-physics/block-51-proof-guide-4f43f6d94bc2.html","sha256":"f9f259aebaeeceae9a43dab653c092314749f070447a332dbbae08814d7d3385","git_blob_sha1":"317db1b2267bcd7db2942c96362ff562062b00ea","size":5045,"source_url":"read/monographs-physics/block-51-proof-guide-4f43f6d94bc2-source.html","passages":0,"macros":{}},{"repository_path":"monographs/physics/formal/continuum/docs/SOURCE_MAP.md","file_url":"documents/files/bba79d18012aea0813ac98dc86a1954a46c865f1/SOURCE_MAP.md","url":"read/monographs-physics/source-map-9723347720ec.html","sha256":"6b70d04fdc3422dd4793975b25b0773c1d7797b509d94317c4509775f9243798","git_blob_sha1":"bba79d18012aea0813ac98dc86a1954a46c865f1","size":88636,"source_url":"read/monographs-physics/source-map-9723347720ec-source.html","passages":0,"macros":{}},{"repository_path":"monographs/physics/source/monographs/Physics_Adic_Monograph.tex","file_url":"documents/files/c1a0b13283fb22c9a959d1c04bcec0137cfdf47d/Physics_Adic_Monograph.tex","url":"read/monographs-physics/physics-adic-monograph-a950584a1959.html","sha256":"94c52ebb1c83ba0b0a8327ddd6b17ee141a45640e40ce57d0f99ac652fa703e9","git_blob_sha1":"c1a0b13283fb22c9a959d1c04bcec0137cfdf47d","size":1423,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/Physics_Dirac_Monograph.tex","file_url":"documents/files/6e81319739e5c381ebeb045319e0410a6d7a927e/Physics_Dirac_Monograph.tex","url":"read/monographs-physics/physics-dirac-monograph-754844b78e43.html","sha256":"41ddd77839c75a454cd5e6cd8687097659d11d02f0265bc520f0716a43e73191","git_blob_sha1":"6e81319739e5c381ebeb045319e0410a6d7a927e","size":827,"passages":0,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian"}},{"repository_path":"monographs/physics/source/monographs/Physics_Foundations_Monograph.tex","file_url":"documents/files/43c9bae76c6cfe4ed64fae6e2edfc19e1858ee18/Physics_Foundations_Monograph.tex","url":"read/monographs-physics/physics-foundations-monograph-5b900a831427.html","sha256":"4622d72ad1f1d75602b8c841fbd8c60e23e5997e9a698750c69c9279da99887a","git_blob_sha1":"43c9bae76c6cfe4ed64fae6e2edfc19e1858ee18","size":1229,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/Physics_Gauge_Monograph.tex","file_url":"documents/files/c3b255f69dd7bf18003b6abb872a63dc77a80fa0/Physics_Gauge_Monograph.tex","url":"read/monographs-physics/physics-gauge-monograph-d0de97bb9896.html","sha256":"f997847956e1a85ed5b92cf1fdc3f89c87371354fe59e50c5264d7e1ae6e60c6","git_blob_sha1":"c3b255f69dd7bf18003b6abb872a63dc77a80fa0","size":1221,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/Physics_Gravity_Monograph.tex","file_url":"documents/files/8d0ca3189e695445ec22a757b6f5a32e3fb4d80e/Physics_Gravity_Monograph.tex","url":"read/monographs-physics/physics-gravity-monograph-dd441fcb4146.html","sha256":"9fb8e428a38a9924229029f26c009faf8797c61762e45dbccd45586a61231439","git_blob_sha1":"8d0ca3189e695445ec22a757b6f5a32e3fb4d80e","size":1285,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/adic/conclusion.tex","file_url":"documents/files/fbe3db01374d48c43e00755715433a54d321b771/conclusion.tex","url":"read/monographs-physics/conclusion-f04a1fc4816b.html","sha256":"0e9be3993ffae18b7e0507a863fba65eac7fdac547ae2d2962436863a5f36668","git_blob_sha1":"fbe3db01374d48c43e00755715433a54d321b771","size":3121,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/adic/finite.tex","file_url":"documents/files/1aaf2d90503af3739b934f4cc80369110191d43f/finite.tex","url":"read/monographs-physics/finite-5615baf947f5.html","sha256":"ec3e1efbedafb2e7a4e1d03cd0a026fb618a884410a647bb52009bed166933a1","git_blob_sha1":"1aaf2d90503af3739b934f4cc80369110191d43f","size":35030,"passages":374,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/adic/limit.tex","file_url":"documents/files/f26611685e6cb05edd38860f3b41394c14d2b3f9/limit.tex","url":"read/monographs-physics/limit-d68952770694.html","sha256":"e96fc6e6b380a61b944ffe2b44c6adc721d9fbc2c39f06797d04162f09709462","git_blob_sha1":"f26611685e6cb05edd38860f3b41394c14d2b3f9","size":41380,"passages":402,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/adic/orientation.tex","file_url":"documents/files/08e01e3b7e95db5864298361dbcc77e211159cef/orientation.tex","url":"read/monographs-physics/orientation-aab477144a67.html","sha256":"13c968ec2a3a07f468c075622f950be7691bcd39d845c5aaabf04e3f0365b8a6","git_blob_sha1":"08e01e3b7e95db5864298361dbcc77e211159cef","size":4837,"passages":10,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement","arraystretch":"1.1"}},{"repository_path":"monographs/physics/source/monographs/adic/primary_context.tex","file_url":"documents/files/0ffcebe006a399995dcd4e5156ad27efd71d79dd/primary_context.tex","url":"read/monographs-physics/primary-context-11f743f336a3.html","sha256":"835c7b335d36c32bd1654a1c700dac0aa53b3cc929febeb0d07c90bcd474aa2b","git_blob_sha1":"0ffcebe006a399995dcd4e5156ad27efd71d79dd","size":3015,"passages":5,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/adic/references.tex","file_url":"documents/files/c0966131f8251a627759500a8bf6b41a2572364b/references.tex","url":"read/monographs-physics/references-73694716926b.html","sha256":"669596d0f21687b135706da96d8f0031a36472c2776d3ea54fedcf72ea7dfabe","git_blob_sha1":"c0966131f8251a627759500a8bf6b41a2572364b","size":1167,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/adic/spectral.tex","file_url":"documents/files/f4a0c81e437f692562f76e4fc05c14d68c227ffc/spectral.tex","url":"read/monographs-physics/spectral-f81b115ca482.html","sha256":"ca07996b0cf10cd2608fa36d1bf9c88d10624db2d972be40e26c9c76e57f80e1","git_blob_sha1":"f4a0c81e437f692562f76e4fc05c14d68c227ffc","size":54059,"passages":564,"macros":{"KomorebiEdition":"","KomorebiTitle":"Adic and Polyphase Quantum Refinement","KomorebiCoverTitle":"Adic and Polyphase\\\\Quantum Refinement","KomorebiSubtitle":"Normalized packets, spectral elimination, and compatible limits","KomorebiKind":"Adic refinement monograph","KomorebiShortTitle":"Adic and polyphase quantum refinement","KomorebiCoverBlurb":"A connected development of prime registers, Haar refinement, radial self-energy, polyphase spectral calculus, and infinite Hilbert-space dynamics under explicit compatibility and domain hypotheses.","KomorebiChapterLabel":"06 / Arithmetic refinement"}},{"repository_path":"monographs/physics/source/monographs/chapters/boundary.tex","file_url":"documents/files/8f0d0038d4b8402b065d0a82842c50ae4b2058ae/boundary.tex","url":"read/monographs-physics/boundary-c25af69483e7.html","sha256":"49873e79c927df03137343dc0d54b5906379567fa31efb703c620fe20265f70d","git_blob_sha1":"8f0d0038d4b8402b065d0a82842c50ae4b2058ae","size":4286,"passages":1,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian"}},{"repository_path":"monographs/physics/source/monographs/chapters/continuum.tex","file_url":"documents/files/54289a1341586b0c03ad80dc79d09aa5a0588a57/continuum.tex","url":"read/monographs-physics/continuum-8fc11e95c83d.html","sha256":"3b933e1b8a5f29892255b8b123001ab8dc67ae53b731332e0dc247c4a5caf247","git_blob_sha1":"54289a1341586b0c03ad80dc79d09aa5a0588a57","size":17811,"passages":159,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian"}},{"repository_path":"monographs/physics/source/monographs/chapters/finite_core.tex","file_url":"documents/files/097ed1720b2e2b038d2a23d347ff48e2d65b8156/finite_core.tex","url":"read/monographs-physics/finite-core-8036200b13d4.html","sha256":"26b5f5f746d574e1b0e249ce1e0bf5d4bb40ae793f593746e82a2f6e6cb4ea8c","git_blob_sha1":"097ed1720b2e2b038d2a23d347ff48e2d65b8156","size":35099,"passages":429,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian"}},{"repository_path":"monographs/physics/source/monographs/chapters/orientation.tex","file_url":"documents/files/9771bbb40f642220a9600a06b8575c30e3e4f4fd/orientation.tex","url":"read/monographs-physics/orientation-68ad7e8cdbae.html","sha256":"0c861d39165e1eedcef1d13b355abf38efaf34a0eee1a02038aad2f86d2dbaa1","git_blob_sha1":"9771bbb40f642220a9600a06b8575c30e3e4f4fd","size":7568,"passages":20,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian"}},{"repository_path":"monographs/physics/source/monographs/foundations/conclusion.tex","file_url":"documents/files/0b4e5d5b776d8a8f221a732c6c27ddf7fd21b4cd/conclusion.tex","url":"read/monographs-physics/conclusion-1dc29c9e913e.html","sha256":"7602b8e812d414be31cb29b7c2758951a317f1e08462ad46b0766b16fdcd0827","git_blob_sha1":"0b4e5d5b776d8a8f221a732c6c27ddf7fd21b4cd","size":3125,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/foundations/core.tex","file_url":"documents/files/782f12ce4cf7c36d33bd699624065ea7d00d1ae6/core.tex","url":"read/monographs-physics/core-450aa2de1d03.html","sha256":"76cd0c4ad10394a57c6dfad6a0483a7fdb821aef178da1bcbeceaec99f7acecf","git_blob_sha1":"782f12ce4cf7c36d33bd699624065ea7d00d1ae6","size":32514,"passages":246,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/foundations/extensions.tex","file_url":"documents/files/5e0cee0b95cc2317c307f9191f238587cf5fc83d/extensions.tex","url":"read/monographs-physics/extensions-60e2da9e1d0a.html","sha256":"0614cebe965d624b840a124a7be938aa79b701b25ca5776ca02395b51ffc1ad4","git_blob_sha1":"5e0cee0b95cc2317c307f9191f238587cf5fc83d","size":29627,"passages":323,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/foundations/orientation.tex","file_url":"documents/files/df90602c233684b71595e2e45e809de460855ade/orientation.tex","url":"read/monographs-physics/orientation-cc9521b55bbb.html","sha256":"2e34fb66ddb5bea712dca267aecbf14611884067507e62aaf45b0b3850abc86b","git_blob_sha1":"df90602c233684b71595e2e45e809de460855ade","size":5367,"passages":24,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/foundations/references.tex","file_url":"documents/files/9dce5b7a71f47d09198286e4a0ecec20d897fe48/references.tex","url":"read/monographs-physics/references-bbaee1bdca5b.html","sha256":"83dd34076b0f3d94ccb698bff9820f695b4ac268672b03fb0f8b18fa4eaab079","git_blob_sha1":"9dce5b7a71f47d09198286e4a0ecec20d897fe48","size":2529,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/foundations/refinement.tex","file_url":"documents/files/8768d239f0f3e70c632d3f19c47a8d86754f2c41/refinement.tex","url":"read/monographs-physics/refinement-48c14f768b57.html","sha256":"3d70acff549562ad60a3dd5b180b35fde5ad8e51668e3c4a227cd9e761414d2b","git_blob_sha1":"8768d239f0f3e70c632d3f19c47a8d86754f2c41","size":4478,"passages":45,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Foundations monograph","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics"}},{"repository_path":"monographs/physics/source/monographs/gauge/conclusion.tex","file_url":"documents/files/e55d4ae907be24bd51363d84eb572036619aeb5b/conclusion.tex","url":"read/monographs-physics/conclusion-e6e63f08dbdf.html","sha256":"25bc071844f43ffb0dd80e7b1e1295d895d8274079c5a3f14880e8feb2db69a9","git_blob_sha1":"e55d4ae907be24bd51363d84eb572036619aeb5b","size":3258,"passages":1,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/gauge/core.tex","file_url":"documents/files/678fa1195559aa4e21d3783cccabd4411a7e94f3/core.tex","url":"read/monographs-physics/core-c295f2f072e4.html","sha256":"f6ae4ad8b9d0d2b114f64d5bfc3aad2c391fa03e3b6417906eb1fbd2e9061334","git_blob_sha1":"678fa1195559aa4e21d3783cccabd4411a7e94f3","size":30491,"passages":264,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/gauge/orientation.tex","file_url":"documents/files/df1147a16081c7414c9dff1a17402e0763e1e062/orientation.tex","url":"read/monographs-physics/orientation-52457df68ec1.html","sha256":"b93775387dca8c68c3380c7e1d5fbec53795097cecd8460db23aa0bc6fb861c5","git_blob_sha1":"df1147a16081c7414c9dff1a17402e0763e1e062","size":4745,"passages":22,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/gauge/references.tex","file_url":"documents/files/bd3276156e8ce3443301dabc55e34927edb26c1d/references.tex","url":"read/monographs-physics/references-f5ba9b8199c4.html","sha256":"f15d1a4ed77897ec17fecc1bf0a1427eb6f435595bc7edd36696a7a12fcafa11","git_blob_sha1":"bd3276156e8ce3443301dabc55e34927edb26c1d","size":3899,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/gauge/refinement.tex","file_url":"documents/files/52bd1404b3803a6d0a8c50772a08df237ac8e3a1/refinement.tex","url":"read/monographs-physics/refinement-40100247231a.html","sha256":"bf94920eb88195362a35469a42c2fadc39681b26ccf91c44f5911327e2015774","git_blob_sha1":"52bd1404b3803a6d0a8c50772a08df237ac8e3a1","size":34021,"passages":261,"macros":{"KomorebiEdition":"","KomorebiTitle":"Gauge Covariance and Geometric Refinement","KomorebiCoverTitle":"Gauge Covariance\\\\and Geometric\\\\Refinement","KomorebiSubtitle":"Ordered transport, curvature, and the control of changing resolution","KomorebiKind":"Gauge monograph","KomorebiShortTitle":"Gauge covariance and geometric refinement","KomorebiCoverBlurb":"From finite link covariance to geometric subdivision and regular background connections, with the exact identities and remaining dynamical obligations kept visible.","KomorebiChapterLabel":"04 / Gauge transport"}},{"repository_path":"monographs/physics/source/monographs/gravity/conclusion.tex","file_url":"documents/files/b1868734abd177d0ebe6515d3572bf2277bb0506/conclusion.tex","url":"read/monographs-physics/conclusion-f0f278395a29.html","sha256":"22509778a436fcee92133473c5300203dd5d3f94f9c688c328fcf56d7bab5979","git_blob_sha1":"b1868734abd177d0ebe6515d3572bf2277bb0506","size":4279,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/gravity/constraints.tex","file_url":"documents/files/eabda6df05a640dc6e33759c0dc14b10a1f40ceb/constraints.tex","url":"read/monographs-physics/constraints-188665800c20.html","sha256":"be8d3d886cc25dce58ee85ee4b018c671cccab33b2eed8c59e8fa765edddd82f","git_blob_sha1":"eabda6df05a640dc6e33759c0dc14b10a1f40ceb","size":53677,"passages":358,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/gravity/core.tex","file_url":"documents/files/35e2833321631cf3df7d35d5aa59eb0fcd0d16ac/core.tex","url":"read/monographs-physics/core-344f99aacc2c.html","sha256":"f667f5cf9c3748119b35ad6a09795846885d7d0ef200b7b0af473913c09e72d3","git_blob_sha1":"35e2833321631cf3df7d35d5aa59eb0fcd0d16ac","size":35215,"passages":328,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/gravity/orientation.tex","file_url":"documents/files/64821941b05b281c6af7aac5b38d18c33392c8f5/orientation.tex","url":"read/monographs-physics/orientation-07c4122a4059.html","sha256":"c1bada1e0ba50345f3fc6be09c5c4e6f47c0f0f2517816cf58d0f1621dd0e53d","git_blob_sha1":"64821941b05b281c6af7aac5b38d18c33392c8f5","size":4881,"passages":5,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/gravity/references.tex","file_url":"documents/files/b4dc219858063bbf00499d3049183c7d53eced9b/references.tex","url":"read/monographs-physics/references-506690cdbb01.html","sha256":"5c9507da0fd4e1fda5d46033fe0b0a47db95ff8ada15202dae8c5c334d275ecf","git_blob_sha1":"b4dc219858063bbf00499d3049183c7d53eced9b","size":5523,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/gravity/strong_fields.tex","file_url":"documents/files/e60bcad6bdf50a36b7ec4f2e859fe4dfe50d4c1d/strong_fields.tex","url":"read/monographs-physics/strong-fields-61effda085f7.html","sha256":"4df57a519b956dacdcaf1c6131390a147fe0eca20987437e82e348fe6f4dbd7c","git_blob_sha1":"e60bcad6bdf50a36b7ec4f2e859fe4dfe50d4c1d","size":39401,"passages":334,"macros":{"KomorebiEdition":"","KomorebiTitle":"Constrained Gravity and Strong Fields","KomorebiCoverTitle":"Constrained Gravity\\\\and Strong Fields","KomorebiSubtitle":"Constraint algebra, regular reduction, and spherical geometry","KomorebiKind":"Gravity monograph","KomorebiShortTitle":"Constrained gravity and strong fields","KomorebiCoverBlurb":"A collected development of gravitational transport, nonlinear constraint constructions, local canonical reduction, and spherical strong field models.","KomorebiChapterLabel":"05 / Constrained geometry"}},{"repository_path":"monographs/physics/source/monographs/preamble.tex","file_url":"documents/files/ed2c857e856527bf20cac9773d06554ec313fca6/preamble.tex","url":"read/monographs-physics/preamble-614553a8a708.html","sha256":"35a053ff14cdcae4bf1e829aa4eaac0c77943bc894b59174c1ad17fb5e02e659","git_blob_sha1":"ed2c857e856527bf20cac9773d06554ec313fca6","size":2197,"passages":0,"macros":{"KomorebiEdition":"","KomorebiTitle":"Projectors, Packets, and Transfer","KomorebiCoverTitle":"Projectors, Packets,\\\\and Transfer","KomorebiSubtitle":"Resolution, conservation, and the passage to infinite systems","KomorebiKind":"Monograph edition","KomorebiShortTitle":"Projectors, packets, and transfer","KomorebiCoverBlurb":"A common mathematical language for resolved dynamics, changing frames, open systems, and controlled reconstruction.","KomorebiChapterLabel":"01 / Resolution and dynamics","C":"\\mathbb C","R":"\\mathbb R","Z":"\\mathbb Z","N":"\\mathbb N","T":"\\mathbb T","id":"\\mathrm{id}","Ran":"\\operatorname{Ran}","Dom":"\\mathcal D","HH":"\\mathcal H","KK":"\\mathcal K","F":"\\mathcal F","arraystretch":"1.16","KomorebiSeries":"Physics monograph series"}},{"repository_path":"monographs/physics/source/monographs/references.tex","file_url":"documents/files/11a326076e8f991ec565457785150a5680f2ecfb/references.tex","url":"read/monographs-physics/references-25e5a462a22d.html","sha256":"f5b56a7d6b293bdf485b5593f5bb07ff689ec2e5d29a13f6daf63ba8d8f4073f","git_blob_sha1":"11a326076e8f991ec565457785150a5680f2ecfb","size":2110,"passages":0,"macros":{"KomorebiTitle":"Fourier Packets and the Continuum Dirac Operator","KomorebiCoverTitle":"Fourier Packets and the\\\\Continuum Dirac Operator","KomorebiSubtitle":"From finite spinor dynamics to a self adjoint Hamiltonian on the three torus","KomorebiKind":"Dirac continuum monograph","KomorebiShortTitle":"Dirac continuum monograph","KomorebiCoverBlurb":"Finite spinor dynamics, Fourier completion, and the exact domain of the physical Hamiltonian.","KomorebiChapterLabel":"02 / The continuum Hamiltonian","refname":"Sources and further reading"}}],"pdfs":[{"repository_path":"monographs/physics/Komorebi_Physics.pdf","url":"read/monographs-physics/komorebi-physics-ace4c44d47e9.html","sha256":"def363fdba2c8cba101f0c31096631b1c5aae10fa2386b729e87b034d5d23e2c"},{"repository_path":"monographs/physics/compiled/Physics_Lean_Companion/Physics_Lean_Companion.pdf","url":"read/monographs-physics/physics-lean-companion-d90dc5c1bec6.html","sha256":"053cec76421d6d482ff9f00f78fd2db3164f6a671bf5ee50c50dfbd853bd7b0a"}],"passages":[{"line":6,"start":271,"end":272,"expression":"X","context":0,"kind":"use","source":10},{"line":7,"start":312,"end":324,"expression":"\\ell^2(X;\\C)","context":0,"kind":"use","source":10},{"line":7,"start":351,"end":360,"expression":"|x\\rangle","context":0,"kind":"use","source":10},{"line":13,"start":694,"end":695,"expression":"p","context":1,"kind":"use","source":10},{"line":14,"start":713,"end":719,"expression":"h\\geq0","context":1,"kind":"use","source":10},{"line":15,"start":728,"end":761,"expression":"\n U_{p,h}=(\\Z/p^{h+1}\\Z)^\\times.\n","context":1,"kind":"use","source":10},{"line":18,"start":769,"end":772,"expression":"p=2","context":1,"kind":"use","source":10},{"line":19,"start":820,"end":853,"expression":"\n U_{2,h}=(\\Z/2^{h+2}\\Z)^\\times.\n","context":1,"kind":"use","source":10},{"line":30,"start":1173,"end":1174,"expression":"p","context":2,"kind":"use","source":10},{"line":30,"start":1192,"end":1203,"expression":"\\zeta_{p,h}","context":2,"kind":"use","source":10},{"line":30,"start":1215,"end":1218,"expression":"p-1","context":2,"kind":"use","source":10},{"line":31,"start":1247,"end":1262,"expression":"(\\Z/p\\Z)^\\times","context":2,"kind":"use","source":10},{"line":32,"start":1287,"end":1539,"expression":"\n (t,a)&\\longmapsto\\zeta_{p,h}^{\\,t}(1+p)^a\n       \\pmod{p^{h+1}},\n & (t,a)&\\in\\Z/(p-1)\\Z\\times\\Z/p^h\\Z,\n \\label{a-finite-odd-coordinate}\\\\\n (s,a)&\\longmapsto(-1)^s5^a\\pmod{2^{h+2}},\n & (s,a)&\\in\\Z/2\\Z\\times\\Z/2^h\\Z\n \\label{a-finite-dyadic-coordinate}\n","context":3,"kind":"use","source":10},{"line":41,"start":1580,"end":1587,"expression":"U_{p,h}","context":4,"kind":"use","source":10},{"line":41,"start":1594,"end":1601,"expression":"U_{2,h}","context":4,"kind":"use","source":10},{"line":45,"start":1659,"end":1660,"expression":"p","context":5,"kind":"use","source":10},{"line":46,"start":1687,"end":1738,"expression":"\n v_p\\bigl((1+p)^{p^k}-1\\bigr)=k+1\n \\qquad(k\\geq0)\n","context":5,"kind":"use","source":10},{"line":50,"start":1753,"end":1756,"expression":"1+p","context":5,"kind":"use","source":10},{"line":50,"start":1769,"end":1772,"expression":"p^h","context":5,"kind":"use","source":10},{"line":50,"start":1782,"end":1789,"expression":"p^{h+1}","context":5,"kind":"use","source":10},{"line":51,"start":1825,"end":1828,"expression":"k=0","context":5,"kind":"use","source":10},{"line":51,"start":1848,"end":1861,"expression":"u=1+c p^{k+1}","context":5,"kind":"use","source":10},{"line":52,"start":1869,"end":1877,"expression":"p\\nmid c","context":5,"kind":"use","source":10},{"line":52,"start":1907,"end":1912,"expression":"u^p-1","context":5,"kind":"use","source":10},{"line":53,"start":1930,"end":1939,"expression":"c p^{k+2}","context":5,"kind":"use","source":10},{"line":53,"start":1984,"end":1985,"expression":"p","context":5,"kind":"use","source":10},{"line":54,"start":2003,"end":2004,"expression":"p","context":5,"kind":"use","source":10},{"line":55,"start":2068,"end":2071,"expression":"1+p","context":6,"kind":"use","source":10},{"line":55,"start":2121,"end":2122,"expression":"p","context":7,"kind":"use","source":10},{"line":56,"start":2154,"end":2157,"expression":"p^h","context":8,"kind":"use","source":10},{"line":58,"start":2218,"end":2221,"expression":"p-1","context":9,"kind":"use","source":10},{"line":59,"start":2296,"end":2305,"expression":"x^{p-1}=1","context":10,"kind":"use","source":10},{"line":60,"start":2331,"end":2343,"expression":"(p-1)x^{p-2}","context":11,"kind":"use","source":10},{"line":60,"start":2353,"end":2354,"expression":"p","context":12,"kind":"use","source":10},{"line":60,"start":2361,"end":2364,"expression":"x_n","context":13,"kind":"use","source":10},{"line":60,"start":2384,"end":2387,"expression":"p^n","context":14,"kind":"use","source":10},{"line":61,"start":2404,"end":2413,"expression":"x_n+c p^n","context":15,"kind":"use","source":10},{"line":61,"start":2423,"end":2430,"expression":"p^{n+1}","context":16,"kind":"use","source":10},{"line":61,"start":2469,"end":2470,"expression":"c","context":17,"kind":"use","source":10},{"line":63,"start":2526,"end":2541,"expression":"(\\Z/p\\Z)^\\times","context":17,"kind":"use","source":10},{"line":63,"start":2553,"end":2564,"expression":"\\zeta_{p,h}","context":17,"kind":"use","source":10},{"line":65,"start":2675,"end":2676,"expression":"d","context":18,"kind":"use","source":10},{"line":65,"start":2715,"end":2716,"expression":"d","context":19,"kind":"use","source":10},{"line":66,"start":2802,"end":2803,"expression":"m","context":20,"kind":"use","source":10},{"line":68,"start":2899,"end":2900,"expression":"m","context":21,"kind":"use","source":10},{"line":70,"start":3030,"end":3035,"expression":"x^m=1","context":22,"kind":"use","source":10},{"line":70,"start":3082,"end":3083,"expression":"m","context":23,"kind":"use","source":10},{"line":73,"start":3163,"end":3189,"expression":"\\zeta_{p,h}^{\\,t}(1+p)^a=1","context":24,"kind":"use","source":10},{"line":73,"start":3210,"end":3211,"expression":"p","context":24,"kind":"use","source":10},{"line":74,"start":3220,"end":3223,"expression":"t=0","context":24,"kind":"use","source":10},{"line":74,"start":3229,"end":3239,"expression":"\\Z/(p-1)\\Z","context":24,"kind":"use","source":10},{"line":75,"start":3279,"end":3282,"expression":"a=0","context":24,"kind":"use","source":10},{"line":75,"start":3288,"end":3296,"expression":"\\Z/p^h\\Z","context":24,"kind":"use","source":10},{"line":76,"start":3361,"end":3379,"expression":"(p-1)p^h=|U_{p,h}|","context":24,"kind":"use","source":10},{"line":79,"start":3459,"end":3491,"expression":"\n v_2\\bigl(5^{2^k}-1\\bigr)=k+2.\n","context":25,"kind":"use","source":10},{"line":82,"start":3502,"end":3505,"expression":"5-1","context":25,"kind":"use","source":10},{"line":83,"start":3563,"end":3572,"expression":"5^{2^k}+1","context":25,"kind":"use","source":10},{"line":83,"start":3599,"end":3600,"expression":"5","context":25,"kind":"use","source":10},{"line":83,"start":3613,"end":3616,"expression":"2^h","context":25,"kind":"use","source":10},{"line":84,"start":3626,"end":3633,"expression":"2^{h+2}","context":25,"kind":"use","source":10},{"line":84,"start":3664,"end":3667,"expression":"2^h","context":25,"kind":"use","source":10},{"line":85,"start":3724,"end":3726,"expression":"-1","context":25,"kind":"use","source":10},{"line":90,"start":3890,"end":3891,"expression":"t","context":26,"kind":"use","source":10},{"line":90,"start":3922,"end":3923,"expression":"a","context":26,"kind":"use","source":10},{"line":91,"start":3982,"end":3983,"expression":"s","context":26,"kind":"use","source":10},{"line":92,"start":4004,"end":4005,"expression":"a","context":26,"kind":"use","source":10},{"line":94,"start":4097,"end":4299,"expression":"\n \\ell^2(U_{p,h})&\\cong\n \\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^h\\Z),\n &&p\\text{ odd},\\nonumber\\\\\n \\ell^2(U_{2,h})&\\cong\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^h\\Z).\n \\label{a-finite-unit-hilbert}\n","context":27,"kind":"use","source":10},{"line":108,"start":4629,"end":4673,"expression":"\\zeta_{p,h+1}\\equiv\\zeta_{p,h}\\pmod{p^{h+1}}","context":28,"kind":"use","source":10},{"line":109,"start":4731,"end":4732,"expression":"t","context":28,"kind":"use","source":10},{"line":109,"start":4747,"end":4748,"expression":"a","context":28,"kind":"use","source":10},{"line":109,"start":4758,"end":4761,"expression":"p^h","context":28,"kind":"use","source":10},{"line":119,"start":5083,"end":5105,"expression":"\\mathcal P_{\\mathrm o}","context":29,"kind":"use","source":10},{"line":120,"start":5147,"end":5205,"expression":"\n M=2^{h_2+2}\\prod_{p\\in\\mathcal P_{\\mathrm o}}p^{h_p+1}.\n","context":30,"kind":"use","source":10},{"line":124,"start":5288,"end":5289,"expression":"M","context":31,"kind":"use","source":10},{"line":126,"start":5428,"end":5695,"expression":"\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n","context":32,"kind":"use","source":10},{"line":135,"start":5734,"end":5768,"expression":"\\mathcal U_{\\mathrm{CRT}}|u\\rangle","context":33,"kind":"use","source":10},{"line":138,"start":5927,"end":5997,"expression":"\n \\varphi(M)=2^{h_2+1}\\prod_{p\\in\\mathcal P_{\\mathrm o}}(p-1)p^{h_p}.\n","context":34,"kind":"use","source":10},{"line":144,"start":6130,"end":6215,"expression":"\n (F_n)_{k x}=n^{-1/2}\\exp(2\\pi\\mathrm{i}kx/n).\n \\label{a-finite-fourier-convention}\n","context":35,"kind":"use","source":10},{"line":149,"start":6312,"end":6522,"expression":"\n F_{U_M}=\\mathcal U_{\\mathrm{CRT}}^*\n \\left[F_2\\otimes F_{2^{h_2}}\\otimes\n \\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}(F_{p-1}\\otimes F_{p^{h_p}})\n \\right]\\mathcal U_{\\mathrm{CRT}}.\n \\label{a-finite-unit-fourier}\n","context":35,"kind":"use","source":10},{"line":159,"start":6754,"end":6755,"expression":"M","context":36,"kind":"use","source":10},{"line":162,"start":6878,"end":6881,"expression":"a,b","context":37,"kind":"use","source":10},{"line":163,"start":6917,"end":6921,"expression":"N=ab","context":37,"kind":"use","source":10},{"line":164,"start":6945,"end":7121,"expression":"\n \\mathcal P_{\\mathrm{in}}|n\\rangle\n &=|n\\bmod a\\rangle\\otimes|n\\bmod b\\rangle,\\\\\n \\mathcal P_{\\mathrm{out}}|k\\rangle\n &=|k b^{-1}\\bmod a\\rangle\\otimes|k a^{-1}\\bmod b\\rangle,\n","context":38,"kind":"use","source":10},{"line":171,"start":7216,"end":7334,"expression":"\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n","context":39,"kind":"use","source":10},{"line":176,"start":7375,"end":7387,"expression":"n_a=n\\bmod a","context":39,"kind":"use","source":10},{"line":176,"start":7394,"end":7406,"expression":"n_b=n\\bmod b","context":39,"kind":"use","source":10},{"line":177,"start":7434,"end":7452,"expression":"k_a=kb^{-1}\\bmod a","context":40,"kind":"use","source":10},{"line":177,"start":7459,"end":7477,"expression":"k_b=ka^{-1}\\bmod b","context":41,"kind":"use","source":10},{"line":179,"start":7504,"end":7604,"expression":"\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n","context":42,"kind":"use","source":10},{"line":183,"start":7622,"end":7633,"expression":"b k_a+a k_b","context":43,"kind":"use","source":10},{"line":183,"start":7652,"end":7653,"expression":"k","context":44,"kind":"use","source":10},{"line":183,"start":7668,"end":7669,"expression":"a","context":45,"kind":"use","source":10},{"line":184,"start":7676,"end":7677,"expression":"b","context":46,"kind":"use","source":10},{"line":184,"start":7694,"end":7696,"expression":"ab","context":47,"kind":"use","source":10},{"line":185,"start":7719,"end":7751,"expression":"N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)","context":48,"kind":"use","source":10},{"line":192,"start":7986,"end":8057,"expression":"\n C_{\\mathrm{phase}}|s,a\\rangle=|a\\rangle\n \\label{a-finite-forgetting}\n","context":49,"kind":"use","source":10},{"line":196,"start":8080,"end":8087,"expression":"2^{h+1}","context":49,"kind":"use","source":10},{"line":197,"start":8130,"end":8133,"expression":"2^h","context":49,"kind":"use","source":10},{"line":198,"start":8204,"end":8207,"expression":"2^h","context":49,"kind":"use","source":10},{"line":198,"start":8218,"end":8241,"expression":"|0,a\\rangle-|1,a\\rangle","context":49,"kind":"use","source":10},{"line":199,"start":8251,"end":8336,"expression":"\n \\operatorname{rank}C_{\\mathrm{phase}}=2^h,\n \\qquad\\dim\\ker C_{\\mathrm{phase}}=2^h.\n","context":49,"kind":"use","source":10},{"line":206,"start":8517,"end":8549,"expression":"(|0,a\\rangle+|1,a\\rangle)/\\sqrt2","context":50,"kind":"use","source":10},{"line":206,"start":8555,"end":8570,"expression":"\\sqrt2|a\\rangle","context":51,"kind":"use","source":10},{"line":210,"start":8709,"end":8757,"expression":"\\rho\\mapsto\\operatorname{Tr}_{\\mathrm{sign}}\\rho","context":52,"kind":"use","source":10},{"line":211,"start":8778,"end":8789,"expression":"|0,a\\rangle","context":52,"kind":"use","source":10},{"line":211,"start":8796,"end":8807,"expression":"|1,a\\rangle","context":52,"kind":"use","source":10},{"line":212,"start":8852,"end":8871,"expression":"|a\\rangle\\langle a|","context":52,"kind":"use","source":10},{"line":213,"start":8900,"end":8969,"expression":"\n Z_{\\mathrm{sign}}=(|0\\rangle\\langle0|-|1\\rangle\\langle1|)\\otimes I\n","context":52,"kind":"use","source":10},{"line":216,"start":8990,"end":8992,"expression":"+1","context":52,"kind":"use","source":10},{"line":216,"start":8999,"end":9001,"expression":"-1","context":52,"kind":"use","source":10},{"line":222,"start":9319,"end":9407,"expression":"\n A=\\begin{pmatrix}1&1\\\\0&-1\\end{pmatrix},\\qquad A^2=I,\n \\qquad E_\\pm=\\frac{I\\pm A}{2}.\n","context":53,"kind":"use","source":10},{"line":226,"start":9426,"end":9439,"expression":"E_\\pm^2=E_\\pm","context":54,"kind":"use","source":10},{"line":226,"start":9443,"end":9451,"expression":"E_+E_-=0","context":54,"kind":"use","source":10},{"line":226,"start":9459,"end":9468,"expression":"E_++E_-=I","context":54,"kind":"use","source":10},{"line":227,"start":9492,"end":9497,"expression":"A^2=I","context":54,"kind":"use","source":10},{"line":228,"start":9566,"end":9569,"expression":"x-1","context":54,"kind":"use","source":10},{"line":228,"start":9576,"end":9579,"expression":"x+1","context":54,"kind":"use","source":10},{"line":229,"start":9643,"end":9652,"expression":"(A-I)^2=0","context":54,"kind":"use","source":10},{"line":229,"start":9662,"end":9665,"expression":"A-I","context":54,"kind":"use","source":10},{"line":232,"start":9813,"end":9818,"expression":"E_\\pm","context":55,"kind":"use","source":10},{"line":233,"start":9877,"end":9878,"expression":"A","context":56,"kind":"use","source":10},{"line":244,"start":10275,"end":10276,"expression":"p","context":57,"kind":"use","source":10},{"line":244,"start":10289,"end":10295,"expression":"h\\geq0","context":57,"kind":"use","source":10},{"line":245,"start":10304,"end":10366,"expression":"\n G_{p,h}=\\Z/p^h\\Z,\\qquad\\mathcal H_{p,h}=\\ell^2(G_{p,h};\\C).\n","context":57,"kind":"use","source":10},{"line":248,"start":10425,"end":10426,"expression":"r","context":57,"kind":"use","source":10},{"line":249,"start":10492,"end":10495,"expression":"p^r","context":57,"kind":"use","source":10},{"line":250,"start":10507,"end":10514,"expression":"p^{h-r}","context":57,"kind":"use","source":10},{"line":251,"start":10612,"end":10613,"expression":"p","context":58,"kind":"use","source":10},{"line":255,"start":10690,"end":10703,"expression":"0\\leq r\\leq h","context":59,"kind":"use","source":10},{"line":256,"start":10750,"end":10865,"expression":"\n (E_r^{(h)}f)(x)=p^{-(h-r)}\n       \\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}f(y).\n \\label{a-finite-cylinder-average}\n","context":59,"kind":"use","source":10},{"line":262,"start":10998,"end":11007,"expression":"E_r^{(h)}","context":59,"kind":"use","source":10},{"line":263,"start":11071,"end":11072,"expression":"r","context":59,"kind":"use","source":10},{"line":264,"start":11110,"end":11113,"expression":"p^r","context":60,"kind":"use","source":10},{"line":270,"start":11256,"end":11317,"expression":"\n E_r^{(h)}E_s^{(h)}=E_{\\min(r,s)}^{(h)},\\qquad E_h^{(h)}=I.\n","context":61,"kind":"use","source":10},{"line":274,"start":11329,"end":11414,"expression":"\n D_0^{(h)}=E_0^{(h)},\\qquad\n D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\\quad(1\\leq r\\leq h).\n","context":61,"kind":"use","source":10},{"line":278,"start":11427,"end":11436,"expression":"D_r^{(h)}","context":61,"kind":"use","source":10},{"line":279,"start":11490,"end":11697,"expression":"\n \\sum_{r=0}^hD_r^{(h)}&=I,\n &D_r^{(h)}D_s^{(h)}&=\\delta_{rs}D_r^{(h)},\\nonumber\\\\\n \\operatorname{rank}D_0^{(h)}&=1,\n &\\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\\quad(r\\geq1).\n \\label{a-finite-detail-ranks}\n","context":62,"kind":"use","source":10},{"line":289,"start":11746,"end":11753,"expression":"r\\leq s","context":63,"kind":"use","source":10},{"line":289,"start":11767,"end":11768,"expression":"s","context":63,"kind":"use","source":10},{"line":289,"start":11802,"end":11803,"expression":"r","context":63,"kind":"use","source":10},{"line":292,"start":11939,"end":11949,"expression":"E_rE_s=E_r","context":63,"kind":"use","source":10},{"line":293,"start":11999,"end":12009,"expression":"E_sE_r=E_r","context":63,"kind":"use","source":10},{"line":295,"start":12048,"end":12054,"expression":"r\\geq1","context":64,"kind":"use","source":10},{"line":296,"start":12058,"end":12098,"expression":"(E_r-E_{r-1})^2=E_r-2E_{r-1}+E_{r-1}=D_r","context":64,"kind":"use","source":10},{"line":297,"start":12106,"end":12115,"expression":"1\\leq r<s","context":64,"kind":"use","source":10},{"line":298,"start":12132,"end":12158,"expression":"(E_r-E_{r-1})(E_s-E_{s-1})","context":64,"kind":"use","source":10},{"line":299,"start":12216,"end":12219,"expression":"D_0","context":64,"kind":"use","source":10},{"line":300,"start":12296,"end":12323,"expression":"\\Ran E_{r-1}\\subset\\Ran E_r","context":64,"kind":"use","source":10},{"line":301,"start":12349,"end":12360,"expression":"p^r-p^{r-1}","context":64,"kind":"use","source":10},{"line":301,"start":12370,"end":12373,"expression":"E_0","context":64,"kind":"use","source":10},{"line":305,"start":12467,"end":12596,"expression":"\n \\mathcal H_{p,h}=\\bigoplus_{r=0}^h\\Ran D_r^{(h)},\\qquad\n \\norm f^2=\\sum_{r=0}^h\\norm{D_r^{(h)}f}^2.\n \\label{a-finite-parseval}\n","context":65,"kind":"use","source":10},{"line":310,"start":12627,"end":12634,"expression":"m\\leq h","context":65,"kind":"use","source":10},{"line":310,"start":12645,"end":12676,"expression":"E_m^{(h)}=\\sum_{r=0}^mD_r^{(h)}","context":65,"kind":"use","source":10},{"line":312,"start":12711,"end":12774,"expression":"\n \\norm{(I-E_m^{(h)})f}^2=\n \\sum_{r=m+1}^h\\norm{D_r^{(h)}f}^2.\n","context":65,"kind":"use","source":10},{"line":317,"start":12872,"end":12873,"expression":"m","context":66,"kind":"use","source":10},{"line":322,"start":13003,"end":13034,"expression":"\\omega_p=\\exp(2\\pi\\mathrm{i}/p)","context":67,"kind":"introduction-passage","source":10},{"line":322,"start":13042,"end":13055,"expression":"1\\leq r\\leq h","context":67,"kind":"introduction-passage","source":10},{"line":323,"start":13073,"end":13099,"expression":"a\\in\\{0,\\ldots,p^{r-1}-1\\}","context":67,"kind":"introduction-passage","source":10},{"line":324,"start":13117,"end":13137,"expression":"q\\in\\{1,\\ldots,p-1\\}","context":67,"kind":"introduction-passage","source":10},{"line":325,"start":13163,"end":13335,"expression":"\n \\psi_{r,a,q}^{(h)}(x)=\n \\begin{cases}\n p^{-(h-r+1)/2}\\omega_p^{q d_r(x)},\n       &x\\equiv a\\pmod{p^{r-1}},\\\\\n 0,&\\text{otherwise},\n \\end{cases}\n \\label{a-finite-wavelet}\n","context":68,"kind":"introduction-passage","source":10},{"line":334,"start":13357,"end":13363,"expression":"d_r(x)","context":69,"kind":"use","source":10},{"line":334,"start":13391,"end":13398,"expression":"p^{r-1}","context":69,"kind":"use","source":10},{"line":335,"start":13438,"end":13462,"expression":"\\psi_0^{(h)}(x)=p^{-h/2}","context":69,"kind":"use","source":10},{"line":337,"start":13501,"end":13502,"expression":"r","context":70,"kind":"use","source":10},{"line":338,"start":13556,"end":13559,"expression":"r-1","context":71,"kind":"use","source":10},{"line":339,"start":13580,"end":13611,"expression":"\\sum_{d=0}^{p-1}\\omega_p^{qd}=0","context":72,"kind":"use","source":10},{"line":339,"start":13626,"end":13627,"expression":"q","context":73,"kind":"use","source":10},{"line":340,"start":13647,"end":13661,"expression":"\\Ran D_r^{(h)}","context":74,"kind":"use","source":10},{"line":340,"start":13686,"end":13695,"expression":"p^{h-r+1}","context":75,"kind":"use","source":10},{"line":342,"start":13820,"end":13824,"expression":"q,q'","context":76,"kind":"use","source":10},{"line":343,"start":13831,"end":13888,"expression":"\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n","context":77,"kind":"use","source":10},{"line":348,"start":14002,"end":14014,"expression":"(p-1)p^{r-1}","context":78,"kind":"use","source":10},{"line":348,"start":14035,"end":14036,"expression":"r","context":79,"kind":"use","source":10},{"line":350,"start":14148,"end":14296,"expression":"\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n","context":80,"kind":"use","source":10},{"line":355,"start":14334,"end":14337,"expression":"q=0","context":81,"kind":"use","source":10},{"line":355,"start":14367,"end":14370,"expression":"p=2","context":82,"kind":"use","source":10},{"line":356,"start":14433,"end":14434,"expression":"p","context":83,"kind":"use","source":10},{"line":357,"start":14462,"end":14465,"expression":"p-1","context":84,"kind":"use","source":10},{"line":361,"start":14665,"end":14668,"expression":"p=3","context":85,"kind":"use","source":10},{"line":364,"start":14844,"end":14856,"expression":"(p-1)p^{r-1}","context":85,"kind":"use","source":10},{"line":365,"start":14905,"end":14916,"expression":"\\mathsf F_r","context":85,"kind":"use","source":10},{"line":366,"start":14993,"end":15002,"expression":"D_r^{(h)}","context":85,"kind":"use","source":10},{"line":366,"start":15009,"end":15016,"expression":"r\\leq h","context":85,"kind":"use","source":10},{"line":372,"start":15102,"end":15103,"expression":"p","context":86,"kind":"use","source":10},{"line":374,"start":15179,"end":15279,"expression":"\n (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),\n \\quad 0\\leq a<p^h,\\quad0\\leq d<p.\n \\label{a-finite-refinement}\n","context":87,"kind":"use","source":10},{"line":379,"start":15323,"end":15351,"expression":"f\\mapsto|+_p\\rangle\\otimes f","context":88,"kind":"use","source":10},{"line":380,"start":15361,"end":15396,"expression":"|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle","context":88,"kind":"use","source":10},{"line":382,"start":15461,"end":15520,"expression":"\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n","context":88,"kind":"use","source":10},{"line":386,"start":15583,"end":15674,"expression":"\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n","context":89,"kind":"use","source":10},{"line":391,"start":15718,"end":15860,"expression":"\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.\n \\label{a-finite-refinement-projectors}\n","context":90,"kind":"use","source":10},{"line":402,"start":16128,"end":16327,"expression":"\n E_r^{(h+1)}V_{p,h}&=V_{p,h}E_r^{(h)}\\quad(0\\leq r\\leq h),\n \\nonumber\\\\\n D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\\quad(0\\leq r\\leq h),\n &D_{h+1}^{(h+1)}V_{p,h}&=0.\n \\label{a-finite-detail-intertwining}\n","context":91,"kind":"use","source":10},{"line":409,"start":16388,"end":16394,"expression":"w\\geq0","context":92,"kind":"use","source":10},{"line":411,"start":16436,"end":16577,"expression":"\n V_{p,h+w-1}\\cdots V_{p,h}|a\\rangle\n =p^{-w/2}\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}|a,\\mathbf d\\rangle,\n \\label{a-finite-nested-refinement}\n","context":92,"kind":"use","source":10},{"line":417,"start":16656,"end":16664,"expression":"p^{-w/2}","context":93,"kind":"use","source":10},{"line":423,"start":16826,"end":16832,"expression":"h\\geq1","context":94,"kind":"use","source":10},{"line":423,"start":16840,"end":16845,"expression":"N=p^h","context":94,"kind":"use","source":10},{"line":423,"start":16852,"end":16861,"expression":"m=p^{h-1}","context":94,"kind":"use","source":10},{"line":425,"start":16930,"end":17185,"expression":"\n \\mathcal A^{\\mathrm{std}}_{p,h}|pk+r\\rangle\n &=|r\\rangle\\otimes|k\\rangle,\n &0\\leq r<p,\\quad0\\leq k<m,\\nonumber\\\\\n \\mathcal A^{\\mathrm{ad}}_{p,h}|a+d p^{h-1}\\rangle\n &=|d\\rangle\\otimes|a\\rangle,\n &0\\leq d<p,\\quad0\\leq a<m.\n \\label{a-finite-digit-splits}\n","context":95,"kind":"use","source":10},{"line":436,"start":17352,"end":17368,"expression":"\\mathcal R_{p,h}","context":96,"kind":"use","source":10},{"line":436,"start":17383,"end":17384,"expression":"h","context":96,"kind":"use","source":10},{"line":436,"start":17392,"end":17393,"expression":"p","context":96,"kind":"use","source":10},{"line":437,"start":17409,"end":17427,"expression":"\\mathcal R_{p,0}=I","context":96,"kind":"use","source":10},{"line":438,"start":17480,"end":17629,"expression":"\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n","context":97,"kind":"use","source":10},{"line":444,"start":17662,"end":17692,"expression":"d_0+d_1p+\\cdots+d_{h-1}p^{h-1}","context":98,"kind":"use","source":10},{"line":445,"start":17701,"end":17708,"expression":"d_{h-1}","context":99,"kind":"use","source":10},{"line":446,"start":17806,"end":17809,"expression":"h-1","context":100,"kind":"use","source":10},{"line":447,"start":17839,"end":17866,"expression":"a=d_0+\\cdots+d_{h-2}p^{h-2}","context":101,"kind":"use","source":10},{"line":451,"start":17953,"end":18009,"expression":"M_{p,h}|a+d p^{h-1}\\rangle=\\omega_p^d|a+d p^{h-1}\\rangle","context":102,"kind":"use","source":10},{"line":452,"start":18041,"end":18042,"expression":"d","context":102,"kind":"use","source":10},{"line":453,"start":18095,"end":18192,"expression":"\n R_d=\\frac1p\\sum_{\\ell=0}^{p-1}\\omega_p^{-d\\ell}M_{p,h}^{\\ell}.\n \\label{a-finite-root-selector}\n","context":102,"kind":"use","source":10},{"line":457,"start":18230,"end":18231,"expression":"e","context":102,"kind":"use","source":10},{"line":458,"start":18254,"end":18301,"expression":"p^{-1}\\sum_\\ell\\omega_p^{(e-d)\\ell}=\\delta_{de}","context":102,"kind":"use","source":10},{"line":459,"start":18315,"end":18336,"expression":"R_dR_e=\\delta_{de}R_d","context":102,"kind":"use","source":10},{"line":459,"start":18340,"end":18349,"expression":"R_d^*=R_d","context":103,"kind":"use","source":10},{"line":459,"start":18357,"end":18368,"expression":"\\sum_dR_d=I","context":104,"kind":"use","source":10},{"line":464,"start":18567,"end":18666,"expression":"\n \\mathcal W_{p,h}=(F_p\\otimes I_m)\\mathcal A^{\\mathrm{ad}}_{p,h}.\n \\label{a-finite-child-fourier}\n","context":105,"kind":"use","source":10},{"line":468,"start":18714,"end":18717,"expression":"F_p","context":105,"kind":"use","source":10},{"line":468,"start":18723,"end":18743,"expression":"p^{-1/2}(1,\\ldots,1)","context":105,"kind":"use","source":10},{"line":470,"start":18768,"end":18984,"expression":"\n\\begin{aligned}\n (\\langle0|\\otimes I_m)\\mathcal W_{p,h}&=V_{p,h-1}^*,\\\\\n \\mathcal W_{p,h}^*\n \\bigl[(I_p-|0\\rangle\\langle0|)\\otimes I_m\\bigr]\n \\mathcal W_{p,h}&=D_h^{(h)}.\n\\end{aligned}\n\\label{a-finite-child-detail}\n","context":106,"kind":"use","source":10},{"line":482,"start":19227,"end":19243,"expression":"\\mathcal T_{p,h}","context":107,"kind":"use","source":10},{"line":482,"start":19271,"end":19276,"expression":"\\Pi_r","context":108,"kind":"use","source":10},{"line":484,"start":19351,"end":19436,"expression":"\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n","context":109,"kind":"use","source":10},{"line":488,"start":19514,"end":19515,"expression":"q","context":110,"kind":"use","source":10},{"line":490,"start":19618,"end":19638,"expression":"q\\leftrightarrow p-q","context":111,"kind":"use","source":10},{"line":495,"start":19841,"end":19842,"expression":"p","context":112,"kind":"use","source":10},{"line":512,"start":20539,"end":20568,"expression":"V:\\mathcal H_c\\to\\mathcal H_f","context":113,"kind":"use","source":10},{"line":513,"start":20621,"end":20627,"expression":"P=VV^*","context":113,"kind":"use","source":10},{"line":513,"start":20634,"end":20639,"expression":"Q=I-P","context":113,"kind":"use","source":10},{"line":514,"start":20670,"end":20677,"expression":"H_c,H_f","context":113,"kind":"use","source":10},{"line":515,"start":20734,"end":20858,"expression":"\n H_fV&=VH_c,\\label{a-finite-hamiltonian-intertwine}\\\\\n QH_fP&=0,\\qquad PH_fP=VH_cV^*.\n \\label{a-finite-hamiltonian-blocks}\n","context":114,"kind":"use","source":10},{"line":520,"start":20898,"end":20916,"expression":"\\Ran P\\oplus\\Ran Q","context":115,"kind":"use","source":10},{"line":521,"start":20935,"end":21047,"expression":"\n H_f=(VH_cV^*)|_{\\Ran P}\\ \\oplus\\ H_Q,\n \\qquad H_Q=(QH_fQ)|_{\\Ran Q}.\n \\label{a-finite-hamiltonian-direct-sum}\n","context":115,"kind":"use","source":10},{"line":526,"start":21083,"end":21086,"expression":"H_Q","context":115,"kind":"use","source":10},{"line":527,"start":21125,"end":21131,"expression":"\\Ran Q","context":115,"kind":"use","source":10},{"line":532,"start":21231,"end":21234,"expression":"V^*","context":116,"kind":"use","source":10},{"line":532,"start":21243,"end":21255,"expression":"H_fP=VH_cV^*","context":116,"kind":"use","source":10},{"line":532,"start":21282,"end":21283,"expression":"Q","context":116,"kind":"use","source":10},{"line":532,"start":21290,"end":21291,"expression":"P","context":116,"kind":"use","source":10},{"line":533,"start":21348,"end":21352,"expression":"PV=V","context":116,"kind":"use","source":10},{"line":534,"start":21359,"end":21365,"expression":"V^*V=I","context":116,"kind":"use","source":10},{"line":535,"start":21375,"end":21412,"expression":"\n H_fV=(PH_fP+QH_fP)V=VH_cV^*V=VH_c.\n","context":116,"kind":"use","source":10},{"line":538,"start":21439,"end":21456,"expression":"PH_fQ=(QH_fP)^*=0","context":116,"kind":"use","source":10},{"line":546,"start":21793,"end":21804,"expression":"H_c=V^*H_fV","context":117,"kind":"use","source":10},{"line":547,"start":21851,"end":21856,"expression":"QH_fV","context":117,"kind":"use","source":10},{"line":555,"start":22266,"end":22267,"expression":"f","context":118,"kind":"use","source":10},{"line":556,"start":22315,"end":22334,"expression":"\n f(H_f)V=Vf(H_c).\n","context":118,"kind":"use","source":10},{"line":559,"start":22362,"end":22363,"expression":"t","context":118,"kind":"use","source":10},{"line":559,"start":22372,"end":22377,"expression":"\\beta","context":118,"kind":"use","source":10},{"line":560,"start":22385,"end":22403,"expression":"z\\notin\\sigma(H_f)","context":118,"kind":"use","source":10},{"line":561,"start":22419,"end":22594,"expression":"\n e^{-\\mathrm{i}tH_f}V&=Ve^{-\\mathrm{i}tH_c},\\nonumber\\\\\n e^{-\\beta H_f}V&=Ve^{-\\beta H_c},\\nonumber\\\\\n (z-H_f)^{-1}V&=V(z-H_c)^{-1}.\n \\label{a-finite-evolution-intertwining}\n","context":119,"kind":"use","source":10},{"line":570,"start":22641,"end":22647,"expression":"\\Ran P","context":120,"kind":"use","source":10},{"line":570,"start":22651,"end":22652,"expression":"V","context":120,"kind":"use","source":10},{"line":570,"start":22686,"end":22689,"expression":"H_c","context":120,"kind":"use","source":10},{"line":571,"start":22716,"end":22719,"expression":"H_f","context":120,"kind":"use","source":10},{"line":573,"start":22863,"end":22864,"expression":"f","context":120,"kind":"use","source":10},{"line":577,"start":23009,"end":23021,"expression":"e^{-\\beta H}","context":121,"kind":"use","source":10},{"line":588,"start":23613,"end":23627,"expression":"\\epsilon_{p,r}","context":122,"kind":"use","source":10},{"line":589,"start":23656,"end":23773,"expression":"\n H_{p,h}^{\\mathrm{Haar}}=\\sum_{r=0}^h\n          \\epsilon_{p,r}D_r^{(h)}.\n \\label{a-finite-hierarchical-hamiltonian}\n","context":123,"kind":"use","source":10},{"line":594,"start":23806,"end":23820,"expression":"\\Ran D_r^{(h)}","context":123,"kind":"use","source":10},{"line":594,"start":23834,"end":23848,"expression":"\\epsilon_{p,r}","context":123,"kind":"use","source":10},{"line":596,"start":23926,"end":23990,"expression":"H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}","context":123,"kind":"use","source":10},{"line":598,"start":24016,"end":24032,"expression":"\\epsilon_{p,h+1}","context":124,"kind":"use","source":10},{"line":600,"start":24147,"end":24242,"expression":"\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n","context":125,"kind":"use","source":10},{"line":604,"start":24276,"end":24277,"expression":"h","context":126,"kind":"use","source":10},{"line":608,"start":24449,"end":24469,"expression":"(S_{p,h}f)(x)=f(x-1)","context":127,"kind":"use","source":10},{"line":609,"start":24497,"end":24500,"expression":"p^h","context":128,"kind":"use","source":10},{"line":610,"start":24572,"end":24575,"expression":"p^h","context":128,"kind":"use","source":10},{"line":611,"start":24580,"end":24615,"expression":"\n S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.\n","context":128,"kind":"use","source":10},{"line":615,"start":24699,"end":24727,"expression":"L_{p,h}=2I-S_{p,h}-S_{p,h}^*","context":128,"kind":"use","source":10},{"line":616,"start":24756,"end":24764,"expression":"\\kappa_p","context":128,"kind":"use","source":10},{"line":617,"start":24793,"end":24883,"expression":"\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n","context":128,"kind":"use","source":10},{"line":622,"start":24973,"end":24974,"expression":"r","context":129,"kind":"use","source":10},{"line":622,"start":25013,"end":25022,"expression":"E_r^{(h)}","context":130,"kind":"use","source":10},{"line":623,"start":25029,"end":25038,"expression":"D_r^{(h)}","context":131,"kind":"use","source":10},{"line":638,"start":25727,"end":25870,"expression":"\n U_{p,h}(t)=e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}\n =\\sum_{r=0}^he^{-\\mathrm{i}t\\epsilon_{p,r}}D_r^{(h)}.\n \\label{a-finite-shell-propagator}\n","context":132,"kind":"use","source":10},{"line":644,"start":25910,"end":26000,"expression":"\n E_r^{(h)}(x,y)=p^{-(h-r)}\n              \\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}.\n","context":132,"kind":"use","source":10},{"line":649,"start":26029,"end":26314,"expression":"\n U_{p,h}(t;x,y)\n &=p^{-h}e^{-\\mathrm{i}t\\epsilon_{p,0}}\\nonumber\\\\\n &\\quad+\\sum_{r=1}^he^{-\\mathrm{i}t\\epsilon_{p,r}}\n \\left[p^{-(h-r)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}\n       -p^{-(h-r+1)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n","context":133,"kind":"use","source":10},{"line":659,"start":26384,"end":26402,"expression":"\\min\\{v_p(x-y),h\\}","context":134,"kind":"use","source":10},{"line":659,"start":26439,"end":26440,"expression":"h","context":134,"kind":"use","source":10},{"line":662,"start":26594,"end":26595,"expression":"y","context":135,"kind":"use","source":10},{"line":662,"start":26622,"end":26625,"expression":"k<h","context":136,"kind":"use","source":10},{"line":663,"start":26637,"end":26651,"expression":"(p-1)p^{h-k-1}","context":137,"kind":"use","source":10},{"line":673,"start":26994,"end":26995,"expression":"P","context":138,"kind":"use","source":10},{"line":674,"start":27003,"end":27010,"expression":"B=QH_fP","context":138,"kind":"use","source":10},{"line":674,"start":27031,"end":27035,"expression":"\\psi","context":138,"kind":"use","source":10},{"line":674,"start":27046,"end":27052,"expression":"\\Ran P","context":138,"kind":"use","source":10},{"line":676,"start":27089,"end":27215,"expression":"\n Qe^{-\\mathrm{i}tH_f}\\psi=-\\mathrm{i}tB\\psi+O(t^2),\n \\qquad\n \\norm{Qe^{-\\mathrm{i}tH_f}\\psi}^2\n =t^2\\norm{B\\psi}^2+O(|t|^3).\n","context":138,"kind":"use","source":10},{"line":682,"start":27264,"end":27267,"expression":"H_f","context":138,"kind":"use","source":10},{"line":682,"start":27274,"end":27278,"expression":"\\psi","context":138,"kind":"use","source":10},{"line":683,"start":27297,"end":27298,"expression":"B","context":138,"kind":"use","source":10},{"line":685,"start":27380,"end":27490,"expression":"\n H_f=\\begin{pmatrix}0&\\overline\\gamma\\\\\\gamma&0\\end{pmatrix},\n \\qquad P=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\n","context":139,"kind":"use","source":10},{"line":690,"start":27549,"end":27566,"expression":"\\sin^2(|\\gamma|t)","context":140,"kind":"use","source":10},{"line":691,"start":27646,"end":27657,"expression":"\\gamma\\neq0","context":140,"kind":"use","source":10},{"line":693,"start":27666,"end":27667,"expression":"z","context":141,"kind":"use","source":10},{"line":693,"start":27694,"end":27715,"expression":"H_Q=(QH_fQ)|_{\\Ran Q}","context":141,"kind":"use","source":10},{"line":695,"start":27766,"end":27834,"expression":"\n \\Sigma_P(z)=PH_fQ(z-H_Q)^{-1}QH_fP.\n \\label{a-finite-self-energy}\n","context":141,"kind":"use","source":10},{"line":699,"start":27901,"end":27904,"expression":"B=0","context":141,"kind":"use","source":10},{"line":718,"start":28894,"end":28901,"expression":"\\hbar=1","context":142,"kind":"use","source":10},{"line":723,"start":29017,"end":29018,"expression":"w","context":143,"kind":"use","source":10},{"line":723,"start":29046,"end":29051,"expression":"n=p^w","context":143,"kind":"use","source":10},{"line":725,"start":29126,"end":29219,"expression":"\n |R_{p,w}\\rangle=n^{-1/2}\\sum_{\\mathbf d}|\\mathbf d\\rangle.\n \\label{a-finite-radial-vector}\n","context":143,"kind":"use","source":10},{"line":729,"start":29247,"end":29272,"expression":"n^{-1}\\sum_{\\mathbf d}1=1","context":143,"kind":"use","source":10},{"line":730,"start":29317,"end":29325,"expression":"p^{-w/2}","context":143,"kind":"use","source":10},{"line":730,"start":29349,"end":29355,"expression":"p^{-w}","context":143,"kind":"use","source":10},{"line":736,"start":29632,"end":29641,"expression":"|c\\rangle","context":144,"kind":"use","source":10},{"line":737,"start":29727,"end":29736,"expression":"|c\\rangle","context":145,"kind":"use","source":10},{"line":738,"start":29768,"end":29780,"expression":"\\gamma_{p,w}","context":146,"kind":"use","source":10},{"line":740,"start":29828,"end":30014,"expression":"\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n","context":147,"kind":"use","source":10},{"line":748,"start":30082,"end":30094,"expression":"\\gamma_{p,w}","context":148,"kind":"use","source":10},{"line":749,"start":30159,"end":30233,"expression":"\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n","context":149,"kind":"use","source":10},{"line":753,"start":30254,"end":30302,"expression":"H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle","context":150,"kind":"use","source":10},{"line":754,"start":30309,"end":30369,"expression":"H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle","context":151,"kind":"use","source":10},{"line":756,"start":30376,"end":30432,"expression":"|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle","context":152,"kind":"use","source":10},{"line":757,"start":30452,"end":30467,"expression":"|R_{p,w}\\rangle","context":153,"kind":"use","source":10},{"line":758,"start":30496,"end":30527,"expression":"\\sum_{\\mathbf d}a_{\\mathbf d}=0","context":154,"kind":"use","source":10},{"line":759,"start":30537,"end":30564,"expression":"H_{\\mathrm{int}}|A\\rangle=0","context":155,"kind":"use","source":10},{"line":766,"start":30948,"end":30957,"expression":"|c\\rangle","context":156,"kind":"use","source":10},{"line":767,"start":30988,"end":30998,"expression":"\\epsilon I","context":156,"kind":"use","source":10},{"line":772,"start":31135,"end":31162,"expression":"(|c\\rangle,|R_{p,w}\\rangle)","context":157,"kind":"use","source":10},{"line":772,"start":31186,"end":31189,"expression":"E_c","context":157,"kind":"use","source":10},{"line":773,"start":31210,"end":31218,"expression":"\\epsilon","context":157,"kind":"use","source":10},{"line":774,"start":31263,"end":31387,"expression":"\n H_{\\mathrm{bright}}=\n \\begin{pmatrix}E_c&\\overline G_{p,w}\\\\G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n","context":158,"kind":"use","source":10},{"line":779,"start":31407,"end":31435,"expression":"\\overline E=(E_c+\\epsilon)/2","context":159,"kind":"use","source":10},{"line":780,"start":31439,"end":31462,"expression":"\\delta=(E_c-\\epsilon)/2","context":160,"kind":"use","source":10},{"line":781,"start":31470,"end":31505,"expression":"\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}","context":161,"kind":"introduction-passage","source":10},{"line":782,"start":31514,"end":31561,"expression":"(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I","context":162,"kind":"introduction-passage","source":10},{"line":783,"start":31587,"end":31812,"expression":"\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n","context":163,"kind":"introduction-passage","source":10},{"line":791,"start":31831,"end":31839,"expression":"\\Omega=0","context":164,"kind":"introduction-passage","source":10},{"line":792,"start":31906,"end":31915,"expression":"|c\\rangle","context":165,"kind":"introduction-passage","source":10},{"line":793,"start":31970,"end":32125,"expression":"\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n","context":166,"kind":"introduction-passage","source":10},{"line":799,"start":32184,"end":32192,"expression":"\\Omega=0","context":167,"kind":"introduction-passage","source":10},{"line":800,"start":32219,"end":32225,"expression":"\\delta","context":168,"kind":"introduction-passage","source":10},{"line":800,"start":32232,"end":32239,"expression":"G_{p,w}","context":169,"kind":"introduction-passage","source":10},{"line":800,"start":32263,"end":32275,"expression":"E_c=\\epsilon","context":170,"kind":"introduction-passage","source":10},{"line":801,"start":32292,"end":32310,"expression":"\\sin^2(|G_{p,w}|t)","context":171,"kind":"introduction-passage","source":10},{"line":802,"start":32358,"end":32376,"expression":"\\Pr(R_{p,w};t)/p^w","context":172,"kind":"introduction-passage","source":10},{"line":805,"start":32477,"end":32587,"expression":"\n \\Sigma_{p,w}(z)=\\frac{|G_{p,w}|^2}{z-\\epsilon},\n \\qquad z\\neq\\epsilon.\n \\label{a-finite-radial-self-energy}\n","context":173,"kind":"use","source":10},{"line":812,"start":32709,"end":32737,"expression":"[z-E_c-\\Sigma_{p,w}(z)]^{-1}","context":173,"kind":"use","source":10},{"line":817,"start":32899,"end":32905,"expression":"g\\in\\C","context":174,"kind":"use","source":10},{"line":818,"start":32928,"end":32956,"expression":"\\gamma_{p,w}=g p^{-\\alpha w}","context":174,"kind":"use","source":10},{"line":818,"start":32979,"end":32985,"expression":"\\alpha","context":174,"kind":"use","source":10},{"line":820,"start":33009,"end":33118,"expression":"\n G_{p,w}=g p^{(1/2-\\alpha)w},\\qquad\n |G_{p,w}|^2=|g|^2p^{(1-2\\alpha)w}.\n \\label{a-finite-coupling-exponent}\n","context":174,"kind":"use","source":10},{"line":825,"start":33187,"end":33194,"expression":"p^{w/2}","context":174,"kind":"use","source":10},{"line":826,"start":33241,"end":33242,"expression":"w","context":174,"kind":"use","source":10},{"line":826,"start":33249,"end":33259,"expression":"\\alpha<1/2","context":174,"kind":"use","source":10},{"line":827,"start":33281,"end":33282,"expression":"w","context":174,"kind":"use","source":10},{"line":827,"start":33289,"end":33299,"expression":"\\alpha=1/2","context":174,"kind":"use","source":10},{"line":828,"start":33318,"end":33328,"expression":"\\alpha>1/2","context":175,"kind":"use","source":10},{"line":830,"start":33435,"end":33447,"expression":"\\pm|G_{p,w}|","context":176,"kind":"use","source":10},{"line":830,"start":33454,"end":33487,"expression":"\\norm{H_{\\mathrm{int}}}=|G_{p,w}|","context":177,"kind":"use","source":10},{"line":837,"start":33670,"end":33676,"expression":"\\alpha","context":178,"kind":"use","source":10},{"line":837,"start":33681,"end":33690,"expression":"|G_{p,w}|","context":178,"kind":"use","source":10},{"line":837,"start":33695,"end":33706,"expression":"|G_{p,w}|^2","context":178,"kind":"use","source":10},{"line":839,"start":33746,"end":33747,"expression":"0","context":179,"kind":"use","source":10},{"line":839,"start":33752,"end":33762,"expression":"|g|p^{w/2}","context":179,"kind":"use","source":10},{"line":839,"start":33767,"end":33775,"expression":"|g|^2p^w","context":179,"kind":"use","source":10},{"line":840,"start":33811,"end":33814,"expression":"1/2","context":180,"kind":"use","source":10},{"line":840,"start":33819,"end":33822,"expression":"|g|","context":180,"kind":"use","source":10},{"line":840,"start":33827,"end":33832,"expression":"|g|^2","context":180,"kind":"use","source":10},{"line":841,"start":33867,"end":33868,"expression":"1","context":181,"kind":"use","source":10},{"line":841,"start":33873,"end":33884,"expression":"|g|p^{-w/2}","context":181,"kind":"use","source":10},{"line":841,"start":33889,"end":33900,"expression":"|g|^2p^{-w}","context":181,"kind":"use","source":10},{"line":846,"start":33969,"end":33977,"expression":"\\alpha=1","context":182,"kind":"use","source":10},{"line":846,"start":33999,"end":34005,"expression":"p^{-w}","context":182,"kind":"use","source":10},{"line":847,"start":34060,"end":34061,"expression":"g","context":182,"kind":"use","source":10},{"line":848,"start":34105,"end":34113,"expression":"p^{-w/2}","context":182,"kind":"use","source":10},{"line":849,"start":34152,"end":34158,"expression":"p^{-w}","context":182,"kind":"use","source":10},{"line":850,"start":34209,"end":34221,"expression":"|z-\\epsilon|","context":182,"kind":"use","source":10},{"line":851,"start":34255,"end":34318,"expression":"\n \\Pr(R_{p,w};t)\\leq |g|^2t^2p^{-w}\n \\quad\\text{when }\\alpha=1\n","context":182,"kind":"use","source":10},{"line":856,"start":34375,"end":34390,"expression":"|\\sin u|\\leq|u|","context":183,"kind":"use","source":10},{"line":857,"start":34449,"end":34460,"expression":"p^{w/2}/|g|","context":184,"kind":"use","source":10},{"line":859,"start":34536,"end":34545,"expression":"|c\\rangle","context":185,"kind":"use","source":10},{"line":860,"start":34600,"end":34608,"expression":"g p^{-w}","context":185,"kind":"use","source":10},{"line":862,"start":34694,"end":34706,"expression":"|g|^2p^{-2w}","context":185,"kind":"use","source":10},{"line":10,"start":432,"end":433,"expression":"p","context":186,"kind":"use","source":11},{"line":10,"start":464,"end":468,"expression":"\\Z_p","context":186,"kind":"use","source":11},{"line":11,"start":499,"end":504,"expression":"\\mu_p","context":186,"kind":"use","source":11},{"line":12,"start":514,"end":584,"expression":"\n B_{m,a}=a+p^m\\Z_p,\\qquad 0\\leq a<p^m,\\qquad\n \\mu_p(B_{m,a})=p^{-m}.\n","context":186,"kind":"use","source":11},{"line":16,"start":594,"end":618,"expression":"\\HH_p=L^2(\\Z_p,\\mu_p;\\C)","context":186,"kind":"use","source":11},{"line":17,"start":655,"end":693,"expression":"\\inner{f}{g}=\\int\\overline f g\\,d\\mu_p","context":186,"kind":"use","source":11},{"line":18,"start":732,"end":733,"expression":"p","context":186,"kind":"use","source":11},{"line":23,"start":922,"end":923,"expression":"m","context":187,"kind":"use","source":11},{"line":23,"start":944,"end":970,"expression":"\\HH_{p,m}=\\ell^2(\\Z/p^m\\Z)","context":188,"kind":"use","source":11},{"line":24,"start":1022,"end":1025,"expression":"e_a","context":189,"kind":"use","source":11},{"line":25,"start":1051,"end":1214,"expression":"\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n","context":190,"kind":"use","source":11},{"line":32,"start":1272,"end":1368,"expression":"\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n","context":190,"kind":"use","source":11},{"line":36,"start":1397,"end":1409,"expression":"\\mathcal V_m","context":191,"kind":"use","source":11},{"line":36,"start":1416,"end":1432,"expression":"\\mathcal U_{p,m}","context":192,"kind":"use","source":11},{"line":37,"start":1470,"end":1485,"expression":"c=\\sum_a c_ae_a","context":193,"kind":"use","source":11},{"line":38,"start":1518,"end":1528,"expression":"p^{m/2}c_a","context":194,"kind":"use","source":11},{"line":38,"start":1534,"end":1541,"expression":"B_{m,a}","context":195,"kind":"use","source":11},{"line":39,"start":1575,"end":1666,"expression":"\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n","context":196,"kind":"use","source":11},{"line":46,"start":1776,"end":1835,"expression":"\n V_{p,m}e_a=\\frac1{\\sqrt p}\n \\sum_{d=0}^{p-1}e_{a+d p^m}.\n","context":197,"kind":"use","source":11},{"line":50,"start":1845,"end":1880,"expression":"B_{m,a}=\\bigsqcup_d B_{m+1,a+d p^m}","context":197,"kind":"use","source":11},{"line":51,"start":1912,"end":1992,"expression":"\n \\mathcal U_{p,m+1}V_{p,m}=\\mathcal U_{p,m}.\n \\label{a-limit-common-embedding}\n","context":197,"kind":"use","source":11},{"line":56,"start":2059,"end":2086,"expression":"p^{-1/2}p^{(m+1)/2}=p^{m/2}","context":197,"kind":"use","source":11},{"line":58,"start":2236,"end":2241,"expression":"\\HH_p","context":198,"kind":"use","source":11},{"line":61,"start":2302,"end":2355,"expression":"\n \\norm{f}_{m,\\mathrm{Haar}}^2=p^{-m}\\sum_a|f(a)|^2.\n","context":199,"kind":"use","source":11},{"line":64,"start":2364,"end":2368,"expression":"f(a)","context":199,"kind":"use","source":11},{"line":65,"start":2451,"end":2456,"expression":"\\HH_p","context":199,"kind":"use","source":11},{"line":65,"start":2465,"end":2466,"expression":"f","context":199,"kind":"use","source":11},{"line":66,"start":2472,"end":2501,"expression":"\\sum_a f(a)\\mathbf1_{B_{m,a}}","context":199,"kind":"use","source":11},{"line":67,"start":2570,"end":2578,"expression":"p^{-1/2}","context":199,"kind":"use","source":11},{"line":68,"start":2604,"end":2619,"expression":"f(a)=p^{m/2}c_a","context":199,"kind":"use","source":11},{"line":69,"start":2666,"end":2672,"expression":"p^{-m}","context":200,"kind":"use","source":11},{"line":69,"start":2694,"end":2701,"expression":"p^{m/2}","context":201,"kind":"use","source":11},{"line":74,"start":2848,"end":2857,"expression":"f\\in\\HH_p","context":202,"kind":"introduction-passage","source":11},{"line":75,"start":2883,"end":2972,"expression":"\n (\\mathsf E_m f)(x)\n =p^m\\int_{x+p^m\\Z_p} f(y)\\,d\\mu_p(y).\n \\label{a-limit-expectation}\n","context":202,"kind":"introduction-passage","source":11},{"line":80,"start":3018,"end":3021,"expression":"L^2","context":202,"kind":"introduction-passage","source":11},{"line":80,"start":3040,"end":3043,"expression":"L^1","context":202,"kind":"introduction-passage","source":11},{"line":81,"start":3119,"end":3120,"expression":"m","context":202,"kind":"introduction-passage","source":11},{"line":86,"start":3232,"end":3243,"expression":"\\mathsf E_m","context":203,"kind":"use","source":11},{"line":86,"start":3280,"end":3292,"expression":"\\mathcal V_m","context":204,"kind":"use","source":11},{"line":87,"start":3300,"end":3308,"expression":"m,n\\geq0","context":205,"kind":"use","source":11},{"line":88,"start":3327,"end":3479,"expression":"\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n","context":206,"kind":"use","source":11},{"line":94,"start":3504,"end":3522,"expression":"\\mathsf E_m f\\to f","context":207,"kind":"use","source":11},{"line":94,"start":3528,"end":3531,"expression":"L^2","context":208,"kind":"use","source":11},{"line":94,"start":3544,"end":3553,"expression":"f\\in\\HH_p","context":209,"kind":"use","source":11},{"line":97,"start":3613,"end":3614,"expression":"g","context":210,"kind":"use","source":11},{"line":97,"start":3634,"end":3649,"expression":"f-\\mathsf E_m f","context":211,"kind":"use","source":11},{"line":98,"start":3668,"end":3669,"expression":"m","context":212,"kind":"use","source":11},{"line":99,"start":3727,"end":3738,"expression":"\\overline g","context":213,"kind":"use","source":11},{"line":100,"start":3765,"end":3792,"expression":"\\inner{g}{f-\\mathsf E_mf}=0","context":214,"kind":"use","source":11},{"line":100,"start":3812,"end":3824,"expression":"\\mathcal V_m","context":215,"kind":"use","source":11},{"line":101,"start":3856,"end":3868,"expression":"\\mathcal V_m","context":216,"kind":"use","source":11},{"line":109,"start":4349,"end":4353,"expression":"\\Z_p","context":217,"kind":"use","source":11},{"line":110,"start":4397,"end":4405,"expression":"f\\in L^2","context":217,"kind":"use","source":11},{"line":111,"start":4478,"end":4501,"expression":"\\nu(A)=\\int_A f\\,d\\mu_p","context":217,"kind":"use","source":11},{"line":112,"start":4578,"end":4581,"expression":"\\nu","context":217,"kind":"use","source":11},{"line":115,"start":4776,"end":4784,"expression":"\\nu(A)=0","context":218,"kind":"use","source":11},{"line":117,"start":4897,"end":4898,"expression":"f","context":219,"kind":"use","source":11},{"line":117,"start":4908,"end":4911,"expression":"f=0","context":220,"kind":"use","source":11},{"line":118,"start":4962,"end":4983,"expression":"\\bigcup_m\\mathcal V_m","context":221,"kind":"use","source":11},{"line":120,"start":5007,"end":5020,"expression":"\\varepsilon>0","context":222,"kind":"use","source":11},{"line":120,"start":5031,"end":5047,"expression":"g\\in\\mathcal V_M","context":222,"kind":"use","source":11},{"line":121,"start":5055,"end":5079,"expression":"\\norm{f-g}_2<\\varepsilon","context":222,"kind":"use","source":11},{"line":121,"start":5087,"end":5094,"expression":"m\\geq M","context":222,"kind":"use","source":11},{"line":121,"start":5119,"end":5120,"expression":"g","context":222,"kind":"use","source":11},{"line":122,"start":5127,"end":5210,"expression":"\\norm{f-\\mathsf E_m f}_2\n=\\norm{(I-\\mathsf E_m)(f-g)}_2\\leq\\norm{f-g}_2<\\varepsilon","context":222,"kind":"use","source":11},{"line":129,"start":5358,"end":5377,"expression":"(\\HH_{p,m},V_{p,m})","context":223,"kind":"use","source":11},{"line":129,"start":5393,"end":5398,"expression":"\\HH_p","context":223,"kind":"use","source":11},{"line":132,"start":5544,"end":5556,"expression":"\\mathcal V_m","context":223,"kind":"use","source":11},{"line":132,"start":5564,"end":5571,"expression":"m\\leq M","context":223,"kind":"use","source":11},{"line":133,"start":5580,"end":5616,"expression":"V_{p,m\\to M}=V_{p,M-1}\\cdots V_{p,m}","context":223,"kind":"use","source":11},{"line":134,"start":5648,"end":5783,"expression":"\n \\mathsf E_m\\big|_{\\mathcal V_M}\n =\\mathcal U_{p,M}V_{p,m\\to M}V_{p,m\\to M}^*\n \\mathcal U_{p,M}^*.\n \\label{a-limit-finite-projection}\n","context":223,"kind":"use","source":11},{"line":140,"start":5843,"end":5855,"expression":"\\mathcal V_M","context":224,"kind":"use","source":11},{"line":141,"start":5863,"end":5875,"expression":"\\mathcal V_m","context":225,"kind":"use","source":11},{"line":146,"start":6037,"end":6060,"expression":"\\mathsf F_0=\\mathsf E_0","context":226,"kind":"use","source":11},{"line":147,"start":6067,"end":6106,"expression":"\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}","context":227,"kind":"use","source":11},{"line":147,"start":6113,"end":6119,"expression":"m\\geq1","context":228,"kind":"use","source":11},{"line":148,"start":6167,"end":6178,"expression":"\\mathsf D_m","context":228,"kind":"use","source":11},{"line":149,"start":6236,"end":6245,"expression":"D_m^{(M)}","context":228,"kind":"use","source":11},{"line":150,"start":6268,"end":6284,"expression":"\\mathcal U_{p,M}","context":228,"kind":"use","source":11},{"line":151,"start":6353,"end":6479,"expression":"\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n","context":228,"kind":"use","source":11},{"line":156,"start":6511,"end":6520,"expression":"1\\leq m<n","context":229,"kind":"use","source":11},{"line":157,"start":6545,"end":6603,"expression":"(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})","context":230,"kind":"use","source":11},{"line":158,"start":6610,"end":6665,"expression":"\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}","context":231,"kind":"use","source":11},{"line":159,"start":6684,"end":6687,"expression":"m=n","context":232,"kind":"use","source":11},{"line":160,"start":6712,"end":6739,"expression":"\\mathsf E_m-\\mathsf E_{m-1}","context":233,"kind":"use","source":11},{"line":160,"start":6763,"end":6766,"expression":"m=0","context":234,"kind":"use","source":11},{"line":161,"start":6773,"end":6807,"expression":"\\mathsf E_0\\mathsf E_n=\\mathsf E_0","context":235,"kind":"use","source":11},{"line":164,"start":6840,"end":6994,"expression":"\n \\HH_p=\\bigoplus_{m=0}^{\\infty}\\operatorname{Ran}\\mathsf F_m,\n \\qquad \\norm{f}_2^2=\\sum_{m=0}^{\\infty}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-parseval}\n","context":236,"kind":"use","source":11},{"line":169,"start":7040,"end":7041,"expression":"I","context":236,"kind":"use","source":11},{"line":170,"start":7098,"end":7103,"expression":"d_0=1","context":236,"kind":"use","source":11},{"line":170,"start":7110,"end":7126,"expression":"d_m=(p-1)p^{m-1}","context":236,"kind":"use","source":11},{"line":170,"start":7133,"end":7139,"expression":"m\\geq1","context":236,"kind":"use","source":11},{"line":171,"start":7177,"end":7188,"expression":"p^m-p^{m-1}","context":236,"kind":"use","source":11},{"line":173,"start":7220,"end":7243,"expression":"\\omega_p=\\exp(2\\pi i/p)","context":237,"kind":"use","source":11},{"line":174,"start":7272,"end":7466,"expression":"\n \\psi_{m,a,q}=p^{(m-1)/2}\\sum_{d=0}^{p-1}\n \\omega_p^{qd}\\mathbf1_{B_{m,a+d p^{m-1}}},\n \\quad\n \\begin{matrix}m\\geq1,\\quad 0\\leq a<p^{m-1},\\\\1\\leq q<p.\\end{matrix}\n \\label{a-limit-haar-wavelets}\n","context":238,"kind":"use","source":11},{"line":182,"start":7527,"end":7548,"expression":"\\sum_d\\omega_p^{qd}=0","context":239,"kind":"use","source":11},{"line":182,"start":7564,"end":7593,"expression":"\\operatorname{Ran}\\mathsf F_m","context":239,"kind":"use","source":11},{"line":183,"start":7617,"end":7635,"expression":"p^{m-1}p\\,p^{-m}=1","context":239,"kind":"use","source":11},{"line":185,"start":7738,"end":7781,"expression":"p^{-1}\\sum_d\\omega_p^{(q'-q)d}=\\delta_{qq'}","context":239,"kind":"use","source":11},{"line":186,"start":7838,"end":7841,"expression":"d_m","context":240,"kind":"use","source":11},{"line":187,"start":7884,"end":7892,"expression":"\\psi_0=1","context":241,"kind":"use","source":11},{"line":192,"start":8014,"end":8017,"expression":"s>0","context":242,"kind":"use","source":11},{"line":193,"start":8054,"end":8206,"expression":"\n \\norm{f}_{\\mathfrak H_p^s}^2\n =\\sum_{m\\geq0}p^{2sm}\\norm{\\mathsf F_mf}_2^2,\n \\qquad\n \\mathfrak H_p^s=\\{f\\in\\HH_p:\\norm{f}_{\\mathfrak H_p^s}<\\infty\\}.\n","context":242,"kind":"use","source":11},{"line":204,"start":8419,"end":8428,"expression":"f\\in\\HH_p","context":243,"kind":"use","source":11},{"line":204,"start":8441,"end":8447,"expression":"M\\geq0","context":244,"kind":"use","source":11},{"line":205,"start":8466,"end":8564,"expression":"\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n","context":245,"kind":"use","source":11},{"line":210,"start":8583,"end":8602,"expression":"f\\in\\mathfrak H_p^s","context":246,"kind":"use","source":11},{"line":211,"start":8626,"end":8727,"expression":"\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n","context":246,"kind":"use","source":11},{"line":216,"start":8759,"end":8775,"expression":"\\mathsf E_M\\to I","context":247,"kind":"use","source":11},{"line":221,"start":8933,"end":8936,"expression":"m>M","context":248,"kind":"use","source":11},{"line":221,"start":8940,"end":8963,"expression":"p^{2sm}\\geq p^{2s(M+1)}","context":249,"kind":"use","source":11},{"line":223,"start":9123,"end":9124,"expression":"M","context":250,"kind":"use","source":11},{"line":224,"start":9153,"end":9186,"expression":"\\operatorname{Ran}\\mathsf F_{M+1}","context":251,"kind":"use","source":11},{"line":225,"start":9199,"end":9212,"expression":"I-\\mathsf E_M","context":252,"kind":"use","source":11},{"line":226,"start":9242,"end":9264,"expression":"\\norm{I-\\mathsf E_M}=1","context":253,"kind":"use","source":11},{"line":226,"start":9277,"end":9278,"expression":"M","context":254,"kind":"use","source":11},{"line":230,"start":9377,"end":9380,"expression":"L^2","context":255,"kind":"use","source":11},{"line":231,"start":9483,"end":9495,"expression":"\\mathcal V_L","context":255,"kind":"use","source":11},{"line":232,"start":9517,"end":9524,"expression":"M\\geq L","context":255,"kind":"use","source":11},{"line":240,"start":9737,"end":9758,"expression":"(\\epsilon_m)_{m\\geq0}","context":256,"kind":"use","source":11},{"line":241,"start":9813,"end":10022,"expression":"\n \\Dom(H_\\epsilon)\n &=\\left\\{f\\in\\HH_p:\n \\sum_{m\\geq0}|\\epsilon_m|^2\\norm{\\mathsf F_mf}_2^2<\\infty\\right\\},\n \\nonumber\\\\\n H_\\epsilon f&=\\sum_{m\\geq0}\\epsilon_m\\mathsf F_mf.\n \\label{a-limit-hamiltonian-domain}\n","context":257,"kind":"use","source":11},{"line":249,"start":10059,"end":10062,"expression":"L^2","context":258,"kind":"use","source":11},{"line":256,"start":10459,"end":10484,"expression":"\\sup_m|\\epsilon_m|<\\infty","context":259,"kind":"use","source":11},{"line":258,"start":10523,"end":10604,"expression":"\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n","context":260,"kind":"use","source":11},{"line":264,"start":10713,"end":10714,"expression":"f","context":261,"kind":"use","source":11},{"line":265,"start":10751,"end":10767,"expression":"f_m=\\mathsf F_mf","context":262,"kind":"use","source":11},{"line":267,"start":10896,"end":10918,"expression":"g\\in\\Dom(H_\\epsilon^*)","context":263,"kind":"use","source":11},{"line":268,"start":10931,"end":10946,"expression":"k=H_\\epsilon^*g","context":264,"kind":"use","source":11},{"line":269,"start":11026,"end":11027,"expression":"m","context":265,"kind":"use","source":11},{"line":269,"start":11036,"end":11054,"expression":"k_m=\\epsilon_m g_m","context":266,"kind":"use","source":11},{"line":269,"start":11064,"end":11065,"expression":"k","context":267,"kind":"use","source":11},{"line":270,"start":11088,"end":11127,"expression":"\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty","context":268,"kind":"use","source":11},{"line":271,"start":11136,"end":11156,"expression":"g\\in\\Dom(H_\\epsilon)","context":269,"kind":"use","source":11},{"line":271,"start":11187,"end":11197,"expression":"H_\\epsilon","context":270,"kind":"use","source":11},{"line":275,"start":11296,"end":11316,"expression":"f\\in\\Dom(H_\\epsilon)","context":271,"kind":"use","source":11},{"line":275,"start":11344,"end":11357,"expression":"\\mathsf E_M f","context":271,"kind":"use","source":11},{"line":277,"start":11375,"end":11403,"expression":"\\norm{f-\\mathsf E_M f}_2\\to0","context":271,"kind":"use","source":11},{"line":278,"start":11410,"end":11450,"expression":"\\norm{H_\\epsilon(f-\\mathsf E_M f)}_2\\to0","context":271,"kind":"use","source":11},{"line":281,"start":11637,"end":11645,"expression":"C<\\infty","context":271,"kind":"use","source":11},{"line":282,"start":11664,"end":11700,"expression":"\\norm{H_\\epsilon f}_2\\leq C\\norm f_2","context":271,"kind":"use","source":11},{"line":283,"start":11739,"end":11740,"expression":"m","context":272,"kind":"use","source":11},{"line":283,"start":11758,"end":11770,"expression":"|\\epsilon_m|","context":273,"kind":"use","source":11},{"line":286,"start":11819,"end":11820,"expression":"z","context":274,"kind":"use","source":11},{"line":287,"start":11870,"end":11889,"expression":"(\\epsilon_m-z)^{-1}","context":274,"kind":"use","source":11},{"line":288,"start":11930,"end":11942,"expression":"H_\\epsilon-z","context":274,"kind":"use","source":11},{"line":289,"start":11983,"end":12027,"expression":"\\epsilon_m/(\\epsilon_m-z)=1+z/(\\epsilon_m-z)","context":274,"kind":"use","source":11},{"line":290,"start":12072,"end":12073,"expression":"z","context":274,"kind":"use","source":11},{"line":291,"start":12111,"end":12121,"expression":"\\epsilon_m","context":274,"kind":"use","source":11},{"line":291,"start":12162,"end":12163,"expression":"z","context":274,"kind":"use","source":11},{"line":292,"start":12189,"end":12192,"expression":"u_j","context":274,"kind":"use","source":11},{"line":292,"start":12200,"end":12230,"expression":"\\norm{(H_\\epsilon-z)u_j}_2\\to0","context":274,"kind":"use","source":11},{"line":303,"start":12616,"end":12625,"expression":"F:\\R\\to\\C","context":275,"kind":"use","source":11},{"line":304,"start":12651,"end":12740,"expression":"\n F(H_\\epsilon)f=\\sum_{m\\geq0}F(\\epsilon_m)\\mathsf F_mf.\n \\label{a-limit-borel-calculus}\n","context":275,"kind":"use","source":11},{"line":309,"start":12793,"end":12850,"expression":"\\norm{F(H_\\epsilon)f}_2\\leq\\sup_m|F(\\epsilon_m)|\\norm f_2","context":275,"kind":"use","source":11},{"line":312,"start":12979,"end":13207,"expression":"\n U(t)&=e^{-itH_\\epsilon}\n =\\sum_{m\\geq0}e^{-it\\epsilon_m}\\mathsf F_m,\n \\label{a-limit-unitary}\\\\\n (z-H_\\epsilon)^{-1}\n &=\\sum_{m\\geq0}(z-\\epsilon_m)^{-1}\\mathsf F_m,\n \\quad z\\notin\\sigma(H_\\epsilon).\n \\label{a-limit-resolvent}\n","context":276,"kind":"use","source":11},{"line":324,"start":13347,"end":13359,"expression":"U(t)^*U(t)=I","context":277,"kind":"use","source":11},{"line":325,"start":13366,"end":13381,"expression":"U(t+s)=U(t)U(s)","context":277,"kind":"use","source":11},{"line":326,"start":13420,"end":13540,"expression":"\n \\norm{(U(t)-U(t_0))f}_2^2\n =\\sum_m|e^{-it\\epsilon_m}-e^{-it_0\\epsilon_m}|^2\n \\norm{\\mathsf F_mf}_2^2\\longrightarrow0.\n","context":277,"kind":"use","source":11},{"line":332,"start":13603,"end":13637,"expression":"4\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2","context":277,"kind":"use","source":11},{"line":333,"start":13694,"end":13695,"expression":"M","context":277,"kind":"use","source":11},{"line":333,"start":13702,"end":13718,"expression":"\\Dom(H_\\epsilon)","context":277,"kind":"use","source":11},{"line":334,"start":13746,"end":13783,"expression":"|(e^{-it\\epsilon}-1)/t|\\leq|\\epsilon|","context":278,"kind":"use","source":11},{"line":335,"start":13795,"end":13843,"expression":"\n \\lim_{t\\to0}\\frac{U(t)f-f}{t}=-iH_\\epsilon f.\n","context":279,"kind":"use","source":11},{"line":339,"start":13915,"end":13918,"expression":"L^2","context":280,"kind":"use","source":11},{"line":343,"start":14051,"end":14067,"expression":"H_\\epsilon^{(M)}","context":281,"kind":"use","source":11},{"line":343,"start":14099,"end":14109,"expression":"H_\\epsilon","context":281,"kind":"use","source":11},{"line":344,"start":14115,"end":14127,"expression":"\\mathcal V_M","context":281,"kind":"use","source":11},{"line":344,"start":14159,"end":14163,"expression":"U(t)","context":281,"kind":"use","source":11},{"line":346,"start":14258,"end":14383,"expression":"\n U_M^{\\mathrm{emb}}(t)\n =\\sum_{m=0}^{M}e^{-it\\epsilon_m}\\mathsf F_m\n =U(t)\\mathsf E_M.\n \\label{a-limit-embedded-propagator}\n","context":281,"kind":"use","source":11},{"line":352,"start":14416,"end":14428,"expression":"\\mathcal V_M","context":281,"kind":"use","source":11},{"line":353,"start":14446,"end":14464,"expression":"\\mathcal V_M^\\perp","context":281,"kind":"use","source":11},{"line":353,"start":14485,"end":14490,"expression":"\\HH_p","context":282,"kind":"use","source":11},{"line":358,"start":14626,"end":14635,"expression":"f\\in\\HH_p","context":283,"kind":"use","source":11},{"line":359,"start":14654,"end":14753,"expression":"\n \\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2\n =\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-unitary-tail}\n","context":283,"kind":"use","source":11},{"line":364,"start":14807,"end":14808,"expression":"t","context":283,"kind":"use","source":11},{"line":365,"start":14816,"end":14835,"expression":"f\\in\\mathfrak H_p^s","context":283,"kind":"use","source":11},{"line":366,"start":14868,"end":14904,"expression":"p^{-s(M+1)}\\norm f_{\\mathfrak H_p^s}","context":283,"kind":"use","source":11},{"line":367,"start":14912,"end":14932,"expression":"f\\in\\Dom(H_\\epsilon)","context":284,"kind":"use","source":11},{"line":368,"start":14991,"end":15199,"expression":"\n &\\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2^2\n +\\norm{H_\\epsilon(U(t)f-U_M^{\\mathrm{emb}}(t)f)}_2^2\n \\nonumber\\\\\n &\\hspace{2em}=\n \\sum_{m>M}(1+|\\epsilon_m|^2)\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-graph-tail}\n","context":285,"kind":"use","source":11},{"line":379,"start":15289,"end":15328,"expression":"\\sum_{m>M}e^{-it\\epsilon_m}\\mathsf F_mf","context":286,"kind":"use","source":11},{"line":381,"start":15423,"end":15433,"expression":"H_\\epsilon","context":286,"kind":"use","source":11},{"line":381,"start":15468,"end":15478,"expression":"\\epsilon_m","context":286,"kind":"use","source":11},{"line":388,"start":15750,"end":15828,"expression":"\n \\widetilde H_M=H_\\epsilon\\mathsf E_M\n =\\sum_{m\\leq M}\\epsilon_m\\mathsf F_m.\n","context":287,"kind":"use","source":11},{"line":392,"start":15882,"end":15887,"expression":"\\HH_p","context":287,"kind":"use","source":11},{"line":394,"start":15972,"end":16162,"expression":"\n e^{-it\\widetilde H_M}\n =U_M^{\\mathrm{emb}}(t)+(I-\\mathsf E_M),\n \\qquad\n \\norm{(e^{-it\\widetilde H_M}-U(t))f}_2\n \\leq2\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-zero-hamiltonian-extension}\n","context":287,"kind":"use","source":11},{"line":402,"start":16200,"end":16201,"expression":"z","context":288,"kind":"use","source":11},{"line":403,"start":16217,"end":16488,"expression":"\n (z-\\widetilde H_M)^{-1}\n &=\\mathsf E_M(z-H_\\epsilon)^{-1}\n   +z^{-1}(I-\\mathsf E_M),\\nonumber\\\\\n \\norm{((z-\\widetilde H_M)^{-1}-(z-H_\\epsilon)^{-1})f}_2\n &\\leq\\left(|z|^{-1}+|\\operatorname{Im}z|^{-1}\\right)\n \\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-strong-resolvent}\n","context":289,"kind":"use","source":11},{"line":412,"start":16506,"end":16520,"expression":"\\widetilde H_M","context":290,"kind":"use","source":11},{"line":412,"start":16536,"end":16546,"expression":"H_\\epsilon","context":290,"kind":"use","source":11},{"line":418,"start":16852,"end":16919,"expression":"\n \\norm{U(t)-U_M^{\\mathrm{emb}}(t)}=1\n \\qquad\\text{for every }M,t.\n","context":291,"kind":"use","source":11},{"line":431,"start":17392,"end":17427,"expression":"u_m\\in\\operatorname{Ran}\\mathsf F_m","context":292,"kind":"use","source":11},{"line":432,"start":17450,"end":17455,"expression":"0<q<1","context":293,"kind":"use","source":11},{"line":433,"start":17468,"end":17533,"expression":"\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n","context":294,"kind":"use","source":11},{"line":437,"start":17542,"end":17556,"expression":"\\norm{f_q}_2=1","context":295,"kind":"use","source":11},{"line":438,"start":17578,"end":17657,"expression":"\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n","context":296,"kind":"use","source":11},{"line":442,"start":17682,"end":17704,"expression":"f_q\\in\\Dom(H_\\epsilon)","context":297,"kind":"use","source":11},{"line":442,"start":17720,"end":17724,"expression":"4q<1","context":298,"kind":"use","source":11},{"line":443,"start":17729,"end":17785,"expression":"\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n","context":299,"kind":"use","source":11},{"line":447,"start":17793,"end":17798,"expression":"q<1/4","context":300,"kind":"use","source":11},{"line":448,"start":17843,"end":17921,"expression":"\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n","context":301,"kind":"use","source":11},{"line":452,"start":17941,"end":17946,"expression":"q=1/8","context":302,"kind":"use","source":11},{"line":453,"start":18014,"end":18019,"expression":"q=1/2","context":303,"kind":"use","source":11},{"line":453,"start":18067,"end":18070,"expression":"L^2","context":304,"kind":"use","source":11},{"line":468,"start":18710,"end":18824,"expression":"\n \\norm{(I-\\mathsf E_M)(z-H_\\epsilon)^{-1}}\n =\\sup_{m>M}|z-\\epsilon_m|^{-1}.\n \\label{a-limit-resolvent-tail-norm}\n","context":305,"kind":"use","source":11},{"line":479,"start":19153,"end":19163,"expression":"H_\\epsilon","context":306,"kind":"use","source":11},{"line":480,"start":19203,"end":19224,"expression":"|\\epsilon_m|\\to\\infty","context":307,"kind":"use","source":11},{"line":490,"start":19806,"end":19809,"expression":"m_j","context":308,"kind":"use","source":11},{"line":491,"start":19817,"end":19839,"expression":"|\\epsilon_{m_j}|\\leq C","context":309,"kind":"use","source":11},{"line":491,"start":19851,"end":19852,"expression":"C","context":310,"kind":"use","source":11},{"line":491,"start":19869,"end":19872,"expression":"u_j","context":311,"kind":"use","source":11},{"line":492,"start":19927,"end":19928,"expression":"z","context":312,"kind":"use","source":11},{"line":493,"start":19998,"end":20010,"expression":"(|z|+C)^{-1}","context":313,"kind":"use","source":11},{"line":498,"start":20109,"end":20130,"expression":"|\\epsilon_m|\\to\\infty","context":314,"kind":"use","source":11},{"line":499,"start":20172,"end":20186,"expression":"\\widetilde H_M","context":314,"kind":"use","source":11},{"line":500,"start":20263,"end":20269,"expression":"z^{-1}","context":314,"kind":"use","source":11},{"line":501,"start":20330,"end":20340,"expression":"H_\\epsilon","context":314,"kind":"use","source":11},{"line":504,"start":20457,"end":20464,"expression":"\\beta>0","context":315,"kind":"use","source":11},{"line":504,"start":20483,"end":20504,"expression":"e^{-\\beta H_\\epsilon}","context":316,"kind":"use","source":11},{"line":507,"start":20656,"end":20720,"expression":"\\{f:\\sum_m e^{-2\\beta\\epsilon_m}\\norm{\\mathsf F_mf}_2^2<\\infty\\}","context":317,"kind":"use","source":11},{"line":511,"start":20898,"end":21044,"expression":"\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n","context":318,"kind":"use","source":11},{"line":519,"start":21222,"end":21236,"expression":"\\epsilon_m=c m","context":319,"kind":"use","source":11},{"line":519,"start":21244,"end":21247,"expression":"c>0","context":320,"kind":"use","source":11},{"line":520,"start":21268,"end":21282,"expression":"\\beta c>\\log p","context":321,"kind":"use","source":11},{"line":521,"start":21320,"end":21343,"expression":"\\epsilon_m=p^{\\alpha m}","context":322,"kind":"use","source":11},{"line":521,"start":21351,"end":21359,"expression":"\\alpha>0","context":323,"kind":"use","source":11},{"line":522,"start":21386,"end":21393,"expression":"\\beta>0","context":324,"kind":"use","source":11},{"line":523,"start":21466,"end":21467,"expression":"m","context":325,"kind":"use","source":11},{"line":533,"start":21794,"end":21804,"expression":"\\mathcal P","context":326,"kind":"use","source":11},{"line":533,"start":21837,"end":21855,"expression":"\\KK_{\\mathrm{tor}}","context":327,"kind":"use","source":11},{"line":536,"start":21967,"end":22102,"expression":"\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n","context":328,"kind":"use","source":11},{"line":543,"start":22162,"end":22221,"expression":"I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}","context":329,"kind":"use","source":11},{"line":544,"start":22243,"end":22258,"expression":"\\mathbf h=(h_p)","context":330,"kind":"use","source":11},{"line":545,"start":22270,"end":22356,"expression":"\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n","context":331,"kind":"use","source":11},{"line":550,"start":22413,"end":22612,"expression":"\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n","context":332,"kind":"use","source":11},{"line":562,"start":23013,"end":23014,"expression":"f","context":333,"kind":"use","source":11},{"line":565,"start":23083,"end":23161,"expression":"\n (V_p\\otimes I)(I\\otimes V_q)\n =(I\\otimes V_q)(V_p\\otimes I)=V_p\\otimes V_q.\n","context":334,"kind":"use","source":11},{"line":569,"start":23181,"end":23199,"expression":"\\KK_{\\mathrm{tor}}","context":334,"kind":"use","source":11},{"line":575,"start":23489,"end":23660,"expression":"\n \\mathsf F_{\\mathbf r}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}\\mathsf F_{p,r_p},\n \\qquad \\mathbf r\\in\\N^{\\mathcal P}.\n \\label{a-limit-joint-details}\n","context":335,"kind":"use","source":11},{"line":583,"start":23764,"end":23785,"expression":"\\mathsf E_{\\mathbf h}","context":335,"kind":"use","source":11},{"line":596,"start":24326,"end":24335,"expression":"\\mathbf r","context":336,"kind":"use","source":11},{"line":596,"start":24343,"end":24356,"expression":"A_{\\mathbf r}","context":336,"kind":"use","source":11},{"line":598,"start":24418,"end":24473,"expression":"\\KK_{\\mathbf r}=\\operatorname{Ran}\\mathsf F_{\\mathbf r}","context":336,"kind":"use","source":11},{"line":599,"start":24480,"end":24542,"expression":"\\HH_{\\mathcal P}^{\\infty}=\\bigoplus_{\\mathbf r}\\KK_{\\mathbf r}","context":336,"kind":"use","source":11},{"line":600,"start":24564,"end":24769,"expression":"\n \\Dom(H_A)&=\\left\\{f:\\sum_{\\mathbf r}\n \\norm{A_{\\mathbf r}\\mathsf F_{\\mathbf r}f}^2<\\infty\\right\\},\n \\nonumber\\\\\n H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n","context":337,"kind":"use","source":11},{"line":607,"start":24787,"end":24790,"expression":"H_A","context":338,"kind":"use","source":11},{"line":608,"start":24890,"end":24893,"expression":"H_A","context":339,"kind":"use","source":11},{"line":609,"start":24924,"end":24941,"expression":"H_A^{(\\mathbf h)}","context":340,"kind":"use","source":11},{"line":610,"start":25044,"end":25045,"expression":"F","context":341,"kind":"use","source":11},{"line":611,"start":25064,"end":25232,"expression":"\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n","context":342,"kind":"use","source":11},{"line":619,"start":25278,"end":25398,"expression":"\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n","context":343,"kind":"use","source":11},{"line":627,"start":25548,"end":25549,"expression":"g","context":344,"kind":"use","source":11},{"line":628,"start":25586,"end":25594,"expression":"k=H_A^*g","context":345,"kind":"use","source":11},{"line":629,"start":25637,"end":25652,"expression":"\\KK_{\\mathbf r}","context":346,"kind":"use","source":11},{"line":630,"start":25661,"end":25719,"expression":"\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g","context":347,"kind":"use","source":11},{"line":631,"start":25745,"end":25746,"expression":"k","context":348,"kind":"use","source":11},{"line":632,"start":25821,"end":25830,"expression":"H_A^*=H_A","context":349,"kind":"use","source":11},{"line":633,"start":25879,"end":25892,"expression":"f_{\\mathbf r}","context":350,"kind":"use","source":11},{"line":634,"start":25899,"end":25925,"expression":"A_{\\mathbf r}f_{\\mathbf r}","context":351,"kind":"use","source":11},{"line":637,"start":26047,"end":26050,"expression":"H_A","context":352,"kind":"use","source":11},{"line":641,"start":26333,"end":26334,"expression":"F","context":352,"kind":"use","source":11},{"line":642,"start":26365,"end":26412,"expression":"\\norm F_\\infty\\norm{(I-\\mathsf E_{\\mathbf h})f}","context":353,"kind":"use","source":11},{"line":649,"start":26634,"end":26789,"expression":"\n A_{\\mathbf r}\n =\\left(\\sum_a\\epsilon_{a,\\mathbf r}T_a\\right)\n \\otimes I_{\\mathrm{phase},\\mathbf r},\n \\qquad \\epsilon_{a,\\mathbf r}\\in\\R,\\quad T_a=T_a^*.\n","context":354,"kind":"use","source":11},{"line":655,"start":26806,"end":26809,"expression":"T_a","context":354,"kind":"use","source":11},{"line":664,"start":27364,"end":27409,"expression":"\\mathcal V_0\\subset\\mathcal V_1\n\\subset\\cdots","context":355,"kind":"use","source":11},{"line":665,"start":27447,"end":27450,"expression":"K_M","context":355,"kind":"use","source":11},{"line":666,"start":27461,"end":27474,"expression":"K_{M+1}f=K_Mf","context":355,"kind":"use","source":11},{"line":666,"start":27481,"end":27497,"expression":"f\\in\\mathcal V_M","context":355,"kind":"use","source":11},{"line":667,"start":27531,"end":27569,"expression":"\\mathcal V_M^\\perp\\cap\\mathcal V_{M+1}","context":355,"kind":"use","source":11},{"line":668,"start":27590,"end":27591,"expression":"g","context":355,"kind":"use","source":11},{"line":668,"start":27604,"end":27620,"expression":"f\\in\\mathcal V_M","context":356,"kind":"use","source":11},{"line":669,"start":27624,"end":27665,"expression":"\\inner{f}{K_{M+1}g}=\\inner{K_{M+1}f}{g}=0","context":357,"kind":"use","source":11},{"line":670,"start":27687,"end":27699,"expression":"\\mathcal V_0","context":358,"kind":"use","source":11},{"line":675,"start":28045,"end":28072,"expression":"K_{M+1}|_{\\mathcal V_M}=K_M","context":359,"kind":"use","source":11},{"line":680,"start":28168,"end":28185,"expression":"Z_{\\mathrm{sign}}","context":360,"kind":"use","source":11},{"line":680,"start":28205,"end":28210,"expression":"+1,-1","context":361,"kind":"use","source":11},{"line":681,"start":28245,"end":28250,"expression":"B=B^*","context":362,"kind":"use","source":11},{"line":682,"start":28319,"end":28342,"expression":"|b_0\\rangle,|b_1\\rangle","context":363,"kind":"use","source":11},{"line":683,"start":28350,"end":28364,"expression":"B|b_0\\rangle=0","context":364,"kind":"use","source":11},{"line":683,"start":28371,"end":28395,"expression":"B|b_1\\rangle=|b_1\\rangle","context":365,"kind":"use","source":11},{"line":685,"start":28526,"end":28528,"expression":"+1","context":366,"kind":"use","source":11},{"line":686,"start":28552,"end":28558,"expression":"g\\neq0","context":367,"kind":"use","source":11},{"line":687,"start":28567,"end":28700,"expression":"\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n","context":368,"kind":"use","source":11},{"line":694,"start":28773,"end":28941,"expression":"\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n","context":369,"kind":"use","source":11},{"line":700,"start":28984,"end":28985,"expression":"b","context":370,"kind":"use","source":11},{"line":701,"start":29082,"end":29090,"expression":"\\cos(gt)","context":371,"kind":"use","source":11},{"line":703,"start":29165,"end":29166,"expression":"c","context":372,"kind":"use","source":11},{"line":703,"start":29185,"end":29192,"expression":"(1+c)/2","context":373,"kind":"use","source":11},{"line":703,"start":29199,"end":29206,"expression":"(1-c)/2","context":374,"kind":"use","source":11},{"line":704,"start":29280,"end":29289,"expression":"(1-c^2)/4","context":375,"kind":"use","source":11},{"line":706,"start":29341,"end":29486,"expression":"\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n","context":376,"kind":"use","source":11},{"line":712,"start":29507,"end":29518,"expression":"0\\log_2 0=0","context":377,"kind":"use","source":11},{"line":712,"start":29525,"end":29540,"expression":"t_* =\\pi/(2|g|)","context":378,"kind":"use","source":11},{"line":728,"start":30254,"end":30255,"expression":"V","context":379,"kind":"use","source":11},{"line":729,"start":30321,"end":30327,"expression":"P=VV^*","context":379,"kind":"use","source":11},{"line":729,"start":30334,"end":30339,"expression":"Q=I-P","context":379,"kind":"use","source":11},{"line":729,"start":30351,"end":30360,"expression":"H_0=H_0^*","context":379,"kind":"use","source":11},{"line":730,"start":30371,"end":30380,"expression":"H_0V=VH_c","context":379,"kind":"use","source":11},{"line":730,"start":30388,"end":30393,"expression":"B=B^*","context":379,"kind":"use","source":11},{"line":731,"start":30421,"end":30443,"expression":"\n PBP=0,\\qquad QBQ=0.\n","context":379,"kind":"use","source":11},{"line":734,"start":30458,"end":30465,"expression":"\\lambda","context":379,"kind":"use","source":11},{"line":734,"start":30476,"end":30499,"expression":"H_\\lambda=H_0+\\lambda B","context":379,"kind":"use","source":11},{"line":735,"start":30523,"end":30537,"expression":"\\lambda=p^{-w}","context":379,"kind":"use","source":11},{"line":735,"start":30563,"end":30569,"expression":"w\\geq1","context":379,"kind":"use","source":11},{"line":740,"start":30710,"end":30884,"expression":"\n \\norm{H_\\lambda V-VH_c}&=|\\lambda|\\norm{BV},\n \\label{a-limit-defect-norm}\\\\\n \\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n","context":380,"kind":"use","source":11},{"line":747,"start":30901,"end":30936,"expression":"z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})","context":381,"kind":"use","source":11},{"line":748,"start":30981,"end":31089,"expression":"\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n","context":382,"kind":"use","source":11},{"line":753,"start":31136,"end":31140,"expression":"Q\\HH","context":383,"kind":"use","source":11},{"line":754,"start":31163,"end":31308,"expression":"\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n","context":384,"kind":"use","source":11},{"line":762,"start":31370,"end":31380,"expression":"\\lambda BV","context":385,"kind":"use","source":11},{"line":763,"start":31440,"end":31443,"expression":"H_0","context":386,"kind":"use","source":11},{"line":764,"start":31495,"end":31509,"expression":"Qe^{-itH_0}P=0","context":387,"kind":"use","source":11},{"line":765,"start":31543,"end":31625,"expression":"\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n","context":388,"kind":"use","source":11},{"line":770,"start":31674,"end":31677,"expression":"H_0","context":389,"kind":"use","source":11},{"line":771,"start":31685,"end":31714,"expression":"B e^{-isH_0}P=QBP e^{-isH_0}P","context":390,"kind":"use","source":11},{"line":773,"start":31839,"end":31842,"expression":"|t|","context":391,"kind":"use","source":11},{"line":777,"start":32016,"end":32019,"expression":"H_0","context":392,"kind":"use","source":11},{"line":778,"start":32033,"end":32057,"expression":"PH_\\lambda Q=\\lambda PBQ","context":392,"kind":"use","source":11},{"line":778,"start":32064,"end":32088,"expression":"QH_\\lambda P=\\lambda QBP","context":392,"kind":"use","source":11},{"line":781,"start":32218,"end":32229,"expression":"QBP=(PBQ)^*","context":392,"kind":"use","source":11},{"line":785,"start":32335,"end":32349,"expression":"-it\\lambda QBP","context":393,"kind":"use","source":11},{"line":786,"start":32384,"end":32543,"expression":"\n \\norm{Qe^{-itH_\\lambda}P+it\\lambda QBP}\n \\leq\\frac{|\\lambda|t^2}{2}\n \\bigl(\\norm{H_\\lambda}+\\norm{H_0}\\bigr)\\norm{QBP}.\n \\label{a-limit-shorttime-remainder}\n","context":393,"kind":"use","source":11},{"line":792,"start":32607,"end":32610,"expression":"QBP","context":393,"kind":"use","source":11},{"line":793,"start":32642,"end":32673,"expression":"\\norm{e^{-iuK}-I}\\leq|u|\\norm K","context":393,"kind":"use","source":11},{"line":794,"start":32702,"end":32703,"expression":"K","context":393,"kind":"use","source":11},{"line":795,"start":32741,"end":32788,"expression":"(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}","context":394,"kind":"use","source":11},{"line":796,"start":32854,"end":32859,"expression":"t^2/2","context":395,"kind":"use","source":11},{"line":797,"start":32867,"end":32881,"expression":"|\\lambda|\\leq1","context":396,"kind":"use","source":11},{"line":797,"start":32903,"end":32918,"expression":"O(t^2|\\lambda|)","context":397,"kind":"use","source":11},{"line":798,"start":32942,"end":32957,"expression":"O(t^2\\lambda^2)","context":398,"kind":"use","source":11},{"line":801,"start":33073,"end":33087,"expression":"\\lambda=p^{-w}","context":399,"kind":"use","source":11},{"line":802,"start":33160,"end":33166,"expression":"p^{-w}","context":399,"kind":"use","source":11},{"line":803,"start":33208,"end":33214,"expression":"p^{-w}","context":399,"kind":"use","source":11},{"line":804,"start":33290,"end":33308,"expression":"QH_\\lambda Q=QH_0Q","context":399,"kind":"use","source":11},{"line":805,"start":33350,"end":33357,"expression":"\\lambda","context":399,"kind":"use","source":11},{"line":808,"start":33593,"end":33599,"expression":"p^{-w}","context":400,"kind":"use","source":11},{"line":816,"start":33756,"end":33762,"expression":"N\\geq3","context":401,"kind":"use","source":11},{"line":816,"start":33769,"end":33775,"expression":"h\\geq0","context":401,"kind":"use","source":11},{"line":816,"start":33783,"end":33786,"expression":"C_N","context":401,"kind":"use","source":11},{"line":817,"start":33819,"end":33826,"expression":"T_{p,h}","context":401,"kind":"use","source":11},{"line":817,"start":33850,"end":33851,"expression":"p","context":401,"kind":"use","source":11},{"line":818,"start":33893,"end":33894,"expression":"h","context":401,"kind":"use","source":11},{"line":819,"start":33920,"end":33953,"expression":"\n \\mathcal G=C_N\\square T_{p,h}.\n","context":401,"kind":"use","source":11},{"line":822,"start":33969,"end":33984,"expression":"v=(x,\\mathbf d)","context":401,"kind":"use","source":11},{"line":822,"start":33993,"end":34003,"expression":"x\\in\\Z/N\\Z","context":401,"kind":"use","source":11},{"line":822,"start":34010,"end":34019,"expression":"\\mathbf d","context":401,"kind":"use","source":11},{"line":823,"start":34055,"end":34056,"expression":"x","context":401,"kind":"use","source":11},{"line":825,"start":34133,"end":34150,"expression":"N\\sum_{r=0}^h p^r","context":401,"kind":"use","source":11},{"line":828,"start":34276,"end":34282,"expression":"\\vec E","context":402,"kind":"introduction-passage","source":11},{"line":828,"start":34318,"end":34326,"expression":"\\tau_2=2","context":403,"kind":"introduction-passage","source":11},{"line":829,"start":34330,"end":34340,"expression":"\\tau_p=p-1","context":404,"kind":"introduction-passage","source":11},{"line":829,"start":34351,"end":34352,"expression":"p","context":405,"kind":"introduction-passage","source":11},{"line":830,"start":34389,"end":34444,"expression":"\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n","context":406,"kind":"introduction-passage","source":11},{"line":833,"start":34474,"end":34477,"expression":"d_v","context":406,"kind":"introduction-passage","source":11},{"line":834,"start":34519,"end":34545,"expression":"s_v=d_v^{-1/2}(1,\\ldots,1)","context":406,"kind":"introduction-passage","source":11},{"line":835,"start":34568,"end":34598,"expression":"G_v=2|s_v\\rangle\\langle s_v|-I","context":406,"kind":"introduction-passage","source":11},{"line":836,"start":34616,"end":34617,"expression":"I","context":406,"kind":"introduction-passage","source":11},{"line":836,"start":34629,"end":34652,"expression":"|s_v\\rangle\\langle s_v|","context":406,"kind":"introduction-passage","source":11},{"line":837,"start":34695,"end":34698,"expression":"R_v","context":407,"kind":"use","source":11},{"line":838,"start":34740,"end":34773,"expression":"\n C=\\bigoplus_v(G_v\\otimes R_v).\n","context":408,"kind":"use","source":11},{"line":841,"start":34796,"end":34797,"expression":"S","context":409,"kind":"use","source":11},{"line":841,"start":34814,"end":34820,"expression":"v\\to u","context":410,"kind":"use","source":11},{"line":841,"start":34826,"end":34832,"expression":"u\\to v","context":411,"kind":"use","source":11},{"line":842,"start":34894,"end":34899,"expression":"S^*=S","context":412,"kind":"use","source":11},{"line":842,"start":34906,"end":34911,"expression":"S^2=I","context":413,"kind":"use","source":11},{"line":844,"start":34948,"end":34993,"expression":"\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n","context":414,"kind":"use","source":11},{"line":848,"start":35028,"end":35064,"expression":"U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I","context":415,"kind":"use","source":11},{"line":853,"start":35202,"end":35213,"expression":"\\psi^n(v,u)","context":416,"kind":"introduction-passage","source":11},{"line":853,"start":35261,"end":35262,"expression":"v","context":417,"kind":"introduction-passage","source":11},{"line":854,"start":35286,"end":35300,"expression":"\\phi^n=C\\psi^n","context":417,"kind":"introduction-passage","source":11},{"line":855,"start":35336,"end":35426,"expression":"\n \\rho_v^n=\\sum_{u\\sim v}\\norm{\\psi^n(v,u)}^2,\n \\qquad F_{v\\to u}^n=\\norm{\\phi^n(v,u)}^2.\n","context":417,"kind":"introduction-passage","source":11},{"line":860,"start":35459,"end":35486,"expression":"\\sum_uF_{v\\to u}^n=\\rho_v^n","context":417,"kind":"introduction-passage","source":11},{"line":862,"start":35515,"end":35547,"expression":"\\rho_v^{n+1}=\\sum_u F_{u\\to v}^n","context":417,"kind":"introduction-passage","source":11},{"line":863,"start":35585,"end":35687,"expression":"\n \\rho_v^{n+1}-\\rho_v^n\n =\\sum_{u\\sim v}(F_{u\\to v}^n-F_{v\\to u}^n).\n \\label{a-limit-walk-continuity}\n","context":417,"kind":"introduction-passage","source":11},{"line":868,"start":35729,"end":35730,"expression":"A","context":418,"kind":"use","source":11},{"line":870,"start":35849,"end":36046,"expression":"\n \\sum_{v\\in A}(\\rho_v^{n+1}-\\rho_v^n)\n =\\sum_{\\substack{u\\notin A,\\ v\\in A\\\\u\\sim v}}F_{u\\to v}^n\n -\\sum_{\\substack{v\\in A,\\ u\\notin A\\\\u\\sim v}}F_{v\\to u}^n.\n \\label{a-limit-walk-coarse-balance}\n","context":419,"kind":"use","source":11},{"line":876,"start":36069,"end":36070,"expression":"A","context":420,"kind":"use","source":11},{"line":882,"start":36358,"end":36361,"expression":"v_0","context":421,"kind":"use","source":11},{"line":883,"start":36385,"end":36513,"expression":"\n d_{C_N}(x,x_0)+d_T(\\mathbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n","context":422,"kind":"use","source":11},{"line":888,"start":36555,"end":36573,"expression":"\\operatorname{lcp}","context":423,"kind":"use","source":11},{"line":890,"start":36624,"end":36725,"expression":"\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n","context":424,"kind":"use","source":11},{"line":895,"start":36809,"end":36852,"expression":"\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n","context":425,"kind":"use","source":11},{"line":906,"start":37342,"end":37345,"expression":"n=0","context":426,"kind":"use","source":11},{"line":909,"start":37550,"end":37551,"expression":"n","context":426,"kind":"use","source":11},{"line":909,"start":37596,"end":37601,"expression":"(n+1)","context":426,"kind":"use","source":11},{"line":932,"start":38669,"end":38675,"expression":"h\\geq1","context":427,"kind":"introduction-passage","source":11},{"line":932,"start":38682,"end":38688,"expression":"N\\geq3","context":428,"kind":"introduction-passage","source":11},{"line":933,"start":38718,"end":38736,"expression":"C_N\\square T_{p,h}","context":429,"kind":"introduction-passage","source":11},{"line":933,"start":38742,"end":38762,"expression":"C_N\\square T_{p,h+1}","context":430,"kind":"introduction-passage","source":11},{"line":934,"start":38843,"end":38844,"expression":"J","context":431,"kind":"introduction-passage","source":11},{"line":935,"start":38882,"end":38885,"expression":"d=3","context":432,"kind":"introduction-passage","source":11},{"line":936,"start":38947,"end":38948,"expression":"p","context":432,"kind":"introduction-passage","source":11},{"line":936,"start":38984,"end":38987,"expression":"d+p","context":433,"kind":"introduction-passage","source":11},{"line":937,"start":39028,"end":39031,"expression":"e_j","context":434,"kind":"introduction-passage","source":11},{"line":938,"start":39071,"end":39078,"expression":"2/(d+p)","context":435,"kind":"introduction-passage","source":11},{"line":938,"start":39096,"end":39097,"expression":"p","context":436,"kind":"introduction-passage","source":11},{"line":941,"start":39320,"end":39324,"expression":"\\chi","context":437,"kind":"use","source":11},{"line":942,"start":39343,"end":39456,"expression":"\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n","context":438,"kind":"use","source":11},{"line":947,"start":39482,"end":39508,"expression":"JU_{p,h,N}(e_j\\otimes\\chi)","context":439,"kind":"use","source":11},{"line":948,"start":39540,"end":39567,"expression":"U_{p,h+1,N}J\\neq JU_{p,h,N}","context":440,"kind":"use","source":11},{"line":962,"start":40069,"end":40072,"expression":"L^2","context":441,"kind":"use","source":11},{"line":964,"start":40202,"end":40205,"expression":"L^2","context":441,"kind":"use","source":11},{"line":971,"start":40523,"end":40554,"expression":"\n (p,M)=(2,7),\\ (3,5),\\ (5,3),\n","context":442,"kind":"use","source":11},{"line":24,"start":3034,"end":3035,"expression":"V","context":443,"kind":"use","source":12},{"line":24,"start":3042,"end":3048,"expression":"V^*V=I","context":443,"kind":"use","source":12},{"line":24,"start":3079,"end":3085,"expression":"P=VV^*","context":443,"kind":"use","source":12},{"line":24,"start":3120,"end":3125,"expression":"Q=I-P","context":443,"kind":"use","source":12},{"line":24,"start":3183,"end":3189,"expression":"H'V=VH","context":443,"kind":"use","source":12},{"line":25,"start":3259,"end":3262,"expression":"E_r","context":444,"kind":"use","source":12},{"line":25,"start":3312,"end":3323,"expression":"\\mathsf E_m","context":444,"kind":"use","source":12},{"line":25,"start":3359,"end":3398,"expression":"\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}","context":444,"kind":"use","source":12},{"line":27,"start":3754,"end":3757,"expression":"z-H","context":445,"kind":"use","source":12},{"line":27,"start":3793,"end":3805,"expression":"(z-QHQ)^{-1}","context":445,"kind":"use","source":12},{"line":16,"start":805,"end":815,"expression":"(J-z)^{-1}","context":446,"kind":"use","source":13},{"line":16,"start":833,"end":843,"expression":"(z-J)^{-1}","context":446,"kind":"use","source":13},{"line":19,"start":889,"end":890,"expression":"p","context":447,"kind":"use","source":13},{"line":23,"start":1116,"end":1127,"expression":"\\mathbb Q_p","context":447,"kind":"use","source":13},{"line":24,"start":1187,"end":1198,"expression":"\\mathbb Z_p","context":447,"kind":"use","source":13},{"line":13,"start":605,"end":612,"expression":"\\hbar=1","context":448,"kind":"use","source":15},{"line":19,"start":882,"end":888,"expression":"p\\geq2","context":449,"kind":"use","source":15},{"line":19,"start":895,"end":901,"expression":"M\\geq3","context":450,"kind":"use","source":15},{"line":19,"start":913,"end":917,"expression":"N=pM","context":451,"kind":"use","source":15},{"line":20,"start":985,"end":994,"expression":"0\\leq r<p","context":452,"kind":"use","source":15},{"line":21,"start":1018,"end":1033,"expression":"J_r:\\C^M\\to\\C^N","context":453,"kind":"use","source":15},{"line":22,"start":1040,"end":1086,"expression":"\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n","context":454,"kind":"use","source":15},{"line":25,"start":1095,"end":1118,"expression":"J_r^*J_s=\\delta_{rs}I_M","context":454,"kind":"use","source":15},{"line":25,"start":1125,"end":1143,"expression":"\\sum_rJ_rJ_r^*=I_N","context":454,"kind":"use","source":15},{"line":26,"start":1173,"end":1183,"expression":"\\mathcal A","context":454,"kind":"use","source":15},{"line":26,"start":1192,"end":1204,"expression":"|pk+r\\rangle","context":454,"kind":"use","source":15},{"line":27,"start":1210,"end":1235,"expression":"|r\\rangle\\otimes|k\\rangle","context":454,"kind":"use","source":15},{"line":28,"start":1287,"end":1288,"expression":"X","context":455,"kind":"use","source":15},{"line":28,"start":1294,"end":1298,"expression":"\\C^N","context":456,"kind":"use","source":15},{"line":28,"start":1321,"end":1337,"expression":"X_{rs}=J_r^*XJ_s","context":457,"kind":"use","source":15},{"line":29,"start":1371,"end":1497,"expression":"\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n","context":458,"kind":"use","source":15},{"line":34,"start":1553,"end":1567,"expression":"\\sum_rJ_rJ_r^*","context":459,"kind":"use","source":15},{"line":35,"start":1587,"end":1588,"expression":"X","context":460,"kind":"use","source":15},{"line":35,"start":1595,"end":1600,"expression":"H=H^*","context":461,"kind":"use","source":15},{"line":35,"start":1623,"end":1638,"expression":"H_{rs}^*=H_{sr}","context":462,"kind":"use","source":15},{"line":35,"start":1645,"end":1646,"expression":"U","context":463,"kind":"use","source":15},{"line":37,"start":1698,"end":1848,"expression":"\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber\\\\\n \\sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\\delta_{rt}I_M.\n \\label{a-spectral-paraunitarity}\n","context":464,"kind":"use","source":15},{"line":44,"start":1943,"end":1944,"expression":"H","context":465,"kind":"use","source":15},{"line":45,"start":1992,"end":2063,"expression":"\n e^{-it\\mathcal A H\\mathcal A^*}\n   =\\mathcal A e^{-itH}\\mathcal A^*.\n","context":465,"kind":"use","source":15},{"line":49,"start":2074,"end":2129,"expression":"(\\mathcal A H\\mathcal A^*)^n=\\mathcal A H^n\\mathcal A^*","context":465,"kind":"use","source":15},{"line":52,"start":2270,"end":2307,"expression":"\\Phi(\\rho)=\\sum_\\mu K_\\mu\\rho K_\\mu^*","context":465,"kind":"use","source":15},{"line":53,"start":2315,"end":2338,"expression":"\\sum_\\mu K_\\mu^*K_\\mu=I","context":465,"kind":"use","source":15},{"line":54,"start":2348,"end":2452,"expression":"\n K_\\mu^{\\rm poly}=\\mathcal A K_\\mu\\mathcal A^*,\\qquad\n \\sum_\\mu(K_\\mu^{\\rm poly})^*K_\\mu^{\\rm poly}=I.\n","context":466,"kind":"use","source":15},{"line":64,"start":2694,"end":2724,"expression":"i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s","context":467,"kind":"use","source":15},{"line":64,"start":2732,"end":2747,"expression":"H_{rs}^*=H_{sr}","context":468,"kind":"use","source":15},{"line":65,"start":2771,"end":2790,"expression":"w_r=\\norm{\\psi_r}^2","context":469,"kind":"use","source":15},{"line":66,"start":2802,"end":3003,"expression":"\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n","context":470,"kind":"use","source":15},{"line":73,"start":3020,"end":3029,"expression":"\\sum_rw_r","context":471,"kind":"use","source":15},{"line":77,"start":3108,"end":3176,"expression":"\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle","context":472,"kind":"use","source":15},{"line":80,"start":3260,"end":3339,"expression":"\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}","context":473,"kind":"use","source":15},{"line":84,"start":3435,"end":3438,"expression":"w_r","context":474,"kind":"use","source":15},{"line":89,"start":3639,"end":3668,"expression":"S|n\\rangle=|n+1\\pmod N\\rangle","context":475,"kind":"use","source":15},{"line":90,"start":3690,"end":3769,"expression":"\n H=\\mu I+\\kappa(S+S^*),\\qquad \\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n","context":475,"kind":"use","source":15},{"line":95,"start":3815,"end":3849,"expression":"(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}","context":475,"kind":"use","source":15},{"line":96,"start":3856,"end":3873,"expression":"\\theta_j=2\\pi j/M","context":475,"kind":"use","source":15},{"line":96,"start":3883,"end":3893,"expression":"\\mathcal A","context":475,"kind":"use","source":15},{"line":96,"start":3923,"end":3924,"expression":"k","context":475,"kind":"use","source":15},{"line":98,"start":4032,"end":4041,"expression":"p\\times p","context":475,"kind":"use","source":15},{"line":99,"start":4085,"end":4086,"expression":"S","context":476,"kind":"use","source":15},{"line":99,"start":4095,"end":4100,"expression":"(r,k)","context":477,"kind":"use","source":15},{"line":99,"start":4106,"end":4113,"expression":"(r+1,k)","context":478,"kind":"use","source":15},{"line":99,"start":4120,"end":4125,"expression":"r<p-1","context":479,"kind":"use","source":15},{"line":100,"start":4138,"end":4145,"expression":"(p-1,k)","context":480,"kind":"use","source":15},{"line":100,"start":4151,"end":4158,"expression":"(0,k+1)","context":481,"kind":"use","source":15},{"line":101,"start":4202,"end":4216,"expression":"e^{-i\\theta_j}","context":482,"kind":"use","source":15},{"line":102,"start":4247,"end":4442,"expression":"\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,\\\\\n e^{-i\\theta}e_0,&r=p-1.\n \\end{cases}\n \\qquad\n \\mathsf H_p(\\theta)=\\mu I_p+\\kappa(P_p(\\theta)+P_p(\\theta)^*).\n \\label{a-spectral-bloch-symbol}\n","context":483,"kind":"use","source":15},{"line":112,"start":4474,"end":4475,"expression":"H","context":484,"kind":"use","source":15},{"line":112,"start":4481,"end":4523,"expression":"\\bigoplus_{j=0}^{M-1}\\mathsf H_p(\\theta_j)","context":484,"kind":"use","source":15},{"line":113,"start":4542,"end":4571,"expression":"P_p(\\theta)^p=e^{-i\\theta}I_p","context":484,"kind":"use","source":15},{"line":117,"start":4662,"end":4676,"expression":"q=0,\\ldots,p-1","context":485,"kind":"use","source":15},{"line":118,"start":4699,"end":4718,"expression":"\\mathsf H_p(\\theta)","context":486,"kind":"use","source":15},{"line":119,"start":4740,"end":4848,"expression":"\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n","context":487,"kind":"use","source":15},{"line":124,"start":4867,"end":4882,"expression":"\\theta=\\theta_j","context":488,"kind":"use","source":15},{"line":124,"start":4905,"end":4907,"expression":"pM","context":489,"kind":"use","source":15},{"line":128,"start":5040,"end":5066,"expression":"\\alpha_q=(\\theta+2\\pi q)/p","context":490,"kind":"use","source":15},{"line":128,"start":5081,"end":5082,"expression":"r","context":491,"kind":"use","source":15},{"line":129,"start":5100,"end":5103,"expression":"v_q","context":492,"kind":"use","source":15},{"line":129,"start":5109,"end":5131,"expression":"p^{-1/2}e^{i\\alpha_qr}","context":493,"kind":"use","source":15},{"line":130,"start":5163,"end":5195,"expression":"P_p(\\theta)v_q=e^{-i\\alpha_q}v_q","context":494,"kind":"use","source":15},{"line":131,"start":5248,"end":5274,"expression":"e^{ip\\alpha_q}=e^{i\\theta}","context":495,"kind":"use","source":15},{"line":131,"start":5282,"end":5285,"expression":"v_q","context":496,"kind":"use","source":15},{"line":133,"start":5342,"end":5365,"expression":"\\mu I+\\kappa(P_p+P_p^*)","context":497,"kind":"use","source":15},{"line":134,"start":5444,"end":5450,"expression":"k=j+qM","context":498,"kind":"use","source":15},{"line":135,"start":5474,"end":5505,"expression":"2\\pi k/(pM)=(\\theta_j+2\\pi q)/p","context":499,"kind":"use","source":15},{"line":136,"start":5554,"end":5579,"expression":"\\mu+2\\kappa\\cos(2\\pi k/N)","context":500,"kind":"use","source":15},{"line":140,"start":5687,"end":5712,"expression":"H=\\sum_\\ell c_\\ell S^\\ell","context":501,"kind":"use","source":15},{"line":141,"start":5728,"end":5761,"expression":"\\sum_\\ell c_\\ell P_p(\\theta)^\\ell","context":501,"kind":"use","source":15},{"line":142,"start":5833,"end":5838,"expression":"N=p^h","context":501,"kind":"use","source":15},{"line":157,"start":6444,"end":6460,"expression":"\\HH_L\\oplus\\HH_D","context":502,"kind":"use","source":15},{"line":158,"start":6519,"end":6626,"expression":"\n H=\\begin{pmatrix}A&B\\\\B^*&D\\end{pmatrix},\\qquad A=A^*,\\quad D=D^*.\n \\label{a-spectral-block-hamiltonian}\n","context":503,"kind":"use","source":15},{"line":162,"start":6647,"end":6662,"expression":"B:\\HH_D\\to\\HH_L","context":504,"kind":"use","source":15},{"line":162,"start":6670,"end":6677,"expression":"\\iota_L","context":504,"kind":"use","source":15},{"line":162,"start":6686,"end":6691,"expression":"\\HH_L","context":504,"kind":"use","source":15},{"line":169,"start":6919,"end":6935,"expression":"z\\notin\\sigma(D)","context":505,"kind":"use","source":15},{"line":170,"start":6961,"end":7062,"expression":"\n \\Sigma(z)=B(z-D)^{-1}B^*,\\qquad\n \\mathcal S(z)=z-A-\\Sigma(z).\n \\label{a-spectral-schur-definition}\n","context":505,"kind":"use","source":15},{"line":175,"start":7083,"end":7086,"expression":"z-H","context":505,"kind":"use","source":15},{"line":175,"start":7116,"end":7129,"expression":"\\mathcal S(z)","context":505,"kind":"use","source":15},{"line":177,"start":7187,"end":7325,"expression":"\n \\iota_L^*(z-H)^{-1}\\iota_L&=\\mathcal S(z)^{-1},\\nonumber\\\\\n \\det(z-H)&=\\det(z-D)\\det\\mathcal S(z).\n \\label{a-spectral-schur-identities}\n","context":506,"kind":"use","source":15},{"line":182,"start":7342,"end":7366,"expression":"E\\in\\R\\setminus\\sigma(D)","context":507,"kind":"use","source":15},{"line":182,"start":7388,"end":7389,"expression":"H","context":507,"kind":"use","source":15},{"line":183,"start":7431,"end":7447,"expression":"\\mathcal S(E)u=0","context":507,"kind":"use","source":15},{"line":184,"start":7473,"end":7489,"expression":"v=(E-D)^{-1}B^*u","context":507,"kind":"use","source":15},{"line":187,"start":7525,"end":7530,"expression":"C=z-D","context":507,"kind":"use","source":15},{"line":188,"start":7569,"end":7719,"expression":"\n z-H=\n \\begin{pmatrix}I&-BC^{-1}\\\\0&I\\end{pmatrix}\n \\begin{pmatrix}\\mathcal S(z)&0\\\\0&C\\end{pmatrix}\n \\begin{pmatrix}I&0\\\\-C^{-1}B^*&I\\end{pmatrix}.\n","context":508,"kind":"use","source":15},{"line":197,"start":7925,"end":7965,"expression":"(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}","context":509,"kind":"use","source":15},{"line":198,"start":7994,"end":8006,"expression":"v=C^{-1}B^*u","context":509,"kind":"use","source":15},{"line":198,"start":8040,"end":8056,"expression":"\\mathcal S(z)u=f","context":509,"kind":"use","source":15},{"line":199,"start":8086,"end":8089,"expression":"f=0","context":509,"kind":"use","source":15},{"line":200,"start":8129,"end":8132,"expression":"u=0","context":510,"kind":"use","source":15},{"line":200,"start":8146,"end":8147,"expression":"E","context":511,"kind":"use","source":15},{"line":200,"start":8168,"end":8171,"expression":"E-D","context":512,"kind":"use","source":15},{"line":200,"start":8181,"end":8184,"expression":"v=0","context":513,"kind":"use","source":15},{"line":201,"start":8229,"end":8234,"expression":"u\\ne0","context":514,"kind":"use","source":15},{"line":204,"start":8254,"end":8274,"expression":"P_L=\\iota_L\\iota_L^*","context":515,"kind":"use","source":15},{"line":205,"start":8308,"end":8359,"expression":"P_L(z-H)^{-1}P_L=\\iota_L\\mathcal S(z)^{-1}\\iota_L^*","context":515,"kind":"use","source":15},{"line":206,"start":8385,"end":8395,"expression":"\\mathcal W","context":515,"kind":"use","source":15},{"line":208,"start":8482,"end":8522,"expression":"\\iota_L=\\mathcal W^*(|0\\rangle\\otimes I)","context":515,"kind":"use","source":15},{"line":209,"start":8541,"end":8546,"expression":"\\HH_L","context":515,"kind":"use","source":15},{"line":213,"start":8723,"end":8732,"expression":"\\det(z-D)","context":516,"kind":"use","source":15},{"line":214,"start":8828,"end":8837,"expression":"\\sigma(D)","context":516,"kind":"use","source":15},{"line":215,"start":8895,"end":8898,"expression":"z-D","context":516,"kind":"use","source":15},{"line":222,"start":9165,"end":9196,"expression":"D=\\sum_\\lambda\\lambda E_\\lambda","context":517,"kind":"use","source":15},{"line":223,"start":9247,"end":9347,"expression":"\n \\Sigma(z)=\\sum_\\lambda\\frac{B E_\\lambda B^*}{z-\\lambda}.\n \\label{a-spectral-self-energy-residues}\n","context":517,"kind":"use","source":15},{"line":227,"start":9377,"end":9392,"expression":"B E_\\lambda B^*","context":517,"kind":"use","source":15},{"line":229,"start":9559,"end":9565,"expression":"\\Sigma","context":517,"kind":"use","source":15},{"line":230,"start":9581,"end":9584,"expression":"z-D","context":518,"kind":"use","source":15},{"line":234,"start":9721,"end":9727,"expression":"z=x+iy","context":519,"kind":"use","source":15},{"line":234,"start":9735,"end":9738,"expression":"y>0","context":520,"kind":"use","source":15},{"line":235,"start":9743,"end":9865,"expression":"\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n","context":521,"kind":"use","source":15},{"line":241,"start":9883,"end":9896,"expression":"\\mathcal S(z)","context":522,"kind":"use","source":15},{"line":242,"start":9917,"end":9953,"expression":"\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}","context":523,"kind":"use","source":15},{"line":243,"start":9966,"end":9982,"expression":"E\\notin\\sigma(D)","context":524,"kind":"use","source":15},{"line":244,"start":9987,"end":10017,"expression":"\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n","context":525,"kind":"use","source":15},{"line":247,"start":10024,"end":10029,"expression":"(u,v)","context":526,"kind":"use","source":15},{"line":247,"start":10068,"end":10069,"expression":"E","context":527,"kind":"use","source":15},{"line":248,"start":10093,"end":10164,"expression":"\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n","context":528,"kind":"use","source":15},{"line":254,"start":10221,"end":10228,"expression":"\\lambda","context":529,"kind":"use","source":15},{"line":255,"start":10232,"end":10288,"expression":"\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)","context":530,"kind":"use","source":15},{"line":257,"start":10381,"end":10408,"expression":"\\operatorname{Im}\\mathcal S","context":531,"kind":"use","source":15},{"line":258,"start":10420,"end":10421,"expression":"u","context":532,"kind":"use","source":15},{"line":259,"start":10449,"end":10531,"expression":"y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}","context":533,"kind":"use","source":15},{"line":261,"start":10540,"end":10553,"expression":"\\mathcal S(z)","context":534,"kind":"use","source":15},{"line":263,"start":10684,"end":10691,"expression":"\\Sigma'","context":535,"kind":"use","source":15},{"line":264,"start":10728,"end":10771,"expression":"\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle","context":536,"kind":"use","source":15},{"line":265,"start":10782,"end":10792,"expression":"\\norm{u}^2","context":537,"kind":"use","source":15},{"line":269,"start":10938,"end":10957,"expression":"[1-\\Sigma'(E)]^{-1}","context":538,"kind":"use","source":15},{"line":279,"start":11357,"end":11370,"expression":"i\\dot u=Au+Bv","context":539,"kind":"use","source":15},{"line":279,"start":11374,"end":11389,"expression":"i\\dot v=Dv+B^*u","context":539,"kind":"use","source":15},{"line":279,"start":11411,"end":11420,"expression":"(u_0,v_0)","context":539,"kind":"use","source":15},{"line":281,"start":11453,"end":11620,"expression":"\n i\\dot u(t)&=Au(t)+B e^{-itD}v_0\n              -i\\int_0^tK(t-s)u(s)\\,ds,\\nonumber\\\\\n K(\\tau)&=B e^{-i\\tau D}B^*,\\qquad \\tau\\geq0,\n \\label{a-spectral-memory-equation}\n","context":540,"kind":"use","source":15},{"line":288,"start":11677,"end":11772,"expression":"\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n","context":541,"kind":"use","source":15},{"line":292,"start":11792,"end":11812,"expression":"\\operatorname{Im}z>0","context":542,"kind":"use","source":15},{"line":293,"start":11881,"end":11972,"expression":"\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n","context":543,"kind":"use","source":15},{"line":299,"start":12030,"end":12041,"expression":"e^{itD}v(t)","context":544,"kind":"use","source":15},{"line":304,"start":12369,"end":12370,"expression":"D","context":545,"kind":"use","source":15},{"line":304,"start":12390,"end":12418,"expression":"\\norm{K(\\tau)}\\leq\\norm{B}^2","context":546,"kind":"use","source":15},{"line":305,"start":12433,"end":12462,"expression":"e^{-\\operatorname{Im}z\\,\\tau}","context":547,"kind":"use","source":15},{"line":306,"start":12536,"end":12537,"expression":"D","context":548,"kind":"use","source":15},{"line":307,"start":12541,"end":12596,"expression":"\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)","context":549,"kind":"use","source":15},{"line":311,"start":12728,"end":12735,"expression":"s\\leq t","context":550,"kind":"use","source":15},{"line":312,"start":12792,"end":12805,"expression":"B e^{-itD}v_0","context":550,"kind":"use","source":15},{"line":313,"start":12850,"end":12851,"expression":"K","context":550,"kind":"use","source":15},{"line":316,"start":13029,"end":13056,"expression":"\\norm{u(t)}^2+\\norm{v(t)}^2","context":550,"kind":"use","source":15},{"line":318,"start":13194,"end":13195,"expression":"H","context":551,"kind":"use","source":15},{"line":324,"start":13321,"end":13328,"expression":"T_{p,h}","context":552,"kind":"use","source":15},{"line":324,"start":13343,"end":13344,"expression":"p","context":552,"kind":"use","source":15},{"line":324,"start":13365,"end":13371,"expression":"h\\geq1","context":552,"kind":"use","source":15},{"line":325,"start":13393,"end":13394,"expression":"r","context":552,"kind":"use","source":15},{"line":325,"start":13416,"end":13431,"expression":"\\epsilon_r\\in\\R","context":552,"kind":"use","source":15},{"line":326,"start":13473,"end":13474,"expression":"r","context":552,"kind":"use","source":15},{"line":326,"start":13480,"end":13483,"expression":"r+1","context":552,"kind":"use","source":15},{"line":326,"start":13504,"end":13520,"expression":"\\gamma_r/\\sqrt p","context":552,"kind":"use","source":15},{"line":327,"start":13530,"end":13543,"expression":"\\gamma_r\\in\\R","context":552,"kind":"use","source":15},{"line":334,"start":13804,"end":13869,"expression":"\n |r\\rangle=p^{-r/2}\\sum_{|v|=r}|v\\rangle,\n \\qquad r=0,\\ldots,h.\n","context":553,"kind":"use","source":15},{"line":339,"start":13942,"end":13943,"expression":"r","context":553,"kind":"use","source":15},{"line":339,"start":13955,"end":13958,"expression":"p^r","context":553,"kind":"use","source":15},{"line":340,"start":14020,"end":14147,"expression":"\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n","context":553,"kind":"use","source":15},{"line":346,"start":14252,"end":14255,"expression":"r+1","context":554,"kind":"use","source":15},{"line":347,"start":14261,"end":14307,"expression":"p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}","context":555,"kind":"use","source":15},{"line":348,"start":14335,"end":14338,"expression":"r-1","context":556,"kind":"use","source":15},{"line":348,"start":14346,"end":14347,"expression":"p","context":557,"kind":"use","source":15},{"line":349,"start":14383,"end":14440,"expression":"p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}","context":558,"kind":"use","source":15},{"line":352,"start":14449,"end":14477,"expression":"R:\\C^{h+1}\\to\\ell^2(T_{p,h})","context":559,"kind":"use","source":15},{"line":352,"start":14485,"end":14488,"expression":"e_r","context":559,"kind":"use","source":15},{"line":352,"start":14494,"end":14503,"expression":"|r\\rangle","context":559,"kind":"use","source":15},{"line":353,"start":14556,"end":14563,"expression":"HR=RJ_h","context":559,"kind":"use","source":15},{"line":354,"start":14588,"end":14812,"expression":"\n J_h=\\begin{pmatrix}\n \\epsilon_0&\\gamma_0&&&\\\\\n \\gamma_0&\\epsilon_1&\\gamma_1&&\\\\\n &\\gamma_1&\\epsilon_2&\\ddots&\\\\\n &&\\ddots&\\ddots&\\gamma_{h-1}\\\\\n &&&\\gamma_{h-1}&\\epsilon_h\n \\end{pmatrix}.\n \\label{a-spectral-jacobi-matrix}\n","context":560,"kind":"use","source":15},{"line":364,"start":14836,"end":14837,"expression":"H","context":561,"kind":"use","source":15},{"line":364,"start":14886,"end":14892,"expression":"\\Ran R","context":561,"kind":"use","source":15},{"line":365,"start":14918,"end":14930,"expression":"w\\perp\\Ran R","context":561,"kind":"use","source":15},{"line":366,"start":14934,"end":14981,"expression":"\\langle Re_r,Hw\\rangle=\\langle HRe_r,w\\rangle=0","context":561,"kind":"use","source":15},{"line":373,"start":15244,"end":15245,"expression":"u","context":562,"kind":"use","source":15},{"line":373,"start":15257,"end":15260,"expression":"a<h","context":563,"kind":"use","source":15},{"line":374,"start":15303,"end":15332,"expression":"c=(c_0,\\ldots,c_{p-1})\\in\\C^p","context":563,"kind":"use","source":15},{"line":374,"start":15340,"end":15351,"expression":"\\sum_dc_d=0","context":563,"kind":"use","source":15},{"line":375,"start":15359,"end":15375,"expression":"s=0,\\ldots,h-a-1","context":563,"kind":"use","source":15},{"line":376,"start":15398,"end":15531,"expression":"\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n","context":563,"kind":"use","source":15},{"line":382,"start":15557,"end":15560,"expression":"u,c","context":563,"kind":"use","source":15},{"line":384,"start":15662,"end":15694,"expression":"\\epsilon_{a+1},\\ldots,\\epsilon_h","context":564,"kind":"use","source":15},{"line":385,"start":15711,"end":15743,"expression":"\\gamma_{a+1},\\ldots,\\gamma_{h-1}","context":565,"kind":"use","source":15},{"line":385,"start":15750,"end":15753,"expression":"s=0","context":566,"kind":"use","source":15},{"line":386,"start":15782,"end":15783,"expression":"u","context":567,"kind":"use","source":15},{"line":386,"start":15805,"end":15814,"expression":"\\sum_dc_d","context":568,"kind":"use","source":15},{"line":386,"start":15837,"end":15840,"expression":"s>0","context":569,"kind":"use","source":15},{"line":393,"start":16163,"end":16178,"expression":"\\ell^2(T_{p,h})","context":570,"kind":"use","source":15},{"line":394,"start":16208,"end":16209,"expression":"r","context":571,"kind":"use","source":15},{"line":394,"start":16223,"end":16228,"expression":"h-r+1","context":572,"kind":"use","source":15},{"line":395,"start":16261,"end":16273,"expression":"(p-1)p^{r-1}","context":573,"kind":"use","source":15},{"line":395,"start":16281,"end":16294,"expression":"1\\leq r\\leq h","context":574,"kind":"use","source":15},{"line":398,"start":16358,"end":16359,"expression":"s","context":575,"kind":"use","source":15},{"line":399,"start":16417,"end":16418,"expression":"u","context":576,"kind":"use","source":15},{"line":400,"start":16471,"end":16492,"expression":"\\langle c,c'\\rangle=0","context":577,"kind":"use","source":15},{"line":407,"start":16965,"end":16966,"expression":"b","context":578,"kind":"use","source":15},{"line":409,"start":17060,"end":17090,"expression":"\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1","context":579,"kind":"use","source":15},{"line":410,"start":17160,"end":17163,"expression":"p^b","context":580,"kind":"use","source":15},{"line":411,"start":17240,"end":17247,"expression":"p^{r-1}","context":581,"kind":"use","source":15},{"line":412,"start":17272,"end":17273,"expression":"u","context":582,"kind":"use","source":15},{"line":412,"start":17285,"end":17288,"expression":"r-1","context":583,"kind":"use","source":15},{"line":412,"start":17295,"end":17298,"expression":"p-1","context":584,"kind":"use","source":15},{"line":416,"start":17374,"end":17386,"expression":"(p-1)p^{r-1}","context":585,"kind":"use","source":15},{"line":416,"start":17422,"end":17423,"expression":"r","context":585,"kind":"use","source":15},{"line":421,"start":17638,"end":17697,"expression":"\n \\sum_{r=1}^h(p-1)p^{r-1}(h-r+1)\n =\\sum_{b=0}^hp^b-(h+1).\n","context":585,"kind":"use","source":15},{"line":430,"start":17908,"end":17915,"expression":"J_{r:h}","context":586,"kind":"use","source":15},{"line":431,"start":17923,"end":17964,"expression":"g_r(z)=\\langle r,(z-J_{r:h})^{-1}r\\rangle","context":586,"kind":"use","source":15},{"line":433,"start":18039,"end":18194,"expression":"\n g_h(z)&=(z-\\epsilon_h)^{-1},\\nonumber\\\\\n g_r(z)&=\\frac1{z-\\epsilon_r-\\gamma_r^2g_{r+1}(z)},\n          \\qquad r<h.\n \\label{a-spectral-continued-fraction}\n","context":587,"kind":"use","source":15},{"line":439,"start":18216,"end":18226,"expression":"\\Sigma_h=0","context":588,"kind":"use","source":15},{"line":440,"start":18233,"end":18284,"expression":"\\Sigma_r=\\gamma_r^2/(z-\\epsilon_{r+1}-\\Sigma_{r+1})","context":588,"kind":"use","source":15},{"line":441,"start":18308,"end":18438,"expression":"\n \\langle\\mathrm{root},(z-H)^{-1}\\mathrm{root}\\rangle\n   =g_0(z)=\\frac1{z-\\epsilon_0-\\Sigma_0(z)}.\n \\label{a-spectral-root-green}\n","context":588,"kind":"use","source":15},{"line":454,"start":18936,"end":18937,"expression":"p","context":589,"kind":"use","source":15},{"line":455,"start":19028,"end":19047,"expression":"\\epsilon_r,\\gamma_r","context":589,"kind":"use","source":15},{"line":456,"start":19099,"end":19100,"expression":"p","context":589,"kind":"use","source":15},{"line":457,"start":19181,"end":19184,"expression":"g_r","context":589,"kind":"use","source":15},{"line":458,"start":19211,"end":19223,"expression":"\\sqrt p\\,g_r","context":589,"kind":"use","source":15},{"line":458,"start":19254,"end":19255,"expression":"p","context":589,"kind":"use","source":15},{"line":465,"start":19469,"end":19499,"expression":"A=H_D(k)=k\\sigma_z+m_0\\sigma_x","context":590,"kind":"use","source":15},{"line":465,"start":19507,"end":19517,"expression":"k,m_0\\in\\R","context":591,"kind":"use","source":15},{"line":466,"start":19524,"end":19544,"expression":"D=J_{1:h}\\otimes I_2","context":592,"kind":"use","source":15},{"line":467,"start":19619,"end":19651,"expression":"B=\\gamma_0(\\langle1|\\otimes I_2)","context":593,"kind":"use","source":15},{"line":469,"start":19697,"end":19787,"expression":"\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n","context":594,"kind":"use","source":15},{"line":474,"start":19877,"end":19990,"expression":"\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n","context":595,"kind":"use","source":15},{"line":480,"start":20076,"end":20140,"expression":"\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n","context":596,"kind":"use","source":15},{"line":489,"start":20447,"end":20450,"expression":"u_*","context":597,"kind":"use","source":15},{"line":490,"start":20510,"end":20529,"expression":"[1-\\sigma'(E)]^{-1}","context":597,"kind":"use","source":15},{"line":490,"start":20536,"end":20543,"expression":"k=m_0=0","context":597,"kind":"use","source":15},{"line":499,"start":21029,"end":21036,"expression":"\\lambda","context":598,"kind":"use","source":15},{"line":500,"start":21068,"end":21085,"expression":"(\\lambda-D)v=B^*u","context":598,"kind":"use","source":15},{"line":501,"start":21158,"end":21170,"expression":"\\gamma_0\\ne0","context":599,"kind":"use","source":15},{"line":502,"start":21213,"end":21216,"expression":"u=0","context":600,"kind":"use","source":15},{"line":502,"start":21260,"end":21264,"expression":"Bv=0","context":601,"kind":"use","source":15},{"line":503,"start":21281,"end":21284,"expression":"v=0","context":602,"kind":"use","source":15},{"line":509,"start":21587,"end":21640,"expression":"K(\\tau)=\\gamma_0^2\\langle1,e^{-i\\tau J_{1:h}}1\\rangle","context":603,"kind":"use","source":15},{"line":510,"start":21656,"end":21659,"expression":"I_2","context":603,"kind":"use","source":15},{"line":522,"start":22211,"end":22221,"expression":"\\kappa\\ne0","context":604,"kind":"use","source":15},{"line":523,"start":22248,"end":22271,"expression":"C=\\{0,p,\\ldots,(M-1)p\\}","context":604,"kind":"use","source":15},{"line":524,"start":22312,"end":22315,"expression":"p-1","context":604,"kind":"use","source":15},{"line":525,"start":22375,"end":22384,"expression":"\\ell^2(C)","context":604,"kind":"use","source":15},{"line":525,"start":22392,"end":22396,"expression":"\\C^M","context":604,"kind":"use","source":15},{"line":525,"start":22407,"end":22436,"expression":"T|j\\rangle=|j+1\\pmod M\\rangle","context":604,"kind":"use","source":15},{"line":527,"start":22459,"end":22548,"expression":"\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n","context":604,"kind":"use","source":15},{"line":532,"start":22570,"end":22573,"expression":"T_n","context":605,"kind":"use","source":15},{"line":533,"start":22644,"end":22647,"expression":"U_n","context":606,"kind":"use","source":15},{"line":533,"start":22658,"end":22666,"expression":"U_{-1}=0","context":607,"kind":"use","source":15},{"line":533,"start":22670,"end":22675,"expression":"U_0=1","context":608,"kind":"use","source":15},{"line":534,"start":22682,"end":22712,"expression":"U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)","context":609,"kind":"use","source":15},{"line":539,"start":22824,"end":22853,"expression":"\\psi_{n+1}=x\\psi_n-\\psi_{n-1}","context":610,"kind":"use","source":15},{"line":539,"start":22881,"end":22884,"expression":"z-H","context":610,"kind":"use","source":15},{"line":540,"start":22902,"end":22919,"expression":"\\kappa K_{p-1}(x)","context":610,"kind":"use","source":15},{"line":540,"start":22929,"end":22935,"expression":"K_n(x)","context":610,"kind":"use","source":15},{"line":540,"start":22951,"end":22952,"expression":"x","context":610,"kind":"use","source":15},{"line":541,"start":22992,"end":22994,"expression":"-1","context":610,"kind":"use","source":15},{"line":545,"start":23098,"end":23104,"expression":"n\\geq1","context":611,"kind":"introduction-passage","source":15},{"line":545,"start":23108,"end":23128,"expression":"\\det K_n(x)=U_n(x/2)","context":612,"kind":"introduction-passage","source":15},{"line":546,"start":23165,"end":23288,"expression":"\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n","context":613,"kind":"introduction-passage","source":15},{"line":551,"start":23296,"end":23299,"expression":"n=1","context":613,"kind":"introduction-passage","source":15},{"line":551,"start":23342,"end":23345,"expression":"1/x","context":613,"kind":"introduction-passage","source":15},{"line":555,"start":23437,"end":23457,"expression":"d_n=xd_{n-1}-d_{n-2}","context":614,"kind":"introduction-passage","source":15},{"line":555,"start":23466,"end":23471,"expression":"d_0=1","context":615,"kind":"introduction-passage","source":15},{"line":555,"start":23478,"end":23483,"expression":"d_1=x","context":616,"kind":"introduction-passage","source":15},{"line":556,"start":23539,"end":23547,"expression":"U_n(x/2)","context":617,"kind":"introduction-passage","source":15},{"line":557,"start":23600,"end":23607,"expression":"K_{n-1}","context":618,"kind":"introduction-passage","source":15},{"line":559,"start":23726,"end":23729,"expression":"n-1","context":619,"kind":"use","source":15},{"line":559,"start":23740,"end":23742,"expression":"-1","context":620,"kind":"use","source":15},{"line":561,"start":23825,"end":23835,"expression":"(-1)^{n+1}","context":621,"kind":"use","source":15},{"line":562,"start":23869,"end":23891,"expression":"(-1)^{n+1}(-1)^{n-1}=1","context":622,"kind":"use","source":15},{"line":562,"start":23907,"end":23910,"expression":"d_n","context":623,"kind":"use","source":15},{"line":567,"start":24034,"end":24050,"expression":"U_{p-1}(x/2)\\ne0","context":624,"kind":"use","source":15},{"line":568,"start":24118,"end":24244,"expression":"\n \\mathcal S_p(z)=\\frac{\\kappa}{U_{p-1}(x/2)}\n          \\bigl[R_p(x)I_M-(T+T^*)\\bigr].\n \\label{a-spectral-decimation-formula}\n","context":624,"kind":"use","source":15},{"line":573,"start":24274,"end":24313,"expression":"\\mathcal S_p(z)=zI-H_{\\rm eff}^{(p)}(z)","context":624,"kind":"use","source":15},{"line":574,"start":24335,"end":24540,"expression":"\n H_{\\rm eff}^{(p)}(z)&=\\mu_p(z)I+\\kappa_p(z)(T+T^*),\\nonumber\\\\\n \\kappa_p(z)&=\\frac{\\kappa}{U_{p-1}(x/2)},\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n","context":625,"kind":"use","source":15},{"line":582,"start":24613,"end":24614,"expression":"M","context":626,"kind":"use","source":15},{"line":583,"start":24652,"end":24655,"expression":"p-1","context":627,"kind":"use","source":15},{"line":584,"start":24701,"end":24717,"expression":"U_{p-1}(x/2)\\ne0","context":628,"kind":"use","source":15},{"line":586,"start":24841,"end":24863,"expression":"\\kappa U_{p-2}/U_{p-1}","context":629,"kind":"use","source":15},{"line":589,"start":25015,"end":25029,"expression":"\\kappa/U_{p-1}","context":630,"kind":"use","source":15},{"line":590,"start":25039,"end":25160,"expression":"\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n","context":631,"kind":"use","source":15},{"line":596,"start":25177,"end":25214,"expression":"xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)","context":632,"kind":"use","source":15},{"line":597,"start":25260,"end":25276,"expression":"U_p-U_{p-2}=2T_p","context":633,"kind":"use","source":15},{"line":598,"start":25318,"end":25332,"expression":"x/2=\\cos\\theta","context":634,"kind":"use","source":15},{"line":599,"start":25341,"end":25405,"expression":"[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)","context":635,"kind":"use","source":15},{"line":600,"start":25451,"end":25452,"expression":"x","context":636,"kind":"use","source":15},{"line":607,"start":25653,"end":25684,"expression":"(z-\\mu_p(z))/\\kappa_p(z)=R_p(x)","context":637,"kind":"use","source":15},{"line":607,"start":25702,"end":25703,"expression":"z","context":637,"kind":"use","source":15},{"line":608,"start":25751,"end":25756,"expression":"\\mu_p","context":637,"kind":"use","source":15},{"line":608,"start":25763,"end":25771,"expression":"\\kappa_p","context":637,"kind":"use","source":15},{"line":610,"start":25882,"end":25885,"expression":"p=2","context":637,"kind":"use","source":15},{"line":610,"start":25889,"end":25901,"expression":"R_2(x)=x^2-2","context":637,"kind":"use","source":15},{"line":610,"start":25908,"end":25918,"expression":"U_1(x/2)=x","context":637,"kind":"use","source":15},{"line":611,"start":25944,"end":26061,"expression":"\n H_{\\rm eff}^{(2)}(z)\n =\\mu I+\\frac{\\kappa^2}{z-\\mu}(2I+T+T^*),\\qquad z\\ne\\mu.\n \\label{a-spectral-dyadic-effective}\n","context":637,"kind":"use","source":15},{"line":621,"start":26257,"end":26301,"expression":"U_{p-1}(\\cos\\theta)=\\sin(p\\theta)/\\sin\\theta","context":638,"kind":"use","source":15},{"line":623,"start":26367,"end":26481,"expression":"\n x_j=2\\cos(j\\pi/p),\\qquad\n z_j=\\mu+2\\kappa\\cos(j\\pi/p),\\quad 1\\leq j<p.\n \\label{a-spectral-dirichlet-resonances}\n","context":638,"kind":"use","source":15},{"line":628,"start":26555,"end":26556,"expression":"j","context":638,"kind":"use","source":15},{"line":629,"start":26597,"end":26611,"expression":"\\sin(j\\pi r/p)","context":638,"kind":"use","source":15},{"line":630,"start":26615,"end":26629,"expression":"r=1,\\ldots,p-1","context":638,"kind":"use","source":15},{"line":631,"start":26685,"end":26688,"expression":"z_j","context":639,"kind":"use","source":15},{"line":631,"start":26716,"end":26721,"expression":"r=0,p","context":640,"kind":"use","source":15},{"line":632,"start":26778,"end":26781,"expression":"p-1","context":641,"kind":"use","source":15},{"line":634,"start":26849,"end":26852,"expression":"R_p","context":642,"kind":"introduction-passage","source":15},{"line":634,"start":26875,"end":26878,"expression":"x_j","context":642,"kind":"introduction-passage","source":15},{"line":642,"start":27301,"end":27302,"expression":"Q","context":643,"kind":"use","source":15},{"line":643,"start":27363,"end":27466,"expression":"\n \\det(z-H)=\\det(z-QHQ)\\det\\mathcal S_p(z),\\qquad\n \\det(z-QHQ)=\\bigl[\\kappa^{p-1}U_{p-1}(x/2)\\bigr]^M.\n","context":643,"kind":"use","source":15},{"line":653,"start":27724,"end":27766,"expression":"A(x)=\\begin{pmatrix}x&-1\\\\1&0\\end{pmatrix}","context":644,"kind":"use","source":15},{"line":655,"start":27828,"end":27971,"expression":"\n A(x)^n=\\begin{pmatrix}\n U_n(x/2)&-U_{n-1}(x/2)\\\\\n U_{n-1}(x/2)&-U_{n-2}(x/2)\n \\end{pmatrix},\\quad n\\geq1.\n \\label{a-spectral-transfer-power}\n","context":645,"kind":"use","source":15},{"line":662,"start":27991,"end":27994,"expression":"n=1","context":646,"kind":"use","source":15},{"line":662,"start":28007,"end":28015,"expression":"U_{-1}=0","context":646,"kind":"use","source":15},{"line":662,"start":28037,"end":28041,"expression":"A(x)","context":646,"kind":"use","source":15},{"line":664,"start":28177,"end":28183,"expression":"R_n(x)","context":646,"kind":"use","source":15},{"line":665,"start":28222,"end":28223,"expression":"p","context":646,"kind":"use","source":15},{"line":666,"start":28278,"end":28284,"expression":"A(x)^p","context":646,"kind":"use","source":15},{"line":667,"start":28333,"end":28369,"expression":"\\chi_{j+1}-R_p(x)\\chi_j+\\chi_{j-1}=0","context":647,"kind":"use","source":15},{"line":668,"start":28376,"end":28392,"expression":"\\chi_j=\\psi_{pj}","context":648,"kind":"use","source":15},{"line":674,"start":28654,"end":28795,"expression":"\n x_{p,q}(\\theta)=2\\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right),\n \\qquad R_p(x_{p,q}(\\theta))=2\\cos\\theta.\n \\label{a-spectral-branch-relation}\n","context":649,"kind":"use","source":15},{"line":679,"start":28839,"end":28868,"expression":"T_p(\\cos\\alpha)=\\cos(p\\alpha)","context":649,"kind":"use","source":15},{"line":680,"start":28924,"end":28930,"expression":"(-2,2)","context":649,"kind":"use","source":15},{"line":681,"start":28954,"end":28961,"expression":"(0,\\pi)","context":650,"kind":"use","source":15},{"line":681,"start":28991,"end":28992,"expression":"p","context":651,"kind":"use","source":15},{"line":682,"start":29018,"end":29036,"expression":"R_p(x)=2\\cos\\theta","context":652,"kind":"use","source":15},{"line":685,"start":29240,"end":29243,"expression":"R_p","context":653,"kind":"use","source":15},{"line":685,"start":29249,"end":29255,"expression":"[-2,2]","context":654,"kind":"use","source":15},{"line":687,"start":29280,"end":29302,"expression":"x_k=2\\cos(2\\pi k/(pM))","context":655,"kind":"use","source":15},{"line":688,"start":29313,"end":29328,"expression":"2\\cos(2\\pi k/M)","context":655,"kind":"use","source":15},{"line":688,"start":29370,"end":29371,"expression":"p","context":655,"kind":"use","source":15},{"line":708,"start":30272,"end":30275,"expression":"m,n","context":656,"kind":"use","source":15},{"line":709,"start":30317,"end":30449,"expression":"\n R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\\nonumber\\\\\n U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n","context":657,"kind":"use","source":15},{"line":714,"start":30472,"end":30474,"expression":"\\C","context":658,"kind":"use","source":15},{"line":714,"start":30499,"end":30504,"expression":"N=mnL","context":659,"kind":"use","source":15},{"line":714,"start":30508,"end":30514,"expression":"L\\geq3","context":660,"kind":"use","source":15},{"line":715,"start":30529,"end":30530,"expression":"m","context":661,"kind":"use","source":15},{"line":715,"start":30539,"end":30540,"expression":"n","context":662,"kind":"use","source":15},{"line":715,"start":30570,"end":30571,"expression":"z","context":663,"kind":"use","source":15},{"line":716,"start":30593,"end":30595,"expression":"mn","context":664,"kind":"use","source":15},{"line":718,"start":30704,"end":30854,"expression":"\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n","context":665,"kind":"use","source":15},{"line":725,"start":30902,"end":30915,"expression":"x=2\\cos\\theta","context":666,"kind":"use","source":15},{"line":726,"start":30948,"end":30985,"expression":"R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)","context":667,"kind":"use","source":15},{"line":728,"start":31076,"end":31077,"expression":"x","context":668,"kind":"use","source":15},{"line":728,"start":31085,"end":31097,"expression":"t=\\cos\\theta","context":669,"kind":"use","source":15},{"line":730,"start":31143,"end":31265,"expression":"\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n","context":670,"kind":"use","source":15},{"line":736,"start":31331,"end":31332,"expression":"t","context":671,"kind":"use","source":15},{"line":738,"start":31400,"end":31429,"expression":"x_m=(z-\\mu_m)/\\kappa_m=R_m(x)","context":672,"kind":"use","source":15},{"line":739,"start":31453,"end":31585,"expression":"\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n","context":673,"kind":"use","source":15},{"line":747,"start":31599,"end":31634,"expression":"\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))","context":674,"kind":"use","source":15},{"line":753,"start":31977,"end":31978,"expression":"z","context":675,"kind":"use","source":15},{"line":761,"start":32381,"end":32382,"expression":"p","context":676,"kind":"use","source":15},{"line":761,"start":32399,"end":32400,"expression":"h","context":677,"kind":"use","source":15},{"line":762,"start":32460,"end":32461,"expression":"p","context":678,"kind":"use","source":15},{"line":762,"start":32467,"end":32470,"expression":"p^h","context":679,"kind":"use","source":15},{"line":770,"start":32809,"end":32824,"expression":"M_0\\in SL_2(\\C)","context":680,"kind":"use","source":15},{"line":770,"start":32831,"end":32853,"expression":"x=\\operatorname{tr}M_0","context":680,"kind":"use","source":15},{"line":772,"start":32923,"end":32937,"expression":"M_0^2-xM_0+I=0","context":680,"kind":"use","source":15},{"line":773,"start":32962,"end":33045,"expression":"\n M_0^n=U_{n-1}(x/2)M_0-U_{n-2}(x/2)I,\\qquad n\\geq1,\n \\label{a-spectral-sl2-power}\n","context":680,"kind":"use","source":15},{"line":777,"start":33066,"end":33074,"expression":"U_{-1}=0","context":680,"kind":"use","source":15},{"line":777,"start":33082,"end":33085,"expression":"n=1","context":680,"kind":"use","source":15},{"line":777,"start":33107,"end":33114,"expression":"M_0=M_0","context":680,"kind":"use","source":15},{"line":778,"start":33126,"end":33129,"expression":"n=2","context":680,"kind":"use","source":15},{"line":778,"start":33171,"end":33174,"expression":"M_0","context":680,"kind":"use","source":15},{"line":779,"start":33191,"end":33196,"expression":"M_0^2","context":680,"kind":"use","source":15},{"line":779,"start":33202,"end":33208,"expression":"xM_0-I","context":680,"kind":"use","source":15},{"line":780,"start":33274,"end":33358,"expression":"\n \\operatorname{tr}(M_0^n)=R_n(\\operatorname{tr}M_0).\n \\label{a-spectral-sl2-trace}\n","context":681,"kind":"use","source":15},{"line":785,"start":33411,"end":33433,"expression":"xU_{n-1}-2U_{n-2}=2T_n","context":682,"kind":"use","source":15},{"line":787,"start":33452,"end":33466,"expression":"w_{k+1}=M_0w_k","context":683,"kind":"use","source":15},{"line":788,"start":33506,"end":33554,"expression":"\n w_{nq+j}=M_0^j(M_0^n)^q w_0,\\qquad 0\\leq j<n.\n","context":683,"kind":"use","source":15},{"line":792,"start":33683,"end":33687,"expression":"\\pm2","context":683,"kind":"use","source":15},{"line":793,"start":33726,"end":33729,"expression":"M_0","context":683,"kind":"use","source":15},{"line":794,"start":33806,"end":33814,"expression":"M_0=A(x)","context":683,"kind":"use","source":15},{"line":800,"start":34019,"end":34025,"expression":"[-2,2]","context":684,"kind":"use","source":15},{"line":801,"start":34074,"end":34148,"expression":"\n d\\nu(x)=\\frac{dx}{\\pi\\sqrt{4-x^2}}.\n \\label{a-spectral-arcsine-measure}\n","context":684,"kind":"use","source":15},{"line":805,"start":34190,"end":34201,"expression":"2\\cos\\theta","context":684,"kind":"use","source":15},{"line":806,"start":34245,"end":34252,"expression":"[0,\\pi]","context":684,"kind":"use","source":15},{"line":807,"start":34359,"end":34360,"expression":"f","context":684,"kind":"use","source":15},{"line":808,"start":34365,"end":34475,"expression":"\n \\frac1N\\sum_{k=0}^{N-1}f(2\\cos(2\\pi k/N))\n \\longrightarrow\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos\\theta)\\,d\\theta.\n","context":684,"kind":"use","source":15},{"line":817,"start":34696,"end":34702,"expression":"p\\geq2","context":685,"kind":"use","source":15},{"line":817,"start":34706,"end":34720,"expression":"(R_p)_*\\nu=\\nu","context":685,"kind":"use","source":15},{"line":818,"start":34732,"end":34756,"expression":"\\mathcal K_pf=f\\circ R_p","context":685,"kind":"use","source":15},{"line":818,"start":34777,"end":34785,"expression":"L^2(\\nu)","context":685,"kind":"use","source":15},{"line":819,"start":34795,"end":34854,"expression":"\n \\phi_0=1,\\qquad \\phi_n(x)=\\sqrt2\\,T_n(x/2),\\quad n\\geq1,\n","context":685,"kind":"use","source":15},{"line":823,"start":34924,"end":35081,"expression":"\n \\mathcal K_p\\phi_n=\\phi_{pn},\\qquad\n \\mathcal K_p^*\\phi_m=\n \\begin{cases}\\phi_{m/p},&p\\mid m,\\\\0,&p\\nmid m.\\end{cases}\n \\label{a-spectral-koopman-routing}\n","context":686,"kind":"use","source":15},{"line":829,"start":35115,"end":35118,"expression":"m=0","context":687,"kind":"use","source":15},{"line":829,"start":35137,"end":35164,"expression":"\\mathcal K_p^*\\phi_0=\\phi_0","context":687,"kind":"use","source":15},{"line":832,"start":35225,"end":35226,"expression":"f","context":687,"kind":"use","source":15},{"line":833,"start":35281,"end":35433,"expression":"\n \\int f(R_p(x))\\,d\\nu(x)\n &=\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos(p\\theta))\\,d\\theta\\\\\n &=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n","context":688,"kind":"use","source":15},{"line":839,"start":35509,"end":35514,"expression":"|f|^2","context":689,"kind":"use","source":15},{"line":841,"start":35598,"end":35606,"expression":"L^2(\\nu)","context":690,"kind":"use","source":15},{"line":841,"start":35626,"end":35633,"expression":"[0,\\pi]","context":691,"kind":"use","source":15},{"line":841,"start":35646,"end":35652,"expression":"\\phi_n","context":692,"kind":"use","source":15},{"line":846,"start":35929,"end":35934,"expression":"[0,1]","context":693,"kind":"use","source":15},{"line":847,"start":36035,"end":36052,"expression":"|k/n-t|\\leq\\delta","context":694,"kind":"use","source":15},{"line":848,"start":36092,"end":36102,"expression":"n\\to\\infty","context":695,"kind":"use","source":15},{"line":848,"start":36106,"end":36116,"expression":"\\delta\\to0","context":696,"kind":"use","source":15},{"line":852,"start":36330,"end":36358,"expression":"\\mathcal K_p\\phi_n=\\phi_{pn}","context":697,"kind":"use","source":15},{"line":853,"start":36372,"end":36380,"expression":"n,m\\geq0","context":698,"kind":"use","source":15},{"line":854,"start":36384,"end":36471,"expression":"\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}","context":699,"kind":"use","source":15},{"line":859,"start":36529,"end":36541,"expression":"\\mathcal K_p","context":700,"kind":"use","source":15},{"line":860,"start":36613,"end":36619,"expression":"\\phi_1","context":700,"kind":"use","source":15},{"line":861,"start":36705,"end":36708,"expression":"\\nu","context":700,"kind":"use","source":15},{"line":862,"start":36780,"end":36783,"expression":"R_p","context":700,"kind":"use","source":15},{"line":868,"start":37084,"end":37162,"expression":"\n \\int_{-2}^2\\log|R_p'(x)|\\,d\\nu(x)=\\log p.\n \\label{a-spectral-log-expansion}\n","context":701,"kind":"use","source":15},{"line":872,"start":37189,"end":37202,"expression":"x=2\\cos\\theta","context":701,"kind":"use","source":15},{"line":873,"start":37206,"end":37239,"expression":"R_p'(x)=p\\sin(p\\theta)/\\sin\\theta","context":701,"kind":"use","source":15},{"line":875,"start":37328,"end":37349,"expression":"\\log|\\theta-\\theta_0|","context":701,"kind":"use","source":15},{"line":876,"start":37395,"end":37401,"expression":"\\log p","context":701,"kind":"use","source":15},{"line":877,"start":37444,"end":37463,"expression":"\\log|\\sin(p\\theta)|","context":702,"kind":"use","source":15},{"line":877,"start":37470,"end":37484,"expression":"\\log\\sin\\theta","context":703,"kind":"use","source":15},{"line":877,"start":37492,"end":37499,"expression":"[0,\\pi]","context":704,"kind":"use","source":15},{"line":878,"start":37520,"end":37529,"expression":"u=p\\theta","context":705,"kind":"use","source":15},{"line":878,"start":37536,"end":37539,"expression":"\\pi","context":706,"kind":"use","source":15},{"line":879,"start":37557,"end":37569,"expression":"\\log|\\sin u|","context":707,"kind":"use","source":15},{"line":883,"start":37742,"end":37745,"expression":"R_2","context":708,"kind":"use","source":15},{"line":883,"start":37749,"end":37752,"expression":"R_3","context":708,"kind":"use","source":15},{"line":883,"start":37760,"end":37763,"expression":"R_5","context":708,"kind":"use","source":15},{"line":899,"start":38415,"end":38421,"expression":"L\\geq2","context":709,"kind":"use","source":15},{"line":899,"start":38431,"end":38452,"expression":"\\omega_L=e^{2\\pi i/L}","context":710,"kind":"use","source":15},{"line":900,"start":38464,"end":38493,"expression":"F(z)=\\sum_{n=0}^\\infty a_nz^n","context":711,"kind":"use","source":15},{"line":900,"start":38512,"end":38520,"expression":"|z|<\\rho","context":712,"kind":"use","source":15},{"line":902,"start":38606,"end":38751,"expression":"\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n","context":713,"kind":"use","source":15},{"line":908,"start":38792,"end":38800,"expression":"|z|<\\rho","context":714,"kind":"use","source":15},{"line":909,"start":38852,"end":39059,"expression":"\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n","context":715,"kind":"use","source":15},{"line":917,"start":39128,"end":39134,"expression":"a_nz^n","context":716,"kind":"use","source":15},{"line":919,"start":39178,"end":39213,"expression":"L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}","context":717,"kind":"use","source":15},{"line":920,"start":39243,"end":39259,"expression":"n\\equiv j\\pmod L","context":718,"kind":"use","source":15},{"line":921,"start":39362,"end":39363,"expression":"F","context":719,"kind":"use","source":15},{"line":926,"start":39605,"end":39608,"expression":"e^z","context":720,"kind":"use","source":15},{"line":927,"start":39634,"end":39725,"expression":"\n E_{L,j}(z)=\\sum_{q=0}^\\infty\\frac{z^{Lq+j}}{(Lq+j)!}.\n \\label{a-spectral-factorial-lane}\n","context":720,"kind":"use","source":15},{"line":931,"start":39745,"end":39756,"expression":"\\alpha\\in\\C","context":720,"kind":"use","source":15},{"line":932,"start":39766,"end":39811,"expression":"(\\alpha)_q=\\alpha(\\alpha+1)\\cdots(\\alpha+q-1)","context":720,"kind":"use","source":15},{"line":933,"start":39818,"end":39830,"expression":"(\\alpha)_0=1","context":720,"kind":"use","source":15},{"line":933,"start":39868,"end":39869,"expression":"L","context":720,"kind":"use","source":15},{"line":934,"start":39898,"end":40064,"expression":"\n \\frac{(Lq+j)!}{j!}\n &=\\prod_{r=1}^{L}\\prod_{k=0}^{q-1}(j+r+Lk)\\nonumber\\\\\n &=L^{Lq}\\prod_{r=1}^{L}\\left(\\frac{j+r}{L}\\right)_q.\n \\label{a-spectral-factorial-split}\n","context":721,"kind":"use","source":15},{"line":940,"start":40117,"end":40122,"expression":"r=L-j","context":722,"kind":"use","source":15},{"line":941,"start":40147,"end":40226,"expression":"\n \\mathcal A_{L,j}\n =\\left\\{\\frac{j+r}{L}:1\\leq r\\leq L\\right\\}\\setminus\\{1\\}.\n","context":722,"kind":"use","source":15},{"line":946,"start":40338,"end":40346,"expression":"(1)_q=q!","context":722,"kind":"use","source":15},{"line":947,"start":40365,"end":40556,"expression":"\n E_{L,j}(z)=\\frac{z^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-\\\\\\mathcal A_{L,j}\\end{matrix};\n                              \\frac{z^L}{L^L}\\right).\n \\label{a-spectral-hypergeometric-lane}\n","context":723,"kind":"introduction-passage","source":15},{"line":953,"start":40577,"end":40603,"expression":"{}_0F_s(;b_1,\\ldots,b_s;w)","context":724,"kind":"introduction-passage","source":15},{"line":954,"start":40620,"end":40655,"expression":"\\sum_{q\\geq0}w^q/[q!\\prod_r(b_r)_q]","context":724,"kind":"introduction-passage","source":15},{"line":955,"start":40703,"end":40728,"expression":"|w|/[(q+1)\\prod_r|q+b_r|]","context":724,"kind":"introduction-passage","source":15},{"line":956,"start":40777,"end":40778,"expression":"w","context":724,"kind":"introduction-passage","source":15},{"line":958,"start":40900,"end":40928,"expression":"\\mathcal A_{3,1}=\\{2/3,4/3\\}","context":725,"kind":"introduction-passage","source":15},{"line":959,"start":40935,"end":40973,"expression":"E_{3,1}(z)=z\\,{}_0F_2(;2/3,4/3;z^3/27)","context":726,"kind":"introduction-passage","source":15},{"line":964,"start":41070,"end":41088,"expression":"E'_{L,j}=E_{L,j-1}","context":727,"kind":"use","source":15},{"line":964,"start":41095,"end":41101,"expression":"j\\geq1","context":728,"kind":"use","source":15},{"line":964,"start":41109,"end":41112,"expression":"j=0","context":729,"kind":"use","source":15},{"line":966,"start":41186,"end":41204,"expression":"E'_{L,0}=E_{L,L-1}","context":730,"kind":"use","source":15},{"line":967,"start":41212,"end":41262,"expression":"\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}","context":731,"kind":"use","source":15},{"line":968,"start":41297,"end":41313,"expression":"(C_Lv)_0=v_{L-1}","context":732,"kind":"use","source":15},{"line":969,"start":41320,"end":41336,"expression":"(C_Lv)_j=v_{j-1}","context":733,"kind":"use","source":15},{"line":969,"start":41343,"end":41349,"expression":"j\\geq1","context":734,"kind":"use","source":15},{"line":970,"start":41373,"end":41519,"expression":"\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n","context":735,"kind":"use","source":15},{"line":977,"start":41578,"end":41599,"expression":"C_L^ne_0=e_{n\\bmod L}","context":736,"kind":"use","source":15},{"line":978,"start":41610,"end":41611,"expression":"j","context":737,"kind":"use","source":15},{"line":980,"start":41731,"end":41861,"expression":"\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n","context":738,"kind":"use","source":15},{"line":990,"start":42160,"end":42163,"expression":"M_1","context":739,"kind":"use","source":15},{"line":991,"start":42194,"end":42220,"expression":"v_{Lq+j}=M_1^j(M_1^L)^qv_0","context":739,"kind":"use","source":15},{"line":1001,"start":42641,"end":42662,"expression":"A_0\\in\\mathcal B(\\HH)","context":740,"kind":"use","source":15},{"line":1001,"start":42669,"end":42675,"expression":"t\\in\\C","context":740,"kind":"use","source":15},{"line":1002,"start":42690,"end":42703,"expression":"E_{L,j}(tA_0)","context":740,"kind":"use","source":15},{"line":1003,"start":42770,"end":43136,"expression":"\n E_{L,j}(tA_0)\n &=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                           e^{\\omega_L^\\ell tA_0},\\nonumber\\\\\n &=\\frac{(tA_0)^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-\\\\\\mathcal A_{L,j}\\end{matrix};\n                      \\frac{(tA_0)^L}{L^L}\\right),\\nonumber\\\\\n \\sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.\n \\label{a-spectral-operator-lane-identities}\n","context":741,"kind":"use","source":15},{"line":1013,"start":43177,"end":43178,"expression":"Q","context":742,"kind":"use","source":15},{"line":1015,"start":43257,"end":43345,"expression":"\n e^a\\frac{a^{Q+1}}{(Q+1)!},\\qquad a=|t|\\norm{A_0}.\n \\label{a-spectral-lane-tail-bound}\n","context":743,"kind":"use","source":15},{"line":1021,"start":43442,"end":43445,"expression":"e^a","context":743,"kind":"use","source":15},{"line":1028,"start":43850,"end":43870,"expression":"(Q+1+k)!\\geq(Q+1)!k!","context":744,"kind":"use","source":15},{"line":1029,"start":43875,"end":43971,"expression":"\n \\sum_{n=Q+1}^\\infty\\frac{a^n}{n!}\n \\leq\\frac{a^{Q+1}}{(Q+1)!}\\sum_{k=0}^\\infty\\frac{a^k}{k!},\n","context":745,"kind":"use","source":15},{"line":1036,"start":44040,"end":44054,"expression":"A_0^L=\\alpha I","context":746,"kind":"use","source":15},{"line":1037,"start":44140,"end":44145,"expression":"A_0^j","context":746,"kind":"use","source":15},{"line":1038,"start":44191,"end":44213,"expression":"I,A_0,\\ldots,A_0^{L-1}","context":746,"kind":"use","source":15},{"line":1043,"start":44439,"end":44456,"expression":"\\mathfrak R_{L,j}","context":747,"kind":"use","source":15},{"line":1045,"start":44559,"end":44572,"expression":"E_{L,j}(tA_0)","context":747,"kind":"use","source":15},{"line":1046,"start":44598,"end":44601,"expression":"L=2","context":747,"kind":"use","source":15},{"line":1046,"start":44605,"end":44630,"expression":"E_{2,0}(tA_0)=\\cosh(tA_0)","context":747,"kind":"use","source":15},{"line":1047,"start":44637,"end":44662,"expression":"E_{2,1}(tA_0)=\\sinh(tA_0)","context":747,"kind":"use","source":15},{"line":1055,"start":44958,"end":45029,"expression":"\n B(y)=\\sum_{n=0}^\\infty\\frac{(-1)^ny^n}{(n!)^2}\n     ={}_0F_1(;1;-y).\n","context":748,"kind":"use","source":15},{"line":1059,"start":45037,"end":45043,"expression":"y\\geq0","context":748,"kind":"use","source":15},{"line":1059,"start":45055,"end":45068,"expression":"J_0(2\\sqrt y)","context":748,"kind":"use","source":15},{"line":1061,"start":45140,"end":45195,"expression":"B_{L,j}(y)=\\sum_{q\\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2","context":748,"kind":"use","source":15},{"line":1063,"start":45264,"end":45461,"expression":"\n B_{L,j}(y)\n =\\frac{(-1)^jy^j}{(j!)^2}\\,\n {}_0F_{2L-1}\\!\\left(\n \\begin{matrix}-\\\\1,\\mathcal A_{L,j},\\mathcal A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n","context":749,"kind":"use","source":15},{"line":1071,"start":45492,"end":45494,"expression":"q!","context":750,"kind":"use","source":15},{"line":1072,"start":45603,"end":45604,"expression":"1","context":751,"kind":"use","source":15},{"line":1074,"start":45727,"end":45742,"expression":"B=\\sum_jB_{L,j}","context":752,"kind":"use","source":15},{"line":1075,"start":45772,"end":45786,"expression":"\\Theta=y\\,d/dy","context":753,"kind":"use","source":15},{"line":1077,"start":45859,"end":45936,"expression":"\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n","context":754,"kind":"use","source":15},{"line":1081,"start":45982,"end":45983,"expression":"n","context":755,"kind":"use","source":15},{"line":1082,"start":46010,"end":46049,"expression":"n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2","context":756,"kind":"use","source":15},{"line":1083,"start":46091,"end":46099,"expression":"\\Theta^2","context":757,"kind":"use","source":15},{"line":1084,"start":46143,"end":46146,"expression":"j=0","context":758,"kind":"use","source":15},{"line":1101,"start":46784,"end":46792,"expression":"H_\\infty","context":759,"kind":"use","source":15},{"line":1101,"start":46833,"end":46834,"expression":"E","context":760,"kind":"use","source":15},{"line":1101,"start":46847,"end":46848,"expression":"t","context":761,"kind":"use","source":15},{"line":1103,"start":46879,"end":46911,"expression":"f_j(\\lambda)=E_{L,j}(-it\\lambda)","context":761,"kind":"use","source":15},{"line":1104,"start":46928,"end":46941,"expression":"f_j(H_\\infty)","context":761,"kind":"use","source":15},{"line":1105,"start":46970,"end":47140,"expression":"\n \\Dom(f_j(H_\\infty))\n =\\left\\{\\psi:\\int_{\\R}|f_j(\\lambda)|^2\n                       \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty\\right\\}.\n \\label{a-spectral-lane-domain}\n","context":761,"kind":"use","source":15},{"line":1111,"start":47176,"end":47209,"expression":"\\sum_jf_j(\\lambda)=e^{-it\\lambda}","context":761,"kind":"use","source":15},{"line":1112,"start":47233,"end":47370,"expression":"\n \\sum_jf_j(H_\\infty)\\psi=e^{-itH_\\infty}\\psi,\n \\qquad \\psi\\in\\bigcap_j\\Dom(f_j(H_\\infty)).\n \\label{a-spectral-unbounded-reconstruction}\n","context":762,"kind":"use","source":15},{"line":1121,"start":47573,"end":47576,"expression":"L=4","context":763,"kind":"use","source":15},{"line":1122,"start":47606,"end":47746,"expression":"\n f_0(\\lambda)=\\tfrac12\\bigl(\\cosh(t\\lambda)+\\cos(t\\lambda)\\bigr),\n \\quad\n f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n","context":763,"kind":"use","source":15},{"line":1127,"start":47753,"end":47761,"expression":"H_\\infty","context":763,"kind":"use","source":15},{"line":1127,"start":47785,"end":47792,"expression":"\\lambda","context":763,"kind":"use","source":15},{"line":1127,"start":47798,"end":47805,"expression":"L^2(\\R)","context":763,"kind":"use","source":15},{"line":1128,"start":47812,"end":47817,"expression":"t\\ne0","context":763,"kind":"use","source":15},{"line":1130,"start":47940,"end":47955,"expression":"e^{-itH_\\infty}","context":763,"kind":"use","source":15},{"line":1130,"start":47992,"end":47999,"expression":"L^2(\\R)","context":764,"kind":"use","source":15},{"line":1131,"start":48007,"end":48010,"expression":"L=2","context":765,"kind":"use","source":15},{"line":1132,"start":48059,"end":48074,"expression":"\\cos(tH_\\infty)","context":766,"kind":"use","source":15},{"line":1132,"start":48081,"end":48098,"expression":"-i\\sin(tH_\\infty)","context":767,"kind":"use","source":15},{"line":1136,"start":48256,"end":48384,"expression":"\n \\int_{\\R} e^{2|t||\\lambda|}\n              \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty,\n \\label{a-spectral-exponential-domain}\n","context":768,"kind":"use","source":15},{"line":1141,"start":48406,"end":48441,"expression":"|f_j(\\lambda)|\\leq e^{|t||\\lambda|}","context":768,"kind":"use","source":15},{"line":1142,"start":48462,"end":48475,"expression":"P_R=E([-R,R])","context":768,"kind":"use","source":15},{"line":1143,"start":48527,"end":48535,"expression":"\\Ran P_R","context":768,"kind":"use","source":15},{"line":1145,"start":48617,"end":48712,"expression":"\n \\sum_jf_j(H_\\infty)P_R\\psi=e^{-itH_\\infty}P_R\\psi\n      \\longrightarrow e^{-itH_\\infty}\\psi.\n","context":769,"kind":"use","source":15},{"line":1150,"start":48790,"end":48807,"expression":"f_j(H_\\infty)\\psi","context":770,"kind":"use","source":15},{"line":1157,"start":49135,"end":49136,"expression":"p","context":771,"kind":"use","source":15},{"line":1157,"start":49153,"end":49157,"expression":"L=mp","context":771,"kind":"use","source":15},{"line":1157,"start":49183,"end":49184,"expression":"m","context":771,"kind":"use","source":15},{"line":1158,"start":49197,"end":49206,"expression":"0\\leq j<p","context":771,"kind":"use","source":15},{"line":1158,"start":49220,"end":49226,"expression":"q\\geq0","context":771,"kind":"use","source":15},{"line":1159,"start":49245,"end":49351,"expression":"\n g_q=\\frac{j!(Lq)!}{(Lq+j)!}\n       =\\binom{Lq+j}{j}^{-1}\\in\\Z_p^\\times.\n \\label{a-spectral-unit-window}\n","context":771,"kind":"use","source":15},{"line":1165,"start":49374,"end":49411,"expression":"\\binom{Lq+j}{j}=\\prod_{r=1}^j(Lq+r)/r","context":771,"kind":"use","source":15},{"line":1166,"start":49451,"end":49452,"expression":"p","context":771,"kind":"use","source":15},{"line":1167,"start":49473,"end":49489,"expression":"Lq\\equiv0\\pmod p","context":771,"kind":"use","source":15},{"line":1167,"start":49496,"end":49505,"expression":"1\\leq r<p","context":771,"kind":"use","source":15},{"line":1167,"start":49513,"end":49516,"expression":"j=0","context":771,"kind":"use","source":15},{"line":1168,"start":49567,"end":49570,"expression":"p=2","context":772,"kind":"use","source":15},{"line":1169,"start":49648,"end":49649,"expression":"p","context":773,"kind":"use","source":15},{"line":1171,"start":49760,"end":49761,"expression":"p","context":774,"kind":"use","source":15},{"line":1174,"start":49906,"end":49912,"expression":"q=pr+a","context":775,"kind":"use","source":15},{"line":1174,"start":49916,"end":49925,"expression":"0\\leq a<p","context":775,"kind":"use","source":15},{"line":1175,"start":49969,"end":50071,"expression":"\n E_{L,j}(z)=\\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\\qquad 0\\leq j<L.\n \\label{a-spectral-analytic-refinement}\n","context":775,"kind":"use","source":15},{"line":1180,"start":50166,"end":50170,"expression":"j+aL","context":775,"kind":"use","source":15},{"line":1180,"start":50184,"end":50190,"expression":"[0,pL)","context":775,"kind":"use","source":15},{"line":1182,"start":50294,"end":50295,"expression":"p","context":776,"kind":"use","source":15},{"line":1187,"start":50463,"end":50493,"expression":"\\widehat N|n\\rangle=n|n\\rangle","context":777,"kind":"use","source":15},{"line":1188,"start":50527,"end":50551,"expression":"V=e^{2\\pi i\\widehat N/L}","context":777,"kind":"use","source":15},{"line":1189,"start":50596,"end":50760,"expression":"\n Q_j^{(L)}=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}V^\\ell\n          =\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n","context":777,"kind":"use","source":15},{"line":1195,"start":50870,"end":50879,"expression":"Q_j^{(L)}","context":778,"kind":"use","source":15},{"line":1196,"start":50917,"end":50938,"expression":"Q_jQ_k=\\delta_{jk}Q_j","context":779,"kind":"use","source":15},{"line":1196,"start":50946,"end":50957,"expression":"\\sum_jQ_j=I","context":780,"kind":"use","source":15},{"line":1198,"start":51063,"end":51064,"expression":"V","context":781,"kind":"use","source":15},{"line":1201,"start":51091,"end":51217,"expression":"\n |\\alpha\\rangle=e^{-|\\alpha|^2/2}\n        \\sum_{n=0}^\\infty\\frac{\\alpha^n}{\\sqrt{n!}}|n\\rangle,\n \\qquad \\mu=|\\alpha|^2\\geq0,\n","context":782,"kind":"use","source":15},{"line":1207,"start":51283,"end":51506,"expression":"\n p_{L,j}(\\mu)\n &=\\langle\\alpha,Q_j^{(L)}\\alpha\\rangle\n   =e^{-\\mu}E_{L,j}(\\mu)\\nonumber\\\\\n &=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                      e^{\\mu(\\omega_L^\\ell-1)}.\n \\label{a-spectral-coherent-packets}\n","context":783,"kind":"use","source":15},{"line":1216,"start":51592,"end":51604,"expression":"p_{L,0}(0)=1","context":784,"kind":"use","source":15},{"line":1216,"start":51611,"end":51623,"expression":"p_{L,j}(0)=0","context":784,"kind":"use","source":15},{"line":1216,"start":51630,"end":51633,"expression":"j>0","context":784,"kind":"use","source":15},{"line":1217,"start":51676,"end":51691,"expression":"\\sum_jp_{L,j}=1","context":784,"kind":"use","source":15},{"line":1218,"start":51735,"end":51743,"expression":"e^{-\\mu}","context":784,"kind":"use","source":15},{"line":1218,"start":51794,"end":51799,"expression":"z=\\mu","context":785,"kind":"use","source":15},{"line":1222,"start":51909,"end":51915,"expression":"L\\geq2","context":786,"kind":"use","source":15},{"line":1222,"start":51922,"end":51930,"expression":"\\mu\\geq0","context":786,"kind":"use","source":15},{"line":1223,"start":51949,"end":52145,"expression":"\n \\left|p_{L,j}(\\mu)-\\frac1L\\right|\n \\leq\\frac1L\\sum_{\\ell=1}^{L-1}\n       e^{-\\mu(1-\\cos(2\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n","context":786,"kind":"use","source":15},{"line":1233,"start":52237,"end":52243,"expression":"\\ell=0","context":787,"kind":"use","source":15},{"line":1233,"start":52254,"end":52257,"expression":"1/L","context":788,"kind":"use","source":15},{"line":1235,"start":52296,"end":52339,"expression":"e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L","context":789,"kind":"use","source":15},{"line":1237,"start":52440,"end":52452,"expression":"\\cos(2\\pi/L)","context":790,"kind":"use","source":15},{"line":1240,"start":52528,"end":52529,"expression":"L","context":791,"kind":"use","source":15},{"line":1241,"start":52550,"end":52560,"expression":"2\\pi^2/L^2","context":791,"kind":"use","source":15},{"line":1241,"start":52568,"end":52569,"expression":"L","context":791,"kind":"use","source":15},{"line":1242,"start":52642,"end":52643,"expression":"j","context":791,"kind":"use","source":15},{"line":1242,"start":52655,"end":52656,"expression":"L","context":791,"kind":"use","source":15},{"line":1246,"start":52809,"end":52984,"expression":"\n Q_r^{(L)}aQ_s^{(L)}=0\\quad\\text{unless }r\\equiv s-1\\pmod L,\n \\qquad\n Q_r^{(L)}a^\\dagger Q_s^{(L)}=0\\quad\\text{unless }r\\equiv s+1\\pmod L.\n \\label{a-spectral-ladder-routing}\n","context":792,"kind":"use","source":15},{"line":1253,"start":53055,"end":53086,"expression":"a|n\\rangle=\\sqrt n\\,|n-1\\rangle","context":793,"kind":"use","source":15},{"line":1254,"start":53093,"end":53135,"expression":"a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle","context":793,"kind":"use","source":15},{"line":1255,"start":53172,"end":53179,"expression":"\\Dom(a)","context":794,"kind":"use","source":15},{"line":1255,"start":53186,"end":53201,"expression":"\\Dom(a^\\dagger)","context":795,"kind":"use","source":15},{"line":1260,"start":53458,"end":53468,"expression":"\\widehat N","context":796,"kind":"use","source":15},{"line":24,"start":1889,"end":1932,"expression":"\n\\mathcal U(t)=e^{-itH_m^{\\mathrm{phys}}}.\n","context":797,"kind":"use","source":16},{"line":5,"start":112,"end":195,"expression":"\n\\KK=\\ell^2(\\Z^3;\\C^4)\n=\\left\\{a=(a_n):\\sum_{n\\in\\Z^3}\\norm{a_n}^2<\\infty\\right\\}.\n","context":798,"kind":"use","source":17},{"line":9,"start":220,"end":255,"expression":"\\inner{a}{b}=\\sum_n\\inner{a_n}{b_n}","context":798,"kind":"use","source":17},{"line":11,"start":511,"end":523,"expression":"S\\subset\\Z^3","context":799,"kind":"use","source":17},{"line":11,"start":549,"end":553,"expression":"P_Sa","context":799,"kind":"use","source":17},{"line":11,"start":569,"end":572,"expression":"a_n","context":799,"kind":"use","source":17},{"line":11,"start":580,"end":586,"expression":"n\\in S","context":799,"kind":"use","source":17},{"line":12,"start":625,"end":672,"expression":"\n\\norm{a-P_Sa}^2=\\sum_{n\\notin S}\\norm{a_n}^2.\n","context":799,"kind":"use","source":17},{"line":15,"start":807,"end":810,"expression":"\\KK","context":799,"kind":"use","source":17},{"line":18,"start":1021,"end":1022,"expression":"n","context":800,"kind":"use","source":17},{"line":18,"start":1026,"end":1041,"expression":"V_n:\\C^4\\to\\C^4","context":800,"kind":"use","source":17},{"line":18,"start":1094,"end":1107,"expression":"(Va)_n=V_na_n","context":800,"kind":"use","source":17},{"line":18,"start":1147,"end":1150,"expression":"\\KK","context":800,"kind":"use","source":17},{"line":18,"start":1162,"end":1165,"expression":"V_n","context":800,"kind":"use","source":17},{"line":18,"start":1192,"end":1195,"expression":"W_n","context":800,"kind":"use","source":17},{"line":18,"start":1211,"end":1212,"expression":"V","context":800,"kind":"use","source":17},{"line":18,"start":1219,"end":1220,"expression":"W","context":800,"kind":"use","source":17},{"line":18,"start":1243,"end":1246,"expression":"\\KK","context":800,"kind":"use","source":17},{"line":21,"start":1287,"end":1288,"expression":"n","context":800,"kind":"use","source":17},{"line":21,"start":1292,"end":1320,"expression":"\\norm{V_na_n}^2=\\norm{a_n}^2","context":800,"kind":"use","source":17},{"line":21,"start":1376,"end":1378,"expression":"Va","context":801,"kind":"use","source":17},{"line":21,"start":1384,"end":1387,"expression":"\\KK","context":802,"kind":"use","source":17},{"line":21,"start":1424,"end":1442,"expression":"\\norm{Va}=\\norm{a}","context":803,"kind":"use","source":17},{"line":26,"start":1623,"end":1676,"expression":"\nV_n(t)=e^{-itH_n},\\qquad V_{h,n}(t)=e^{-itH_{h,n}}.\n","context":804,"kind":"introduction-passage","source":17},{"line":30,"start":1769,"end":1824,"expression":"\n(U(t)a)_n=V_n(t)a_n,\\qquad (U_h(t)a)_n=V_{h,n}(t)a_n.\n","context":804,"kind":"introduction-passage","source":17},{"line":33,"start":1877,"end":1880,"expression":"\\KK","context":804,"kind":"introduction-passage","source":17},{"line":33,"start":1920,"end":1922,"expression":"-t","context":804,"kind":"introduction-passage","source":17},{"line":34,"start":1962,"end":2009,"expression":"\nU(s+t)=U(s)U(t),\\qquad U_h(s+t)=U_h(s)U_h(t).\n","context":804,"kind":"introduction-passage","source":17},{"line":41,"start":2359,"end":2404,"expression":"\nU(t)P_S=P_SU(t),\\qquad U_h(t)P_S=P_SU_h(t).\n","context":805,"kind":"use","source":17},{"line":52,"start":3330,"end":3331,"expression":"V","context":806,"kind":"use","source":17},{"line":52,"start":3338,"end":3339,"expression":"W","context":806,"kind":"use","source":17},{"line":52,"start":3390,"end":3391,"expression":"a","context":806,"kind":"use","source":17},{"line":52,"start":3416,"end":3417,"expression":"b","context":806,"kind":"use","source":17},{"line":53,"start":3422,"end":3465,"expression":"\n\\norm{Va-Wa}\\le 2\\norm{a-b}+\\norm{Vb-Wb}.\n","context":806,"kind":"use","source":17},{"line":58,"start":3517,"end":3538,"expression":"V(a-b)+(Vb-Wb)+W(b-a)","context":806,"kind":"use","source":17},{"line":58,"start":3638,"end":3648,"expression":"\\norm{a-b}","context":806,"kind":"use","source":17},{"line":61,"start":3706,"end":3714,"expression":"V=U_h(t)","context":807,"kind":"use","source":17},{"line":61,"start":3718,"end":3724,"expression":"W=U(t)","context":807,"kind":"use","source":17},{"line":61,"start":3732,"end":3738,"expression":"b=P_Sa","context":807,"kind":"use","source":17},{"line":61,"start":3761,"end":3762,"expression":"S","context":807,"kind":"use","source":17},{"line":62,"start":3806,"end":3850,"expression":"\n\\norm{U_h(t)P_Sa-U(t)P_Sa}\\le K_{S,a,T}h^2\n","context":807,"kind":"use","source":17},{"line":65,"start":3862,"end":3870,"expression":"|t|\\le T","context":807,"kind":"use","source":17},{"line":65,"start":3905,"end":3906,"expression":"h","context":807,"kind":"use","source":17},{"line":66,"start":4028,"end":4082,"expression":"\n\\norm{U_h(t)a-U(t)a}\\le\n2\\norm{a-P_Sa}+K_{S,a,T}h^2.\n","context":807,"kind":"use","source":17},{"line":72,"start":4146,"end":4153,"expression":"a\\in\\KK","context":808,"kind":"use","source":17},{"line":72,"start":4163,"end":4168,"expression":"T\\ge0","context":808,"kind":"use","source":17},{"line":72,"start":4182,"end":4195,"expression":"\\varepsilon>0","context":808,"kind":"use","source":17},{"line":72,"start":4212,"end":4220,"expression":"\\delta>0","context":808,"kind":"use","source":17},{"line":73,"start":4234,"end":4321,"expression":"\n0<h<\\delta,\\quad |t|\\le T\n\\quad\\Longrightarrow\\quad\n\\norm{U_h(t)a-U(t)a}<\\varepsilon.\n","context":808,"kind":"use","source":17},{"line":80,"start":4369,"end":4370,"expression":"S","context":808,"kind":"use","source":17},{"line":80,"start":4383,"end":4410,"expression":"\\norm{a-P_Sa}<\\varepsilon/4","context":808,"kind":"use","source":17},{"line":80,"start":4476,"end":4482,"expression":"\\delta","context":808,"kind":"use","source":17},{"line":80,"start":4526,"end":4539,"expression":"\\varepsilon/2","context":809,"kind":"use","source":17},{"line":80,"start":4635,"end":4646,"expression":"\\varepsilon","context":810,"kind":"use","source":17},{"line":80,"start":4664,"end":4665,"expression":"S","context":811,"kind":"use","source":17},{"line":80,"start":4691,"end":4692,"expression":"h","context":812,"kind":"use","source":17},{"line":84,"start":4996,"end":5002,"expression":"O(h^2)","context":813,"kind":"use","source":17},{"line":84,"start":5019,"end":5026,"expression":"a\\in\\KK","context":813,"kind":"use","source":17},{"line":89,"start":5451,"end":5477,"expression":"e_n(x)=e^{2\\pi i n\\cdot x}","context":814,"kind":"use","source":17},{"line":90,"start":5660,"end":5714,"expression":"\n\\F:\\HH\\longrightarrow\\KK,\n\\qquad \\HH=L^2(\\T^3;\\C^4).\n","context":814,"kind":"use","source":17},{"line":95,"start":5853,"end":5929,"expression":"\n\\norm{\\F f}_{\\KK}=\\norm{f}_{\\HH},\\qquad\n\\F^{-1}\\F f=f,\\qquad \\F\\F^{-1}a=a.\n","context":815,"kind":"use","source":17},{"line":99,"start":5977,"end":5980,"expression":"L^2","context":816,"kind":"use","source":17},{"line":102,"start":6176,"end":6201,"expression":"\n\\F^{-1}a=\\sum_n e_na_n.\n","context":817,"kind":"use","source":17},{"line":109,"start":6462,"end":6532,"expression":"\n\\mathcal U(t)=\\F^{-1}U(t)\\F,\n\\qquad\n\\mathcal U_h(t)=\\F^{-1}U_h(t)\\F.\n","context":818,"kind":"definition-passage","definition_title":"Physical propagators","source":17},{"line":115,"start":6580,"end":6583,"expression":"L^2","context":818,"kind":"use","source":17},{"line":115,"start":6647,"end":6658,"expression":"\\F\\F^{-1}=I","context":818,"kind":"use","source":17},{"line":116,"start":6735,"end":6816,"expression":"\n\\norm{\\mathcal U_h(t)f-\\mathcal U(t)f}_{\\HH}\n=\\norm{U_h(t)\\F f-U(t)\\F f}_{\\KK}.\n","context":818,"kind":"use","source":17},{"line":120,"start":6886,"end":6893,"expression":"f\\in\\HH","context":819,"kind":"use","source":17},{"line":130,"start":7629,"end":7632,"expression":"|n|","context":820,"kind":"use","source":17},{"line":130,"start":7663,"end":7666,"expression":"a_n","context":820,"kind":"use","source":17},{"line":130,"start":7725,"end":7728,"expression":"H_n","context":820,"kind":"use","source":17},{"line":133,"start":7838,"end":7945,"expression":"\n\\Dom(\\widehat H_m)\n=\\left\\{a\\in\\KK:\\sum_n\\norm{H_na_n}^2<\\infty\\right\\},\n\\qquad\n(\\widehat H_ma)_n=H_na_n.\n","context":821,"kind":"use","source":17},{"line":139,"start":8151,"end":8189,"expression":"\\norm{x+y}^2\\le2\\norm{x}^2+2\\norm{y}^2","context":821,"kind":"use","source":17},{"line":139,"start":8214,"end":8217,"expression":"H_n","context":821,"kind":"use","source":17},{"line":141,"start":8456,"end":8474,"expression":"\\Dom(\\widehat H_m)","context":822,"kind":"use","source":17},{"line":141,"start":8480,"end":8483,"expression":"\\KK","context":822,"kind":"use","source":17},{"line":147,"start":8934,"end":8937,"expression":"a,b","context":823,"kind":"use","source":17},{"line":148,"start":8982,"end":9088,"expression":"\n\\inner{\\widehat H_ma}{b}\n=\\sum_n\\inner{H_na_n}{b_n}\n=\\sum_n\\inner{a_n}{H_nb_n}\n=\\inner{a}{\\widehat H_mb}\n","context":823,"kind":"use","source":17},{"line":154,"start":9163,"end":9198,"expression":"\\widehat H_m\\subseteq\\widehat H_m^*","context":823,"kind":"use","source":17},{"line":159,"start":9514,"end":9526,"expression":"\\widehat H_m","context":824,"kind":"use","source":17},{"line":162,"start":9589,"end":9613,"expression":"b\\in\\Dom(\\widehat H_m^*)","context":825,"kind":"use","source":17},{"line":162,"start":9626,"end":9643,"expression":"c=\\widehat H_m^*b","context":826,"kind":"use","source":17},{"line":163,"start":9673,"end":9744,"expression":"\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n","context":826,"kind":"use","source":17},{"line":167,"start":9759,"end":9760,"expression":"n","context":826,"kind":"use","source":17},{"line":167,"start":9774,"end":9775,"expression":"a","context":826,"kind":"use","source":17},{"line":167,"start":9814,"end":9819,"expression":"a_n=v","context":826,"kind":"use","source":17},{"line":167,"start":9829,"end":9837,"expression":"v\\in\\C^4","context":826,"kind":"use","source":17},{"line":168,"start":9928,"end":9963,"expression":"\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n","context":827,"kind":"use","source":17},{"line":171,"start":9985,"end":10017,"expression":"\\inner{v}{H_nb_n}=\\inner{v}{c_n}","context":828,"kind":"use","source":17},{"line":171,"start":10030,"end":10031,"expression":"v","context":829,"kind":"use","source":17},{"line":171,"start":10042,"end":10054,"expression":"v=H_nb_n-c_n","context":830,"kind":"use","source":17},{"line":171,"start":10063,"end":10073,"expression":"c_n=H_nb_n","context":831,"kind":"use","source":17},{"line":171,"start":10085,"end":10092,"expression":"c\\in\\KK","context":832,"kind":"use","source":17},{"line":171,"start":10109,"end":10119,"expression":"(H_nb_n)_n","context":833,"kind":"use","source":17},{"line":171,"start":10147,"end":10169,"expression":"b\\in\\Dom(\\widehat H_m)","context":834,"kind":"use","source":17},{"line":171,"start":10226,"end":10261,"expression":"\\widehat H_m^*\\subseteq\\widehat H_m","context":835,"kind":"use","source":17},{"line":176,"start":10649,"end":10661,"expression":"a^{(r)}\\to a","context":836,"kind":"use","source":17},{"line":176,"start":10668,"end":10692,"expression":"\\widehat H_ma^{(r)}\\to b","context":836,"kind":"use","source":17},{"line":176,"start":10698,"end":10701,"expression":"\\KK","context":836,"kind":"use","source":17},{"line":176,"start":10772,"end":10775,"expression":"H_n","context":836,"kind":"use","source":17},{"line":177,"start":10794,"end":10826,"expression":"\nb_n=\\lim_rH_na_n^{(r)}=H_na_n.\n","context":836,"kind":"use","source":17},{"line":180,"start":10857,"end":10864,"expression":"b\\in\\KK","context":836,"kind":"use","source":17},{"line":180,"start":10871,"end":10872,"expression":"a","context":836,"kind":"use","source":17},{"line":180,"start":10909,"end":10924,"expression":"\\widehat H_ma=b","context":836,"kind":"use","source":17},{"line":183,"start":11067,"end":11068,"expression":"a","context":837,"kind":"use","source":17},{"line":184,"start":11123,"end":11159,"expression":"\n\\widehat H_mP_Sa=P_S\\widehat H_ma.\n","context":838,"kind":"use","source":17},{"line":187,"start":11168,"end":11169,"expression":"a","context":838,"kind":"use","source":17},{"line":187,"start":11176,"end":11189,"expression":"\\widehat H_ma","context":838,"kind":"use","source":17},{"line":187,"start":11219,"end":11232,"expression":"\\varepsilon>0","context":838,"kind":"use","source":17},{"line":187,"start":11342,"end":11343,"expression":"S","context":838,"kind":"use","source":17},{"line":188,"start":11357,"end":11443,"expression":"\n\\norm{a-P_Sa}<\\varepsilon,\n\\qquad\n\\norm{\\widehat H_ma-\\widehat H_mP_Sa}<\\varepsilon.\n","context":838,"kind":"use","source":17},{"line":194,"start":11575,"end":11640,"expression":"\n\\norm{a}_{\\mathrm{graph}}^2\n=\\norm{a}^2+\\norm{\\widehat H_ma}^2.\n","context":839,"kind":"use","source":17},{"line":205,"start":12455,"end":12464,"expression":"H_n^*=H_n","context":840,"kind":"use","source":17},{"line":205,"start":12471,"end":12494,"expression":"H_n^2=(m^2+|p(n)|^2)I_4","context":840,"kind":"use","source":17},{"line":206,"start":12499,"end":12563,"expression":"\n\\norm{H_nv}^2\n=\\inner{v}{H_n^*H_nv}\n=(m^2+|p(n)|^2)\\norm{v}^2.\n","context":840,"kind":"use","source":17},{"line":212,"start":12627,"end":12722,"expression":"\na\\in\\Dom(\\widehat H_m)\n\\quad\\Longleftrightarrow\\quad\n\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2<\\infty.\n","context":840,"kind":"use","source":17},{"line":217,"start":12768,"end":12775,"expression":"a\\in\\KK","context":841,"kind":"use","source":17},{"line":217,"start":12832,"end":12858,"expression":"\\sum_n|p(n)|^2\\norm{a_n}^2","context":842,"kind":"use","source":17},{"line":220,"start":12933,"end":12934,"expression":"m","context":843,"kind":"use","source":17},{"line":221,"start":12939,"end":13023,"expression":"\n\\Dom(\\widehat H_m)\n=\\left\\{a\\in\\KK:\n\\sum_n(1+|p(n)|^2)\\norm{a_n}^2<\\infty\\right\\}.\n","context":843,"kind":"use","source":17},{"line":226,"start":13040,"end":13085,"expression":"\\Dom(\\widehat H_{m_1})=\\Dom(\\widehat H_{m_2})","context":843,"kind":"use","source":17},{"line":229,"start":13220,"end":13241,"expression":"m^2\\sum_n\\norm{a_n}^2","context":843,"kind":"use","source":17},{"line":233,"start":13576,"end":13579,"expression":"m=0","context":844,"kind":"use","source":17},{"line":233,"start":13801,"end":13804,"expression":"H^1","context":844,"kind":"use","source":17},{"line":236,"start":14046,"end":14055,"expression":"n=(k,0,0)","context":845,"kind":"use","source":17},{"line":236,"start":14059,"end":14064,"expression":"k\\ge1","context":845,"kind":"use","source":17},{"line":236,"start":14076,"end":14087,"expression":"a_n=k^{-s}v","context":845,"kind":"use","source":17},{"line":236,"start":14114,"end":14115,"expression":"v","context":845,"kind":"use","source":17},{"line":237,"start":14162,"end":14203,"expression":"\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n","context":845,"kind":"use","source":17},{"line":240,"start":14236,"end":14241,"expression":"s>1/2","context":845,"kind":"use","source":17},{"line":241,"start":14280,"end":14309,"expression":"\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n","context":845,"kind":"use","source":17},{"line":244,"start":14342,"end":14347,"expression":"s>3/2","context":846,"kind":"use","source":17},{"line":244,"start":14367,"end":14370,"expression":"s=1","context":847,"kind":"use","source":17},{"line":244,"start":14382,"end":14385,"expression":"L^2","context":848,"kind":"use","source":17},{"line":244,"start":14432,"end":14435,"expression":"s=2","context":849,"kind":"use","source":17},{"line":249,"start":14860,"end":14882,"expression":"a\\in\\Dom(\\widehat H_m)","context":850,"kind":"use","source":17},{"line":250,"start":14925,"end":14984,"expression":"\n\\norm{\\widehat H_ma}^2\n=\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2.\n","context":850,"kind":"use","source":17},{"line":254,"start":15105,"end":15129,"expression":"\\inner{a}{\\widehat H_ma}","context":850,"kind":"use","source":17},{"line":257,"start":15329,"end":15395,"expression":"\n\\norm{a}_{\\mathrm{graph}}^2\n=\\sum_n(1+m^2+|p(n)|^2)\\norm{a_n}^2.\n","context":851,"kind":"use","source":17},{"line":261,"start":15501,"end":15516,"expression":"w(n)=1+|p(n)|^2","context":851,"kind":"use","source":17},{"line":262,"start":15529,"end":15567,"expression":"\nw(n)\\le1+m^2+|p(n)|^2\\le(1+m^2)w(n).\n","context":851,"kind":"use","source":17},{"line":269,"start":15782,"end":15889,"expression":"\n\\Dom(H_m^{\\mathrm{phys}})=\\F^{-1}\\Dom(\\widehat H_m),\n\\qquad\nH_m^{\\mathrm{phys}}f=\\F^{-1}\\widehat H_m\\F f.\n","context":852,"kind":"use","source":17},{"line":274,"start":15929,"end":15930,"expression":"f","context":852,"kind":"use","source":17},{"line":275,"start":15996,"end":16042,"expression":"\n\\F(H_m^{\\mathrm{phys}}f)=\\widehat H_m(\\F f).\n","context":852,"kind":"use","source":17},{"line":280,"start":16104,"end":16123,"expression":"H_m^{\\mathrm{phys}}","context":853,"kind":"use","source":17},{"line":280,"start":16185,"end":16203,"expression":"\\HH=L^2(\\T^3;\\C^4)","context":853,"kind":"use","source":17},{"line":281,"start":16257,"end":16318,"expression":"\n\\sum_{n\\in\\Z^3}(1+|2\\pi n|^2)\\norm{\\widehat f(n)}^2<\\infty,\n","context":853,"kind":"use","source":17},{"line":285,"start":16352,"end":16436,"expression":"\n\\norm{H_m^{\\mathrm{phys}}f}_{\\HH}^2\n=\\sum_n(m^2+|2\\pi n|^2)\\norm{\\widehat f(n)}^2.\n","context":853,"kind":"use","source":17},{"line":291,"start":16543,"end":16547,"expression":"\\F f","context":854,"kind":"use","source":17},{"line":293,"start":16781,"end":16782,"expression":"g","context":855,"kind":"use","source":17},{"line":293,"start":16813,"end":16814,"expression":"z","context":855,"kind":"use","source":17},{"line":293,"start":16854,"end":16855,"expression":"a","context":855,"kind":"use","source":17},{"line":293,"start":16897,"end":16905,"expression":"\\F^{-1}a","context":855,"kind":"use","source":17},{"line":294,"start":16933,"end":16979,"expression":"\n\\inner{\\widehat H_ma}{\\F g}=\\inner{a}{\\F z}.\n","context":855,"kind":"use","source":17},{"line":297,"start":16988,"end":16992,"expression":"\\F g","context":855,"kind":"use","source":17},{"line":297,"start":17056,"end":17060,"expression":"\\F z","context":855,"kind":"use","source":17},{"line":297,"start":17098,"end":17102,"expression":"\\F g","context":855,"kind":"use","source":17},{"line":297,"start":17151,"end":17152,"expression":"g","context":856,"kind":"use","source":17},{"line":301,"start":17453,"end":17486,"expression":"-i\\sum_j\\alpha_j\\partial_j+m\\beta","context":857,"kind":"use","source":17},{"line":301,"start":17646,"end":17649,"expression":"L^2","context":857,"kind":"use","source":17},{"line":5,"start":166,"end":175,"expression":"\\hbar=c=1","context":858,"kind":"use","source":18},{"line":7,"start":211,"end":249,"expression":"\n\\mathbb T^3=(\\mathbb R/\\mathbb Z)^3,\n","context":859,"kind":"use","source":18},{"line":11,"start":293,"end":295,"expression":"dx","context":860,"kind":"use","source":18},{"line":12,"start":304,"end":329,"expression":"\\int_{\\mathbb T^3}1\\,dx=1","context":860,"kind":"use","source":18},{"line":13,"start":407,"end":414,"expression":"[0,1)^3","context":860,"kind":"use","source":18},{"line":14,"start":460,"end":489,"expression":"n=(n_1,n_2,n_3)\\in\\mathbb Z^3","context":860,"kind":"use","source":18},{"line":15,"start":511,"end":522,"expression":"p(n)=2\\pi n","context":860,"kind":"use","source":18},{"line":17,"start":558,"end":605,"expression":"\ne_n(x)=e^{2\\pi i n\\cdot x}=e^{i p(n)\\cdot x}.\n","context":861,"kind":"use","source":18},{"line":21,"start":647,"end":654,"expression":"0,1,2,3","context":862,"kind":"use","source":18},{"line":22,"start":713,"end":730,"expression":"v,w\\in\\mathbb C^4","context":862,"kind":"use","source":18},{"line":24,"start":737,"end":830,"expression":"\n\\langle v,w\\rangle=\\sum_{r=0}^{3}\\overline{v_r}w_r,\n\\qquad\n\\|v\\|_2^2=\\sum_{r=0}^{3}|v_r|^2.\n","context":863,"kind":"use","source":18},{"line":35,"start":1110,"end":1184,"expression":"\n\\langle f,g\\rangle_{L^2}\n=\\int_{\\mathbb T^3}\\langle f(x),g(x)\\rangle\\,dx\n","context":864,"kind":"use","source":18},{"line":40,"start":1193,"end":1221,"expression":"L^2(\\mathbb T^3;\\mathbb C^4)","context":865,"kind":"use","source":18},{"line":41,"start":1244,"end":1263,"expression":"S\\subset\\mathbb Z^3","context":865,"kind":"use","source":18},{"line":41,"start":1294,"end":1311,"expression":"a_n\\in\\mathbb C^4","context":865,"kind":"use","source":18},{"line":44,"start":1320,"end":1352,"expression":"\nf_a(x)=\\sum_{n\\in S}e_n(x)a_n.\n","context":866,"kind":"use","source":18},{"line":50,"start":1425,"end":1526,"expression":"\n\\mathcal K_S=\\bigoplus_{n\\in S}\\mathbb C^4,\n\\qquad\n\\|a\\|_{\\mathcal K_S}^2=\\sum_{n\\in S}\\|a_n\\|_2^2.\n","context":867,"kind":"use","source":18},{"line":56,"start":1549,"end":1586,"expression":"\\mathcal F_S=\\{f_a:a\\in\\mathcal K_S\\}","context":868,"kind":"use","source":18},{"line":57,"start":1629,"end":1641,"expression":"\\mathcal K_S","context":868,"kind":"use","source":18},{"line":61,"start":1882,"end":1885,"expression":"0=0","context":868,"kind":"use","source":18},{"line":63,"start":1907,"end":1912,"expression":"m,h,t","context":869,"kind":"use","source":18},{"line":64,"start":1978,"end":1983,"expression":"m\\ge0","context":869,"kind":"use","source":18},{"line":65,"start":2082,"end":2083,"expression":"E","context":869,"kind":"use","source":18},{"line":66,"start":2135,"end":2138,"expression":"m<0","context":869,"kind":"use","source":18},{"line":67,"start":2186,"end":2193,"expression":"h\\to0^+","context":869,"kind":"use","source":18},{"line":68,"start":2255,"end":2256,"expression":"h","context":869,"kind":"use","source":18},{"line":68,"start":2299,"end":2300,"expression":"h","context":870,"kind":"use","source":18},{"line":76,"start":2506,"end":2674,"expression":"\n\\alpha_x=\n\\begin{pmatrix}\n0&0&0&1\\\\\n0&0&1&0\\\\\n0&1&0&0\\\\\n1&0&0&0\n\\end{pmatrix},\n\\qquad\n\\alpha_y=\n\\begin{pmatrix}\n0&0&0&-i\\\\\n0&0&i&0\\\\\n0&-i&0&0\\\\\ni&0&0&0\n\\end{pmatrix},\n","context":871,"kind":"use","source":18},{"line":94,"start":2680,"end":2847,"expression":"\n\\alpha_z=\n\\begin{pmatrix}\n0&0&1&0\\\\\n0&0&0&-1\\\\\n1&0&0&0\\\\\n0&-1&0&0\n\\end{pmatrix},\n\\qquad\n\\beta=\n\\begin{pmatrix}\n1&0&0&0\\\\\n0&1&0&0\\\\\n0&0&-1&0\\\\\n0&0&0&-1\n\\end{pmatrix}.\n","context":872,"kind":"use","source":18},{"line":112,"start":2868,"end":2901,"expression":"\\alpha_yv=(-iv_3,iv_2,-iv_1,iv_0)","context":873,"kind":"use","source":18},{"line":117,"start":3121,"end":3240,"expression":"\n\\alpha_j=\n\\begin{pmatrix}0&\\sigma_j\\\\\\sigma_j&0\\end{pmatrix},\n\\qquad\n\\beta=\\begin{pmatrix}I_2&0\\\\0&-I_2\\end{pmatrix},\n","context":874,"kind":"use","source":18},{"line":126,"start":3253,"end":3409,"expression":"\n\\sigma_x=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\n\\quad\n\\sigma_y=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix},\n\\quad\n\\sigma_z=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}.\n","context":875,"kind":"use","source":18},{"line":138,"start":3641,"end":3646,"expression":"\\beta","context":876,"kind":"use","source":18},{"line":140,"start":3653,"end":3724,"expression":"\n\\alpha_j^*=\\alpha_j,\\quad \\beta^*=\\beta,\\quad\n\\alpha_j^2=\\beta^2=I_4,\n","context":877,"kind":"use","source":18},{"line":145,"start":3730,"end":3819,"expression":"\n\\alpha_j\\alpha_k+\\alpha_k\\alpha_j=0\\quad(j\\ne k),\n\\qquad\n\\alpha_j\\beta+\\beta\\alpha_j=0.\n","context":878,"kind":"use","source":18},{"line":151,"start":3868,"end":3873,"expression":"\\beta","context":879,"kind":"use","source":18},{"line":152,"start":3882,"end":3891,"expression":"1,1,-1,-1","context":879,"kind":"use","source":18},{"line":153,"start":3951,"end":3959,"expression":"\\sigma_j","context":879,"kind":"use","source":18},{"line":153,"start":3992,"end":3995,"expression":"I_2","context":879,"kind":"use","source":18},{"line":156,"start":4047,"end":4144,"expression":"\n\\sigma_x\\sigma_y=i\\sigma_z,\\qquad\n\\sigma_y\\sigma_z=i\\sigma_x,\\qquad\n\\sigma_z\\sigma_x=i\\sigma_y,\n","context":880,"kind":"use","source":18},{"line":164,"start":4213,"end":4339,"expression":"\n\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\n\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}\n=\\begin{pmatrix}i&0\\\\0&-i\\end{pmatrix}=i\\sigma_z.\n","context":881,"kind":"use","source":18},{"line":173,"start":4464,"end":4635,"expression":"\n\\alpha_j^*=\n\\begin{pmatrix}0&\\sigma_j^*\\\\\\sigma_j^*&0\\end{pmatrix}\n=\\alpha_j,\\qquad\n\\alpha_j\\alpha_k=\n\\begin{pmatrix}\\sigma_j\\sigma_k&0\\\\0&\\sigma_j\\sigma_k\\end{pmatrix}.\n","context":882,"kind":"use","source":18},{"line":181,"start":4646,"end":4660,"expression":"\\alpha_j^2=I_4","context":883,"kind":"use","source":18},{"line":183,"start":4715,"end":4859,"expression":"\n\\alpha_j\\beta=\n\\begin{pmatrix}0&-\\sigma_j\\\\\\sigma_j&0\\end{pmatrix},\n\\qquad\n\\beta\\alpha_j=\n\\begin{pmatrix}0&\\sigma_j\\\\-\\sigma_j&0\\end{pmatrix},\n","context":884,"kind":"use","source":18},{"line":192,"start":4944,"end":4951,"expression":"\\square","context":885,"kind":"use","source":18},{"line":198,"start":5119,"end":5134,"expression":"p\\in\\mathbb R^3","context":886,"kind":"use","source":18},{"line":200,"start":5148,"end":5200,"expression":"\nH(m,p)=m\\beta+p_x\\alpha_x+p_y\\alpha_y+p_z\\alpha_z.\n","context":887,"kind":"use","source":18},{"line":206,"start":5227,"end":5345,"expression":"\nH(m,p)=\n\\begin{pmatrix}\nm&0&p_z&p_x-ip_y\\\\\n0&m&p_x+ip_y&-p_z\\\\\np_z&p_x-ip_y&-m&0\\\\\np_x+ip_y&-p_z&0&-m\n\\end{pmatrix}.\n","context":888,"kind":"use","source":18},{"line":216,"start":5359,"end":5360,"expression":"n","context":889,"kind":"use","source":18},{"line":216,"start":5395,"end":5408,"expression":"H_n=H(m,p(n))","context":889,"kind":"use","source":18},{"line":219,"start":5463,"end":5506,"expression":"\n-i\\sum_{j=x,y,z}\\alpha_j\\partial_j+m\\beta\n","context":890,"kind":"use","source":18},{"line":223,"start":5540,"end":5564,"expression":"\\partial_je_n=ip_j(n)e_n","context":891,"kind":"use","source":18},{"line":224,"start":5600,"end":5606,"expression":"e_na_n","context":891,"kind":"use","source":18},{"line":224,"start":5617,"end":5626,"expression":"e_nH_na_n","context":891,"kind":"use","source":18},{"line":225,"start":5649,"end":5650,"expression":"h","context":891,"kind":"use","source":18},{"line":227,"start":5661,"end":5731,"expression":"\nq_j(h,p)=\n\\begin{cases}\n\\sin(hp_j)/h,&h\\ne0,\\\\\np_j,&h=0.\n\\end{cases}\n","context":892,"kind":"use","source":18},{"line":235,"start":5743,"end":5759,"expression":"q_h(n)=q(h,p(n))","context":893,"kind":"use","source":18},{"line":235,"start":5768,"end":5787,"expression":"H_{h,n}=H(m,q_h(n))","context":893,"kind":"use","source":18},{"line":236,"start":5806,"end":5809,"expression":"q_j","context":893,"kind":"use","source":18},{"line":236,"start":5825,"end":5826,"expression":"h","context":893,"kind":"use","source":18},{"line":237,"start":5870,"end":5882,"expression":"\\sin z/z\\to1","context":893,"kind":"use","source":18},{"line":237,"start":5892,"end":5897,"expression":"h\\ne0","context":893,"kind":"use","source":18},{"line":240,"start":5971,"end":6018,"expression":"\n(D_{h,j}f)(x)=\\frac{f(x+he_j)-f(x-he_j)}{2h}.\n","context":894,"kind":"use","source":18},{"line":244,"start":6029,"end":6032,"expression":"e_j","context":895,"kind":"use","source":18},{"line":245,"start":6107,"end":6110,"expression":"e_n","context":895,"kind":"use","source":18},{"line":248,"start":6174,"end":6258,"expression":"\nD_{h,j}e_n\n=e_n\\frac{e^{ihp_j(n)}-e^{-ihp_j(n)}}{2h}\n=i\\frac{\\sin(hp_j(n))}{h}e_n.\n","context":896,"kind":"use","source":18},{"line":254,"start":6301,"end":6331,"expression":"-i\\sum_j\\alpha_jD_{h,j}+m\\beta","context":897,"kind":"use","source":18},{"line":255,"start":6357,"end":6364,"expression":"H_{h,n}","context":897,"kind":"use","source":18},{"line":255,"start":6392,"end":6403,"expression":"-i\\cdot i=1","context":897,"kind":"use","source":18},{"line":256,"start":6412,"end":6415,"expression":"h=0","context":897,"kind":"use","source":18},{"line":256,"start":6439,"end":6442,"expression":"H_n","context":897,"kind":"use","source":18},{"line":261,"start":6647,"end":6648,"expression":"m","context":898,"kind":"use","source":18},{"line":261,"start":6657,"end":6672,"expression":"p\\in\\mathbb R^3","context":898,"kind":"use","source":18},{"line":263,"start":6679,"end":6728,"expression":"\nH(m,p)^*=H(m,p),\\qquad\nH(m,p)^2=(m^2+|p|^2)I_4.\n","context":899,"kind":"use","source":18},{"line":268,"start":6749,"end":6760,"expression":"H_n,H_{h,n}","context":900,"kind":"use","source":18},{"line":268,"start":6792,"end":6795,"expression":"m,h","context":900,"kind":"use","source":18},{"line":270,"start":6806,"end":6853,"expression":"\nH_n^2=E_n^2I_4,\\quad E_n=\\sqrt{m^2+|p(n)|^2},\n","context":901,"kind":"use","source":18},{"line":274,"start":6859,"end":6920,"expression":"\nH_{h,n}^2=E_{h,n}^2I_4,\\quad\nE_{h,n}=\\sqrt{m^2+|q_h(n)|^2}.\n","context":902,"kind":"use","source":18},{"line":281,"start":7103,"end":7135,"expression":"m^2\\beta^2+\\sum_jp_j^2\\alpha_j^2","context":903,"kind":"use","source":18},{"line":282,"start":7151,"end":7184,"expression":"mp_j(\\beta\\alpha_j+\\alpha_j\\beta)","context":903,"kind":"use","source":18},{"line":283,"start":7204,"end":7245,"expression":"p_jp_k(\\alpha_j\\alpha_k+\\alpha_k\\alpha_j)","context":903,"kind":"use","source":18},{"line":283,"start":7254,"end":7257,"expression":"j<k","context":903,"kind":"use","source":18},{"line":284,"start":7315,"end":7318,"expression":"I_4","context":903,"kind":"use","source":18},{"line":285,"start":7331,"end":7351,"expression":"(m^2+\\sum_jp_j^2)I_4","context":904,"kind":"use","source":18},{"line":286,"start":7399,"end":7406,"expression":"\\square","context":905,"kind":"use","source":18},{"line":289,"start":7476,"end":7487,"expression":"\\mathbb C^4","context":906,"kind":"use","source":18},{"line":297,"start":7856,"end":7864,"expression":"H=H(m,p)","context":907,"kind":"use","source":18},{"line":297,"start":7873,"end":7891,"expression":"E=\\sqrt{m^2+|p|^2}","context":907,"kind":"use","source":18},{"line":299,"start":7926,"end":7983,"expression":"\nU_H(t)=e^{-itH}=\\sum_{k=0}^{\\infty}\\frac{(-itH)^k}{k!}.\n","context":908,"kind":"use","source":18},{"line":304,"start":8058,"end":8079,"expression":"\\sum_k(|t|\\|H\\|)^k/k!","context":909,"kind":"use","source":18},{"line":309,"start":8267,"end":8270,"expression":"E>0","context":910,"kind":"use","source":18},{"line":311,"start":8284,"end":8367,"expression":"\nP_+=\\frac12\\left(I_4+\\frac HE\\right),\\qquad\nP_-=\\frac12\\left(I_4-\\frac HE\\right).\n","context":911,"kind":"use","source":18},{"line":316,"start":8378,"end":8391,"expression":"P_\\pm^*=P_\\pm","context":912,"kind":"use","source":18},{"line":316,"start":8397,"end":8410,"expression":"P_\\pm^2=P_\\pm","context":912,"kind":"use","source":18},{"line":316,"start":8416,"end":8431,"expression":"P_+P_-=P_-P_+=0","context":912,"kind":"use","source":18},{"line":317,"start":8437,"end":8448,"expression":"P_++P_-=I_4","context":912,"kind":"use","source":18},{"line":317,"start":8458,"end":8475,"expression":"HP_\\pm=\\pm EP_\\pm","context":912,"kind":"use","source":18},{"line":320,"start":8517,"end":8583,"expression":"\nU_H(t)=e^{-itE}P_++e^{itE}P_-\n=\\cos(tE)I_4-i\\frac{\\sin(tE)}{E}H.\n","context":913,"kind":"use","source":18},{"line":325,"start":8592,"end":8595,"expression":"E=0","context":914,"kind":"use","source":18},{"line":325,"start":8606,"end":8609,"expression":"H=0","context":914,"kind":"use","source":18},{"line":325,"start":8618,"end":8628,"expression":"U_H(t)=I_4","context":914,"kind":"use","source":18},{"line":326,"start":8710,"end":8722,"expression":"\\sin(tE)/E=t","context":914,"kind":"use","source":18},{"line":327,"start":8730,"end":8733,"expression":"E=0","context":914,"kind":"use","source":18},{"line":329,"start":8758,"end":8761,"expression":"E>0","context":915,"kind":"use","source":18},{"line":329,"start":8788,"end":8793,"expression":"K=H/E","context":915,"kind":"use","source":18},{"line":330,"start":8808,"end":8815,"expression":"K^2=I_4","context":915,"kind":"use","source":18},{"line":332,"start":8827,"end":8885,"expression":"\nP_\\pm^2=\\tfrac14(I_4\\pm2K+K^2)=\\tfrac12(I_4\\pm K)=P_\\pm,\n","context":916,"kind":"use","source":18},{"line":336,"start":8895,"end":8921,"expression":"P_+P_-=\\tfrac14(I_4-K^2)=0","context":917,"kind":"use","source":18},{"line":337,"start":9010,"end":9013,"expression":"I_4","context":917,"kind":"use","source":18},{"line":338,"start":9024,"end":9058,"expression":"K(I_4\\pm K)=K\\pm I_4=\\pm(I_4\\pm K)","context":917,"kind":"use","source":18},{"line":338,"start":9072,"end":9089,"expression":"HP_\\pm=\\pm EP_\\pm","context":917,"kind":"use","source":18},{"line":340,"start":9148,"end":9168,"expression":"\\operatorname{tr}H=0","context":917,"kind":"use","source":18},{"line":340,"start":9177,"end":9201,"expression":"\\operatorname{tr}P_\\pm=2","context":917,"kind":"use","source":18},{"line":344,"start":9372,"end":9383,"expression":"I_4=P_++P_-","context":918,"kind":"use","source":18},{"line":345,"start":9442,"end":9462,"expression":"H^k=E^kP_++(-E)^kP_-","context":918,"kind":"use","source":18},{"line":345,"start":9477,"end":9482,"expression":"k\\ge0","context":918,"kind":"use","source":18},{"line":349,"start":9641,"end":9644,"expression":"E=0","context":919,"kind":"use","source":18},{"line":349,"start":9658,"end":9667,"expression":"m^2+|p|^2","context":919,"kind":"use","source":18},{"line":350,"start":9714,"end":9721,"expression":"m=0,p=0","context":919,"kind":"use","source":18},{"line":350,"start":9731,"end":9734,"expression":"H=0","context":919,"kind":"use","source":18},{"line":350,"start":9759,"end":9762,"expression":"I_4","context":919,"kind":"use","source":18},{"line":351,"start":9781,"end":9791,"expression":"\\sin(tE)/E","context":919,"kind":"use","source":18},{"line":351,"start":9805,"end":9806,"expression":"t","context":919,"kind":"use","source":18},{"line":352,"start":9852,"end":9855,"expression":"H=0","context":919,"kind":"use","source":18},{"line":353,"start":9914,"end":9921,"expression":"\\square","context":919,"kind":"use","source":18},{"line":355,"start":9958,"end":9965,"expression":"H_{h,n}","context":920,"kind":"use","source":18},{"line":355,"start":9983,"end":9990,"expression":"E_{h,n}","context":920,"kind":"use","source":18},{"line":356,"start":10038,"end":10047,"expression":"E_{h,n}=0","context":920,"kind":"use","source":18},{"line":357,"start":10078,"end":10081,"expression":"m=0","context":920,"kind":"use","source":18},{"line":363,"start":10365,"end":10368,"expression":"s,t","context":921,"kind":"use","source":18},{"line":365,"start":10375,"end":10417,"expression":"\nU_H(0)=I_4,\\qquad U_H(s+t)=U_H(s)U_H(t),\n","context":922,"kind":"use","source":18},{"line":369,"start":10423,"end":10496,"expression":"\nU_H(-t)=U_H(t)^{-1}=U_H(t)^*,\n\\qquad U_H(t)^*U_H(t)=U_H(t)U_H(t)^*=I_4.\n","context":923,"kind":"use","source":18},{"line":376,"start":10517,"end":10597,"expression":"\n\\langle U_H(t)v,U_H(t)w\\rangle=\\langle v,w\\rangle,\\qquad\n\\|U_H(t)v\\|_2=\\|v\\|_2\n","context":924,"kind":"use","source":18},{"line":382,"start":10683,"end":10688,"expression":"m,h,t","context":925,"kind":"use","source":18},{"line":384,"start":10714,"end":10717,"expression":"E>0","context":926,"kind":"use","source":18},{"line":385,"start":10762,"end":10767,"expression":"P_\\pm","context":926,"kind":"use","source":18},{"line":387,"start":10778,"end":10834,"expression":"\nU_H(s)U_H(t)\n=e^{-i(s+t)E}P_++e^{i(s+t)E}P_-=U_H(s+t).\n","context":927,"kind":"use","source":18},{"line":392,"start":10864,"end":10867,"expression":"I_4","context":928,"kind":"use","source":18},{"line":393,"start":10918,"end":10920,"expression":"tE","context":928,"kind":"use","source":18},{"line":393,"start":10930,"end":10969,"expression":"U_H(t)^*=e^{itE}P_++e^{-itE}P_-=U_H(-t)","context":928,"kind":"use","source":18},{"line":394,"start":10983,"end":10987,"expression":"s=-t","context":928,"kind":"use","source":18},{"line":394,"start":11028,"end":11031,"expression":"E=0","context":928,"kind":"use","source":18},{"line":395,"start":11066,"end":11076,"expression":"U_H(t)=I_4","context":928,"kind":"use","source":18},{"line":397,"start":11098,"end":11099,"expression":"A","context":929,"kind":"use","source":18},{"line":398,"start":11156,"end":11197,"expression":"\\langle Av,w\\rangle=\\langle v,A^*w\\rangle","context":929,"kind":"use","source":18},{"line":400,"start":11209,"end":11295,"expression":"\n\\langle U_H(t)v,U_H(t)w\\rangle\n=\\langle v,U_H(t)^*U_H(t)w\\rangle=\\langle v,w\\rangle.\n","context":930,"kind":"use","source":18},{"line":405,"start":11306,"end":11309,"expression":"w=v","context":931,"kind":"use","source":18},{"line":406,"start":11407,"end":11414,"expression":"\\square","context":931,"kind":"use","source":18},{"line":409,"start":11497,"end":11501,"expression":"-itH","context":932,"kind":"use","source":18},{"line":416,"start":11806,"end":11853,"expression":"J_n:\\mathbb C^4\\to L^2(\\mathbb T^3;\\mathbb C^4)","context":933,"kind":"use","source":18},{"line":417,"start":11861,"end":11870,"expression":"J_nv=e_nv","context":933,"kind":"use","source":18},{"line":421,"start":11986,"end":11989,"expression":"n,k","context":934,"kind":"use","source":18},{"line":421,"start":12006,"end":12009,"expression":"v,w","context":934,"kind":"use","source":18},{"line":423,"start":12016,"end":12110,"expression":"\n\\langle J_nv,J_kw\\rangle_{L^2}\n=\\begin{cases}\\langle v,w\\rangle,&n=k,\\\\0,&n\\ne k.\\end{cases}\n","context":935,"kind":"use","source":18},{"line":428,"start":12131,"end":12134,"expression":"J_n","context":936,"kind":"use","source":18},{"line":430,"start":12165,"end":12236,"expression":"\n\\left\\|\\sum_{n\\in S}J_na_n\\right\\|_{L^2}^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2.\n","context":937,"kind":"use","source":18},{"line":437,"start":12313,"end":12410,"expression":"\n\\langle J_nv,J_kw\\rangle_{L^2}\n=\\langle v,w\\rangle\\int_{\\mathbb T^3}e^{2\\pi i(k-n)\\cdot x}\\,dx.\n","context":938,"kind":"use","source":18},{"line":443,"start":12492,"end":12493,"expression":"r","context":939,"kind":"use","source":18},{"line":443,"start":12515,"end":12527,"expression":"e^{2\\pi irx}","context":939,"kind":"use","source":18},{"line":443,"start":12537,"end":12542,"expression":"[0,1)","context":939,"kind":"use","source":18},{"line":443,"start":12550,"end":12551,"expression":"1","context":939,"kind":"use","source":18},{"line":444,"start":12561,"end":12564,"expression":"r=0","context":939,"kind":"use","source":18},{"line":444,"start":12574,"end":12579,"expression":"r\\ne0","context":939,"kind":"use","source":18},{"line":444,"start":12591,"end":12618,"expression":"(e^{2\\pi ir}-1)/(2\\pi ir)=0","context":939,"kind":"use","source":18},{"line":445,"start":12649,"end":12650,"expression":"1","context":939,"kind":"use","source":18},{"line":445,"start":12668,"end":12671,"expression":"n=k","context":939,"kind":"use","source":18},{"line":446,"start":12713,"end":12720,"expression":"k=n,w=v","context":939,"kind":"use","source":18},{"line":448,"start":12885,"end":12896,"expression":"\\|a_n\\|_2^2","context":940,"kind":"use","source":18},{"line":449,"start":12902,"end":12909,"expression":"\\square","context":941,"kind":"use","source":18},{"line":453,"start":12943,"end":13038,"expression":"\nf(t,x)=\\sum_{n\\in S}e_n(x)U_{H_n}(t)a_n,\\qquad\nf_h(t,x)=\\sum_{n\\in S}e_n(x)U_{H_{h,n}}(t)a_n.\n","context":942,"kind":"use","source":18},{"line":458,"start":13059,"end":13071,"expression":"\\mathcal F_S","context":943,"kind":"use","source":18},{"line":458,"start":13093,"end":13110,"expression":"U_S(t),U_{h,S}(t)","context":943,"kind":"use","source":18},{"line":460,"start":13178,"end":13203,"expression":"U_{H_n}(t),U_{H_{h,n}}(t)","context":943,"kind":"use","source":18},{"line":464,"start":13315,"end":13320,"expression":"m,h,t","context":944,"kind":"use","source":18},{"line":464,"start":13333,"end":13334,"expression":"S","context":944,"kind":"use","source":18},{"line":464,"start":13363,"end":13364,"expression":"a","context":944,"kind":"use","source":18},{"line":466,"start":13371,"end":13451,"expression":"\n\\|f(t)\\|_{L^2}^2=\\|f_h(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2=\\|f_a\\|_{L^2}^2.\n","context":945,"kind":"use","source":18},{"line":471,"start":13493,"end":13496,"expression":"L^2","context":946,"kind":"use","source":18},{"line":472,"start":13538,"end":13550,"expression":"\\mathcal F_S","context":946,"kind":"use","source":18},{"line":478,"start":13686,"end":13773,"expression":"\n\\|f_h(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|U_{H_{h,n}}(t)a_n\\|_2^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2.\n","context":947,"kind":"use","source":18},{"line":484,"start":13841,"end":13844,"expression":"a,b","context":948,"kind":"use","source":18},{"line":487,"start":13882,"end":14030,"expression":"\n\\langle f_{h,a}(t),f_{h,b}(t)\\rangle_{L^2}\n=\\sum_{n\\in S}\n\\langle U_{H_{h,n}}(t)a_n,U_{H_{h,n}}(t)b_n\\rangle\n=\\sum_{n\\in S}\\langle a_n,b_n\\rangle.\n","context":949,"kind":"use","source":18},{"line":496,"start":14167,"end":14179,"expression":"\\mathcal F_S","context":950,"kind":"use","source":18},{"line":496,"start":14208,"end":14215,"expression":"\\square","context":950,"kind":"use","source":18},{"line":499,"start":14328,"end":14329,"expression":"m","context":951,"kind":"use","source":18},{"line":500,"start":14360,"end":14363,"expression":"E_n","context":951,"kind":"use","source":18},{"line":510,"start":14751,"end":14857,"expression":"\n\\Delta_{h,n}:=H_{h,n}-H_n\n=\\sum_{j=x,y,z}\\delta_j(h,n)\\alpha_j,\n\\qquad\n\\delta_j(h,n)=q_j(h,p(n))-p_j(n).\n","context":952,"kind":"use","source":18},{"line":519,"start":14871,"end":14929,"expression":"\nC(n)=\\frac16\\left(\\sum_{j=x,y,z}|p_j(n)|^6\\right)^{1/2}.\n","context":953,"kind":"use","source":18},{"line":528,"start":15179,"end":15180,"expression":"h","context":954,"kind":"use","source":18},{"line":528,"start":15200,"end":15201,"expression":"n","context":954,"kind":"use","source":18},{"line":530,"start":15208,"end":15252,"expression":"\n\\|\\Delta_{h,n}\\|=|\\delta(h,n)|\\le h^2C(n).\n","context":955,"kind":"use","source":18},{"line":536,"start":15290,"end":15388,"expression":"\n\\left\\|\\sum_{n\\in S}e_n\\Delta_{h,n}a_n\\right\\|_{L^2}^2\n=\\sum_{n\\in S}|\\delta(h,n)|^2\\|a_n\\|_2^2,\n","context":956,"kind":"use","source":18},{"line":543,"start":15399,"end":15511,"expression":"\n\\left\\|\\sum_{n\\in S}e_n\\Delta_{h,n}a_n\\right\\|_{L^2}\n\\le h^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}.\n","context":957,"kind":"use","source":18},{"line":549,"start":15593,"end":15594,"expression":"h","context":958,"kind":"use","source":18},{"line":550,"start":15642,"end":15656,"expression":"|\\sin u|\\le|u|","context":958,"kind":"use","source":18},{"line":550,"start":15668,"end":15693,"expression":"\\sin u=\\int_0^u\\cos v\\,dv","context":958,"kind":"use","source":18},{"line":551,"start":15702,"end":15714,"expression":"|\\cos v|\\le1","context":958,"kind":"use","source":18},{"line":553,"start":15731,"end":15766,"expression":"\n|1-\\cos u|=2\\sin^2(u/2)\\le u^2/2.\n","context":959,"kind":"use","source":18},{"line":557,"start":15776,"end":15781,"expression":"z\\ge0","context":960,"kind":"use","source":18},{"line":558,"start":15811,"end":15842,"expression":"\\sin z-z=\\int_0^z(\\cos u-1)\\,du","context":960,"kind":"use","source":18},{"line":558,"start":15857,"end":15876,"expression":"|\\sin z-z|\\le z^3/6","context":960,"kind":"use","source":18},{"line":559,"start":15913,"end":15933,"expression":"|\\sin z-z|\\le|z|^3/6","context":960,"kind":"use","source":18},{"line":560,"start":15951,"end":15952,"expression":"z","context":960,"kind":"use","source":18},{"line":560,"start":15962,"end":15967,"expression":"h\\ne0","context":960,"kind":"use","source":18},{"line":560,"start":15984,"end":15993,"expression":"z=hp_j(n)","context":960,"kind":"use","source":18},{"line":560,"start":16012,"end":16015,"expression":"|h|","context":960,"kind":"use","source":18},{"line":562,"start":16022,"end":16067,"expression":"\n|\\delta_j(h,n)|\\le \\frac{h^2|p_j(n)|^3}{6}.\n","context":961,"kind":"use","source":18},{"line":566,"start":16077,"end":16080,"expression":"h=0","context":962,"kind":"use","source":18},{"line":567,"start":16175,"end":16199,"expression":"|\\delta(h,n)|\\le h^2C(n)","context":962,"kind":"use","source":18},{"line":569,"start":16227,"end":16239,"expression":"\\Delta_{h,n}","context":963,"kind":"use","source":18},{"line":570,"start":16254,"end":16287,"expression":"\\Delta_{h,n}^2=|\\delta(h,n)|^2I_4","context":963,"kind":"use","source":18},{"line":572,"start":16308,"end":16404,"expression":"\n\\|\\Delta_{h,n}v\\|_2^2\n=\\langle v,\\Delta_{h,n}^*\\Delta_{h,n}v\\rangle\n=|\\delta(h,n)|^2\\|v\\|_2^2.\n","context":964,"kind":"use","source":18},{"line":578,"start":16447,"end":16460,"expression":"|\\delta(h,n)|","context":965,"kind":"use","source":18},{"line":581,"start":16675,"end":16682,"expression":"\\square","context":965,"kind":"use","source":18},{"line":583,"start":16738,"end":16744,"expression":"O(h^2)","context":966,"kind":"use","source":18},{"line":584,"start":16762,"end":16769,"expression":"h\\to0^+","context":966,"kind":"use","source":18},{"line":585,"start":16866,"end":16877,"expression":"r(h)=O(h^2)","context":966,"kind":"use","source":18},{"line":586,"start":16898,"end":16913,"expression":"M,\\varepsilon>0","context":966,"kind":"use","source":18},{"line":586,"start":16926,"end":16942,"expression":"\\|r(h)\\|\\le Mh^2","context":966,"kind":"use","source":18},{"line":587,"start":16951,"end":16966,"expression":"0<h<\\varepsilon","context":966,"kind":"use","source":18},{"line":591,"start":17129,"end":17140,"expression":"\\mathbb R^4","context":967,"kind":"use","source":18},{"line":592,"start":17157,"end":17167,"expression":"(m,q_h(n))","context":967,"kind":"use","source":18},{"line":592,"start":17176,"end":17184,"expression":"(m,p(n))","context":967,"kind":"use","source":18},{"line":594,"start":17197,"end":17240,"expression":"\n|E_{h,n}-E_n|\\le|q_h(n)-p(n)|\\le h^2C(n).\n","context":968,"kind":"use","source":18},{"line":598,"start":17256,"end":17278,"expression":"|\\sin(hp_j)/h|\\le|p_j|","context":969,"kind":"use","source":18},{"line":598,"start":17287,"end":17292,"expression":"h\\ne0","context":969,"kind":"use","source":18},{"line":599,"start":17342,"end":17361,"expression":"0\\le E_{h,n}\\le E_n","context":969,"kind":"use","source":18},{"line":602,"start":17409,"end":17459,"expression":"\nq_j(h,p(n))=p_j(n)-\\frac{h^2p_j(n)^3}{6}+O(h^4),\n","context":970,"kind":"use","source":18},{"line":606,"start":17465,"end":17518,"expression":"\nE_{h,n}^2=E_n^2-\\frac{h^2}{3}\\sum_jp_j(n)^4+O(h^4).\n","context":971,"kind":"use","source":18},{"line":612,"start":17657,"end":17669,"expression":"|hp_j|^5/120","context":972,"kind":"use","source":18},{"line":612,"start":17687,"end":17690,"expression":"|h|","context":972,"kind":"use","source":18},{"line":612,"start":17701,"end":17707,"expression":"O(h^4)","context":972,"kind":"use","source":18},{"line":614,"start":17797,"end":17802,"expression":"E_n>0","context":972,"kind":"use","source":18},{"line":614,"start":17839,"end":17844,"expression":"E_n^2","context":972,"kind":"use","source":18},{"line":616,"start":17854,"end":17906,"expression":"\nE_{h,n}=E_n-\\frac{h^2}{6E_n}\\sum_jp_j(n)^4+O(h^4).\n","context":973,"kind":"use","source":18},{"line":620,"start":17915,"end":17920,"expression":"E_n=0","context":974,"kind":"use","source":18},{"line":620,"start":17931,"end":17938,"expression":"m=0,n=0","context":974,"kind":"use","source":18},{"line":620,"start":17947,"end":17956,"expression":"E_{h,n}=0","context":974,"kind":"use","source":18},{"line":621,"start":17979,"end":17980,"expression":"h","context":974,"kind":"use","source":18},{"line":621,"start":18011,"end":18016,"expression":"1/E_n","context":974,"kind":"use","source":18},{"line":625,"start":18098,"end":18157,"expression":"\n|e^{\\pm itE_{h,n}}-e^{\\pm itE_n}|\n\\le |t|\\,|E_{h,n}-E_n|,\n","context":975,"kind":"use","source":18},{"line":630,"start":18196,"end":18216,"expression":"s\\mapsto e^{\\pm its}","context":976,"kind":"use","source":18},{"line":636,"start":18545,"end":18548,"expression":"m=1","context":977,"kind":"use","source":18},{"line":636,"start":18552,"end":18558,"expression":"h=1/12","context":977,"kind":"use","source":18},{"line":644,"start":18899,"end":18902,"expression":"m,t","context":978,"kind":"use","source":18},{"line":644,"start":18917,"end":18918,"expression":"n","context":978,"kind":"use","source":18},{"line":646,"start":18925,"end":18974,"expression":"\nU_{H_{h,n}}(t)-U_{H_n}(t)=O(h^2)\\quad(h\\to0^+).\n","context":979,"kind":"use","source":18},{"line":650,"start":18997,"end":18998,"expression":"S","context":980,"kind":"use","source":18},{"line":650,"start":19018,"end":19019,"expression":"a","context":980,"kind":"use","source":18},{"line":652,"start":19026,"end":19091,"expression":"\nf_h(t)-f(t)=O(h^2)\n\\quad\\hbox{in }L^2(\\mathbb T^3;\\mathbb C^4),\n","context":981,"kind":"use","source":18},{"line":657,"start":19115,"end":19140,"expression":"\\|f_h(t)-f(t)\\|_{L^2}\\to0","context":982,"kind":"use","source":18},{"line":660,"start":19178,"end":19257,"expression":"\n\\|f_h(t)-f(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\\|_2^2.\n","context":983,"kind":"use","source":18},{"line":666,"start":19373,"end":19376,"expression":"X,Y","context":984,"kind":"use","source":18},{"line":669,"start":19409,"end":19466,"expression":"\nX^k-Y^k=\\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r\n\\quad(k\\ge1).\n","context":985,"kind":"use","source":18},{"line":675,"start":19551,"end":19578,"expression":"X^{k-r}Y^r-X^{k-1-r}Y^{r+1}","context":986,"kind":"use","source":18},{"line":676,"start":19601,"end":19617,"expression":"\\|X\\|,\\|Y\\|\\le R","context":986,"kind":"use","source":18},{"line":678,"start":19629,"end":19662,"expression":"\n\\|X^k-Y^k\\|\\le kR^{k-1}\\|X-Y\\|.\n","context":987,"kind":"use","source":18},{"line":684,"start":19732,"end":19807,"expression":"\n\\|e^X-e^Y\\|\n\\le\\|X-Y\\|\\sum_{k=1}^{\\infty}\\frac{kR^{k-1}}{k!}\n=e^R\\|X-Y\\|.\n","context":988,"kind":"use","source":18},{"line":690,"start":19824,"end":19855,"expression":"\\|H_n\\|=E_n,\\|H_{h,n}\\|=E_{h,n}","context":989,"kind":"use","source":18},{"line":691,"start":19905,"end":19919,"expression":"E_{h,n}\\le E_n","context":989,"kind":"use","source":18},{"line":691,"start":19930,"end":19942,"expression":"X=-itH_{h,n}","context":989,"kind":"use","source":18},{"line":692,"start":19948,"end":19956,"expression":"Y=-itH_n","context":989,"kind":"use","source":18},{"line":692,"start":19966,"end":19974,"expression":"R=|t|E_n","context":989,"kind":"use","source":18},{"line":694,"start":19995,"end":20052,"expression":"\n\\|U_{H_{h,n}}(t)-U_{H_n}(t)\\|\n\\le |t|e^{|t|E_n}h^2C(n).\n","context":990,"kind":"use","source":18},{"line":699,"start":20100,"end":20103,"expression":"t=0","context":991,"kind":"use","source":18},{"line":703,"start":20225,"end":20324,"expression":"\n\\|f_h(t)-f(t)\\|_{L^2}\n\\le h^2\\left(\n\\sum_{n\\in S}|t|^2e^{2|t|E_n}C(n)^2\\|a_n\\|_2^2\n\\right)^{1/2}.\n","context":992,"kind":"use","source":18},{"line":710,"start":20363,"end":20368,"expression":"t,S,a","context":993,"kind":"use","source":18},{"line":711,"start":20409,"end":20416,"expression":"\\square","context":993,"kind":"use","source":18},{"line":717,"start":20714,"end":20717,"expression":"L^2","context":994,"kind":"use","source":18},{"line":725,"start":21072,"end":21082,"expression":"\\mathcal A","context":995,"kind":"use","source":18},{"line":725,"start":21111,"end":21134,"expression":"\\{0,1,2,3\\}\\to\\mathbb C","context":995,"kind":"use","source":18},{"line":726,"start":21157,"end":21181,"expression":"\\|v\\|_\\infty=\\max_r|v_r|","context":995,"kind":"use","source":18},{"line":726,"start":21195,"end":21205,"expression":"\\mathcal H","context":995,"kind":"use","source":18},{"line":729,"start":21277,"end":21334,"expression":"\nT:\\mathcal A\\longrightarrow\\mathcal H,\\qquad (Tv)_r=v_r\n","context":996,"kind":"use","source":18},{"line":737,"start":21486,"end":21527,"expression":"\n\\|v\\|_\\infty\\le\\|Tv\\|_2\\le2\\|v\\|_\\infty\n","context":997,"kind":"use","source":18},{"line":743,"start":21698,"end":21712,"expression":"\\|v\\|_\\infty^2","context":998,"kind":"use","source":18},{"line":747,"start":21831,"end":21832,"expression":"A","context":999,"kind":"use","source":18},{"line":748,"start":21873,"end":21883,"expression":"\\mathcal A","context":999,"kind":"use","source":18},{"line":748,"start":21897,"end":21920,"expression":"A^{\\mathcal H}=TAT^{-1}","context":999,"kind":"use","source":18},{"line":749,"start":21944,"end":21945,"expression":"z","context":999,"kind":"use","source":18},{"line":751,"start":21952,"end":21988,"expression":"\ne^{zA^{\\mathcal H}}=Te^{zA}T^{-1}.\n","context":1000,"kind":"use","source":18},{"line":758,"start":22208,"end":22209,"expression":"T","context":1001,"kind":"use","source":18},{"line":759,"start":22279,"end":22280,"expression":"T","context":1001,"kind":"use","source":18},{"line":762,"start":22382,"end":22405,"expression":"(TAT^{-1})^k=TA^kT^{-1}","context":1002,"kind":"use","source":18},{"line":763,"start":22415,"end":22418,"expression":"k=0","context":1002,"kind":"use","source":18},{"line":763,"start":22428,"end":22437,"expression":"TT^{-1}=I","context":1002,"kind":"use","source":18},{"line":764,"start":22487,"end":22494,"expression":"T^{-1}T","context":1002,"kind":"use","source":18},{"line":766,"start":22551,"end":22653,"expression":"\n\\sum_{k=0}^{N}\\frac{z^k(A^{\\mathcal H})^k}{k!}\n=T\\left(\\sum_{k=0}^{N}\\frac{z^kA^k}{k!}\\right)T^{-1}.\n","context":1003,"kind":"use","source":18},{"line":774,"start":22795,"end":22863,"expression":"\n\\|TBT^{-1}\\|_2\\le\\|T\\|\\,\\|B\\|_\\infty\\,\\|T^{-1}\\|\n\\le2\\|B\\|_\\infty.\n","context":1004,"kind":"use","source":18},{"line":782,"start":22967,"end":22989,"expression":"v\\mapsto T(A(T^{-1}v))","context":1005,"kind":"introduction-passage","source":18},{"line":785,"start":23179,"end":23184,"expression":"z=-it","context":1005,"kind":"introduction-passage","source":18},{"line":786,"start":23294,"end":23301,"expression":"\\square","context":1005,"kind":"introduction-passage","source":18},{"line":800,"start":23925,"end":23931,"expression":"h=1/12","context":1006,"kind":"use","source":18},{"line":801,"start":23943,"end":23946,"expression":"m=1","context":1006,"kind":"use","source":18},{"line":801,"start":23952,"end":23958,"expression":"h=1/12","context":1006,"kind":"use","source":18},{"line":803,"start":23969,"end":24044,"expression":"\nS=\\{n_0,n_A,n_D\\},\\qquad\nn_0=(0,0,0),\\quad n_A=(1,0,0),\\quad n_D=(1,1,1).\n","context":1007,"kind":"use","source":18},{"line":808,"start":24054,"end":24069,"expression":"e_0=(1,0,0,0)^T","context":1008,"kind":"introduction-passage","source":18},{"line":810,"start":24124,"end":24204,"expression":"\na_{n_0}=\\frac1{\\sqrt2}e_0,\\qquad\na_{n_A}=\\frac12e_0,\\qquad\na_{n_D}=\\frac12e_0.\n","context":1009,"kind":"use","source":18},{"line":816,"start":24234,"end":24245,"expression":"1/2,1/4,1/4","context":1010,"kind":"use","source":18},{"line":823,"start":24405,"end":24409,"expression":"p(n)","context":1011,"kind":"use","source":18},{"line":823,"start":24416,"end":24422,"expression":"q_h(n)","context":1011,"kind":"use","source":18},{"line":826,"start":24440,"end":24443,"expression":"n_0","context":1012,"kind":"use","source":18},{"line":826,"start":24450,"end":24457,"expression":"(0,0,0)","context":1012,"kind":"use","source":18},{"line":826,"start":24464,"end":24471,"expression":"(0,0,0)","context":1012,"kind":"use","source":18},{"line":827,"start":24479,"end":24482,"expression":"n_A","context":1013,"kind":"use","source":18},{"line":827,"start":24489,"end":24499,"expression":"(2\\pi,0,0)","context":1013,"kind":"use","source":18},{"line":827,"start":24506,"end":24513,"expression":"(6,0,0)","context":1013,"kind":"use","source":18},{"line":828,"start":24521,"end":24524,"expression":"n_D","context":1014,"kind":"use","source":18},{"line":828,"start":24531,"end":24547,"expression":"(2\\pi,2\\pi,2\\pi)","context":1014,"kind":"use","source":18},{"line":828,"start":24554,"end":24561,"expression":"(6,6,6)","context":1014,"kind":"use","source":18},{"line":833,"start":24636,"end":24646,"expression":"hp_j=\\pi/6","context":1015,"kind":"use","source":18},{"line":834,"start":24655,"end":24682,"expression":"\\sin(hp_j)/h=(1/2)/(1/12)=6","context":1015,"kind":"use","source":18},{"line":840,"start":24779,"end":24782,"expression":"E_n","context":1016,"kind":"use","source":18},{"line":840,"start":24789,"end":24796,"expression":"E_{h,n}","context":1016,"kind":"use","source":18},{"line":843,"start":24814,"end":24817,"expression":"n_0","context":1017,"kind":"use","source":18},{"line":843,"start":24824,"end":24825,"expression":"1","context":1017,"kind":"use","source":18},{"line":843,"start":24832,"end":24833,"expression":"1","context":1017,"kind":"use","source":18},{"line":844,"start":24841,"end":24844,"expression":"n_A","context":1018,"kind":"use","source":18},{"line":844,"start":24851,"end":24866,"expression":"\\sqrt{1+4\\pi^2}","context":1018,"kind":"use","source":18},{"line":844,"start":24873,"end":24882,"expression":"\\sqrt{37}","context":1018,"kind":"use","source":18},{"line":845,"start":24890,"end":24893,"expression":"n_D","context":1019,"kind":"use","source":18},{"line":845,"start":24900,"end":24916,"expression":"\\sqrt{1+12\\pi^2}","context":1019,"kind":"use","source":18},{"line":845,"start":24923,"end":24933,"expression":"\\sqrt{109}","context":1019,"kind":"use","source":18},{"line":850,"start":25041,"end":25042,"expression":"6","context":1020,"kind":"use","source":18},{"line":851,"start":25087,"end":25088,"expression":"1","context":1020,"kind":"use","source":18},{"line":852,"start":25116,"end":25120,"expression":"2\\pi","context":1020,"kind":"use","source":18},{"line":854,"start":25155,"end":25156,"expression":"r","context":1021,"kind":"use","source":18},{"line":856,"start":25186,"end":25219,"expression":"\nH(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.\n","context":1022,"kind":"use","source":18},{"line":860,"start":25229,"end":25247,"expression":"E_r=\\sqrt{1+|r|^2}","context":1023,"kind":"use","source":18},{"line":860,"start":25253,"end":25270,"expression":"c_r(t)=\\cos(E_rt)","context":1023,"kind":"use","source":18},{"line":861,"start":25280,"end":25301,"expression":"s_r(t)=\\sin(E_rt)/E_r","context":1023,"kind":"use","source":18},{"line":863,"start":25319,"end":25428,"expression":"\nu_r(t)=U_{H(1,r)}(t)e_0=\n\\begin{pmatrix}\nc_r(t)-is_r(t)\\\\\n0\\\\\n-ir_zs_r(t)\\\\\n(r_y-ir_x)s_r(t)\n\\end{pmatrix}.\n","context":1024,"kind":"use","source":18},{"line":873,"start":25449,"end":25466,"expression":"u_0(t)=e^{-it}e_0","context":1025,"kind":"use","source":18},{"line":876,"start":25515,"end":25640,"expression":"\nu_A(t)=\n\\begin{pmatrix}\n\\cos(\\sqrt{37}t)-i\\sin(\\sqrt{37}t)/\\sqrt{37}\\\\\n0\\\\\n0\\\\\n-6i\\sin(\\sqrt{37}t)/\\sqrt{37}\n\\end{pmatrix},\n","context":1026,"kind":"use","source":18},{"line":886,"start":25646,"end":25809,"expression":"\nu_D(t)=\n\\begin{pmatrix}\n\\cos(\\sqrt{109}t)-i\\sin(\\sqrt{109}t)/\\sqrt{109}\\\\\n0\\\\\n-6i\\sin(\\sqrt{109}t)/\\sqrt{109}\\\\\n6(1-i)\\sin(\\sqrt{109}t)/\\sqrt{109}\n\\end{pmatrix}.\n","context":1027,"kind":"use","source":18},{"line":898,"start":25848,"end":25951,"expression":"\nf_h(t,x)=\\frac{e^{-it}}{\\sqrt2}e_0\n+\\frac12e^{2\\pi ix_1}u_A(t)\n+\\frac12e^{2\\pi i(x_1+x_2+x_3)}u_D(t).\n","context":1028,"kind":"use","source":18},{"line":904,"start":26001,"end":26008,"expression":"u_A,u_D","context":1029,"kind":"use","source":18},{"line":905,"start":26025,"end":26039,"expression":"u_{(2\\pi,0,0)}","context":1029,"kind":"use","source":18},{"line":905,"start":26048,"end":26068,"expression":"u_{(2\\pi,2\\pi,2\\pi)}","context":1029,"kind":"use","source":18},{"line":905,"start":26077,"end":26080,"expression":"t=0","context":1029,"kind":"use","source":18},{"line":907,"start":26098,"end":26191,"expression":"\nf_a(x)=\\left(\\frac1{\\sqrt2}+\\frac12e^{2\\pi ix_1}\n+\\frac12e^{2\\pi i(x_1+x_2+x_3)}\\right)e_0.\n","context":1030,"kind":"use","source":18},{"line":914,"start":26246,"end":26332,"expression":"\n\\|u_r(t)\\|_2^2\n=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)\n=\\cos^2(E_rt)+\\sin^2(E_rt)=1.\n","context":1031,"kind":"use","source":18},{"line":920,"start":26343,"end":26359,"expression":"|c-is|^2=c^2+s^2","context":1032,"kind":"use","source":18},{"line":920,"start":26368,"end":26392,"expression":"|r_y-ir_x|^2=r_x^2+r_y^2","context":1032,"kind":"use","source":18},{"line":923,"start":26460,"end":26536,"expression":"\n\\|f_h(t)\\|_{L^2}^2=\\tfrac12+\\tfrac14+\\tfrac14=1,\\qquad\n\\|f(t)\\|_{L^2}^2=1.\n","context":1033,"kind":"use","source":18},{"line":929,"start":26625,"end":26628,"expression":"x=0","context":1034,"kind":"use","source":18},{"line":929,"start":26640,"end":26665,"expression":"(1+1/\\sqrt2)^2=3/2+\\sqrt2","context":1034,"kind":"use","source":18},{"line":933,"start":26866,"end":26872,"expression":"r=p(n)","context":1035,"kind":"use","source":18},{"line":934,"start":26878,"end":26886,"expression":"q=q_h(n)","context":1035,"kind":"use","source":18},{"line":934,"start":26915,"end":26930,"expression":"c_r,s_r,c_q,s_q","context":1035,"kind":"use","source":18},{"line":937,"start":26966,"end":27048,"expression":"\n\\|u_q(t)-u_r(t)\\|_2^2\n=(c_q-c_r)^2+(s_q-s_r)^2\n+\\sum_{j=x,y,z}(q_js_q-r_js_r)^2.\n","context":1036,"kind":"use","source":18},{"line":945,"start":27119,"end":27285,"expression":"\n\\begin{aligned}\n\\|f_h(t)-f(t)\\|_{L^2}^2\n={}&\\frac14\\|u_{(6,0,0)}(t)-u_{(2\\pi,0,0)}(t)\\|_2^2\\\\\n&+\\frac14\\|u_{(6,6,6)}(t)-u_{(2\\pi,2\\pi,2\\pi)}(t)\\|_2^2.\n\\end{aligned}\n","context":1037,"kind":"use","source":18},{"line":957,"start":27523,"end":27526,"expression":"m=1","context":1038,"kind":"use","source":18},{"line":957,"start":27530,"end":27536,"expression":"h=1/12","context":1038,"kind":"use","source":18},{"line":966,"start":28208,"end":28211,"expression":"A,B","context":1039,"kind":"use","source":18},{"line":968,"start":28308,"end":28309,"expression":"t","context":1039,"kind":"use","source":18},{"line":970,"start":28316,"end":28356,"expression":"\n\\|e^{-itA}-e^{-itB}\\|\\le |t|\\,\\|A-B\\|.\n","context":1040,"kind":"use","source":18},{"line":976,"start":28466,"end":28474,"expression":"|s|\\le R","context":1041,"kind":"use","source":18},{"line":977,"start":28495,"end":28503,"expression":"e^{-isB}","context":1041,"kind":"use","source":18},{"line":980,"start":28577,"end":28642,"expression":"\n\\sum_{k=1}^{\\infty}\\frac{kR^{k-1}\\|B\\|^k}{k!}\n=\\|B\\|e^{R\\|B\\|}.\n","context":1042,"kind":"use","source":18},{"line":987,"start":28690,"end":28738,"expression":"\n\\frac{d}{ds}e^{-isB}=-iB e^{-isB}=-ie^{-isB}B.\n","context":1043,"kind":"use","source":18},{"line":991,"start":28763,"end":28764,"expression":"A","context":1044,"kind":"use","source":18},{"line":991,"start":28776,"end":28777,"expression":"A","context":1044,"kind":"use","source":18},{"line":992,"start":28836,"end":28844,"expression":"e^{-iuA}","context":1044,"kind":"use","source":18},{"line":992,"start":28854,"end":28855,"expression":"t","context":1044,"kind":"use","source":18},{"line":993,"start":28871,"end":28896,"expression":"F(s)=e^{-i(t-s)A}e^{-isB}","context":1044,"kind":"use","source":18},{"line":995,"start":28926,"end":28962,"expression":"\nF'(s)=i e^{-i(t-s)A}(A-B)e^{-isB}.\n","context":1045,"kind":"use","source":18},{"line":999,"start":28986,"end":28999,"expression":"F(0)=e^{-itA}","context":1046,"kind":"use","source":18},{"line":999,"start":29008,"end":29021,"expression":"F(t)=e^{-itB}","context":1046,"kind":"use","source":18},{"line":1003,"start":29114,"end":29175,"expression":"\ne^{-itB}-e^{-itA}\n=i\\int_0^t e^{-i(t-s)A}(A-B)e^{-isB}\\,ds.\n","context":1047,"kind":"use","source":18},{"line":1010,"start":29324,"end":29344,"expression":"(e^{-iuA})^*=e^{iuA}","context":1048,"kind":"use","source":18},{"line":1011,"start":29419,"end":29436,"expression":"e^{iuA}e^{-iuA}=I","context":1048,"kind":"use","source":18},{"line":1012,"start":29469,"end":29472,"expression":"A^k","context":1048,"kind":"use","source":18},{"line":1012,"start":29481,"end":29484,"expression":"k>0","context":1048,"kind":"use","source":18},{"line":1013,"start":29492,"end":29535,"expression":"(iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0","context":1048,"kind":"use","source":18},{"line":1014,"start":29573,"end":29574,"expression":"I","context":1048,"kind":"use","source":18},{"line":1015,"start":29689,"end":29696,"expression":"\\|A-B\\|","context":1049,"kind":"use","source":18},{"line":1016,"start":29706,"end":29711,"expression":"t\\ge0","context":1050,"kind":"use","source":18},{"line":1016,"start":29735,"end":29743,"expression":"t\\|A-B\\|","context":1051,"kind":"use","source":18},{"line":1016,"start":29753,"end":29756,"expression":"t<0","context":1052,"kind":"use","source":18},{"line":1017,"start":29791,"end":29802,"expression":"(-t)\\|A-B\\|","context":1053,"kind":"use","source":18},{"line":1018,"start":29854,"end":29861,"expression":"\\square","context":1054,"kind":"use","source":18},{"line":1022,"start":29968,"end":29973,"expression":"m,h,t","context":1055,"kind":"use","source":18},{"line":1022,"start":29990,"end":29991,"expression":"n","context":1055,"kind":"use","source":18},{"line":1024,"start":30023,"end":30092,"expression":"\n\\|U_{H_{h,n}}(t)-U_{H_n}(t)\\|\n\\le |t|\\,|\\delta(h,n)|\\le |t|h^2C(n),\n","context":1056,"kind":"use","source":18},{"line":1029,"start":30098,"end":30282,"expression":"\n\\begin{aligned}\n\\|f_h(t)-f(t)\\|_{L^2}\n&\\le |t|\\left(\\sum_{n\\in S}|\\delta(h,n)|^2\\|a_n\\|_2^2\\right)^{1/2}\\\\\n&\\le |t|h^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}.\n\\end{aligned}\n","context":1057,"kind":"use","source":18},{"line":1037,"start":30312,"end":30317,"expression":"T\\ge0","context":1058,"kind":"use","source":18},{"line":1039,"start":30324,"end":30421,"expression":"\n\\sup_{|t|\\le T}\\|f_h(t)-f(t)\\|_{L^2}\n\\le Th^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2},\n","context":1059,"kind":"use","source":18},{"line":1047,"start":30578,"end":30587,"expression":"A=H_{h,n}","context":1060,"kind":"use","source":18},{"line":1047,"start":30593,"end":30598,"expression":"B=H_n","context":1060,"kind":"use","source":18},{"line":1051,"start":30697,"end":30794,"expression":"\n\\|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\\|_2\n\\le |t|\\,|\\delta(h,n)|\\,\\|a_n\\|_2\n\\le |t|h^2C(n)\\|a_n\\|_2.\n","context":1061,"kind":"use","source":18},{"line":1057,"start":30820,"end":30821,"expression":"S","context":1062,"kind":"use","source":18},{"line":1061,"start":31038,"end":31083,"expression":"\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4","context":1062,"kind":"use","source":18},{"line":1063,"start":31224,"end":31227,"expression":"|t|","context":1063,"kind":"use","source":18},{"line":1063,"start":31235,"end":31236,"expression":"T","context":1064,"kind":"use","source":18},{"line":1064,"start":31303,"end":31308,"expression":"h\\to0","context":1065,"kind":"use","source":18},{"line":1065,"start":31314,"end":31321,"expression":"\\square","context":1066,"kind":"use","source":18},{"line":1067,"start":31337,"end":31345,"expression":"C(n_0)=0","context":1067,"kind":"use","source":18},{"line":1067,"start":31351,"end":31368,"expression":"C(n_A)=(2\\pi)^3/6","context":1067,"kind":"use","source":18},{"line":1068,"start":31378,"end":31401,"expression":"C(n_D)=\\sqrt3(2\\pi)^3/6","context":1067,"kind":"use","source":18},{"line":1070,"start":31411,"end":31500,"expression":"\n\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}\n=\\frac{(2\\pi)^3}{6}=\\frac{4\\pi^3}{3}.\n","context":1068,"kind":"use","source":18},{"line":1075,"start":31526,"end":31536,"expression":"d=2\\pi-6>0","context":1069,"kind":"use","source":18},{"line":1076,"start":31574,"end":31575,"expression":"d","context":1069,"kind":"use","source":18},{"line":1076,"start":31617,"end":31624,"expression":"\\sqrt3d","context":1069,"kind":"use","source":18},{"line":1077,"start":31670,"end":31677,"expression":"1/4,1/4","context":1069,"kind":"use","source":18},{"line":1080,"start":31722,"end":31772,"expression":"\n\\left(\\tfrac14d^2+\\tfrac14\\,3d^2\\right)^{1/2}=d,\n","context":1070,"kind":"use","source":18},{"line":1086,"start":31803,"end":31870,"expression":"\n\\|f_{1/12}(t)-f(t)\\|_{L^2}\n\\le(2\\pi-6)|t|\\le\\frac{\\pi^3}{108}|t|.\n","context":1071,"kind":"use","source":18},{"line":1091,"start":31911,"end":31936,"expression":"\\|f_h(t)-f(t)\\|_{L^2}\\le2","context":1072,"kind":"use","source":18},{"line":1095,"start":32200,"end":32203,"expression":"L^2","context":1072,"kind":"use","source":18},{"line":1102,"start":32534,"end":32545,"expression":"H_S,H_{h,S}","context":1073,"kind":"introduction-passage","source":18},{"line":1106,"start":32681,"end":32773,"expression":"\nN(a)=\\sum_{n\\in S}\\|a_n\\|_2^2,\\qquad\n\\mathcal E(a)=\\sum_{n\\in S}\\langle a_n,H_na_n\\rangle,\n","context":1074,"kind":"use","source":18},{"line":1111,"start":32805,"end":32820,"expression":"\\mathcal E_h(a)","context":1075,"kind":"use","source":18},{"line":1112,"start":32847,"end":32850,"expression":"H_n","context":1075,"kind":"use","source":18},{"line":1112,"start":32860,"end":32867,"expression":"H_{h,n}","context":1075,"kind":"use","source":18},{"line":1114,"start":32939,"end":32990,"expression":"\\langle H_na_n,a_n\\rangle=\\langle a_n,H_na_n\\rangle","context":1075,"kind":"use","source":18},{"line":1121,"start":33198,"end":33283,"expression":"\n\\mathcal E(U_S(t)a)=\\mathcal E(a),\\qquad\n\\mathcal E_h(U_{h,S}(t)a)=\\mathcal E_h(a).\n","context":1076,"kind":"use","source":18},{"line":1127,"start":33356,"end":33385,"expression":"\\mathcal K_S\\cong\\mathcal F_S","context":1077,"kind":"use","source":18},{"line":1128,"start":33459,"end":33463,"expression":"N(a)","context":1077,"kind":"use","source":18},{"line":1130,"start":33494,"end":33564,"expression":"\n|\\mathcal E_h(a)-\\mathcal E(a)|\n\\le h^2\\sum_{n\\in S}C(n)\\|a_n\\|_2^2.\n","context":1078,"kind":"use","source":18},{"line":1138,"start":33709,"end":33800,"expression":"\n\\langle U_H(t)a,H U_H(t)a\\rangle\n=\\langle a,U_H(t)^*HU_H(t)a\\rangle\n=\\langle a,Ha\\rangle.\n","context":1079,"kind":"use","source":18},{"line":1147,"start":33907,"end":34004,"expression":"\n|\\langle a_n,(H_{h,n}-H_n)a_n\\rangle|\n\\le\\|a_n\\|_2\\|\\Delta_{h,n}a_n\\|_2\n\\le h^2C(n)\\|a_n\\|_2^2.\n","context":1080,"kind":"use","source":18},{"line":1153,"start":34068,"end":34075,"expression":"\\square","context":1081,"kind":"use","source":18},{"line":1155,"start":34127,"end":34130,"expression":"P_-","context":1082,"kind":"use","source":18},{"line":1156,"start":34172,"end":34174,"expression":"-E","context":1082,"kind":"use","source":18},{"line":1157,"start":34239,"end":34240,"expression":"E","context":1082,"kind":"use","source":18},{"line":1158,"start":34309,"end":34312,"expression":"e_0","context":1082,"kind":"use","source":18},{"line":1159,"start":34335,"end":34365,"expression":"\\langle e_0,H(m,p)e_0\\rangle=m","context":1082,"kind":"use","source":18},{"line":1160,"start":34460,"end":34461,"expression":"1","context":1082,"kind":"use","source":18},{"line":1161,"start":34500,"end":34520,"expression":"\\sqrt{37},\\sqrt{109}","context":1083,"kind":"use","source":18},{"line":30,"start":4059,"end":4071,"expression":"\\sin(hp_j)/h","context":1084,"kind":"notation-table","source":19},{"line":42,"start":5228,"end":5229,"expression":"V","context":1085,"kind":"use","source":19},{"line":42,"start":5293,"end":5307,"expression":"(P_a)_{a\\in A}","context":1085,"kind":"use","source":19},{"line":43,"start":5322,"end":5372,"expression":"\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n","context":1085,"kind":"use","source":19},{"line":46,"start":5390,"end":5394,"expression":"P_av","context":1085,"kind":"use","source":19},{"line":46,"start":5414,"end":5415,"expression":"v","context":1085,"kind":"use","source":19},{"line":46,"start":5430,"end":5431,"expression":"a","context":1085,"kind":"use","source":19},{"line":46,"start":5452,"end":5460,"expression":"L:V\\to V","context":1086,"kind":"use","source":19},{"line":46,"start":5495,"end":5496,"expression":"a","context":1087,"kind":"use","source":19},{"line":46,"start":5509,"end":5510,"expression":"b","context":1088,"kind":"use","source":19},{"line":47,"start":5517,"end":5534,"expression":"\nL_{ba}=P_bLP_a.\n","context":1089,"kind":"use","source":19},{"line":50,"start":5557,"end":5575,"expression":"L=\\sum_{a,b}L_{ba}","context":1090,"kind":"use","source":19},{"line":50,"start":5613,"end":5620,"expression":"P_aLP_a","context":1091,"kind":"use","source":19},{"line":52,"start":5696,"end":5697,"expression":"L","context":1092,"kind":"use","source":19},{"line":52,"start":5704,"end":5705,"expression":"M","context":1092,"kind":"use","source":19},{"line":53,"start":5746,"end":5789,"expression":"\nP_cMLP_a=\\sum_{b\\in A}(P_cMP_b)(P_bLP_a).\n","context":1092,"kind":"use","source":19},{"line":58,"start":6076,"end":6099,"expression":"P_A=\\sum_{a\\in C(A)}P_a","context":1093,"kind":"use","source":19},{"line":58,"start":6106,"end":6129,"expression":"P_B=\\sum_{b\\in C(B)}P_b","context":1093,"kind":"use","source":19},{"line":59,"start":6139,"end":6189,"expression":"\nP_BLP_A=\\sum_{b\\in C(B)}\\sum_{a\\in C(A)}P_bLP_a.\n","context":1093,"kind":"use","source":19},{"line":67,"start":6859,"end":6870,"expression":"\\sum_aP_a=I","context":1094,"kind":"use","source":19},{"line":5,"start":488,"end":491,"expression":"\\HH","context":1095,"kind":"use","source":21},{"line":5,"start":525,"end":526,"expression":"I","context":1095,"kind":"use","source":21},{"line":5,"start":674,"end":689,"expression":"\\mathcal B(\\HH)","context":1095,"kind":"use","source":21},{"line":5,"start":733,"end":736,"expression":"\\HH","context":1095,"kind":"use","source":21},{"line":5,"start":757,"end":760,"expression":"\\HH","context":1095,"kind":"use","source":21},{"line":8,"start":1005,"end":1019,"expression":"(P_a)_{a\\in I}","context":1096,"kind":"definition-passage","definition_title":"Finite orthogonal atlas","source":21},{"line":9,"start":1048,"end":1145,"expression":"\\label{f:atlas-relations}\n P_a^*=P_a,\\qquad P_aP_b=\\delta_{ab}P_a,\n \\qquad \\sum_{a\\in I}P_a=\\id.\n","context":1096,"kind":"definition-passage","definition_title":"Finite orthogonal atlas","source":21},{"line":13,"start":1184,"end":1188,"expression":"\\psi","context":1096,"kind":"definition-passage","definition_title":"Finite orthogonal atlas","source":21},{"line":13,"start":1194,"end":1208,"expression":"\\psi_a=P_a\\psi","context":1096,"kind":"definition-passage","definition_title":"Finite orthogonal atlas","source":21},{"line":13,"start":1233,"end":1252,"expression":"p_a=\\norm{\\psi_a}^2","context":1096,"kind":"definition-passage","definition_title":"Finite orthogonal atlas","source":21},{"line":16,"start":1432,"end":1445,"expression":"\\norm{\\psi}=1","context":1097,"kind":"use","source":21},{"line":19,"start":1834,"end":1849,"expression":"\\psi,\\phi\\in\\HH","context":1098,"kind":"use","source":21},{"line":20,"start":1868,"end":2012,"expression":"\\label{f:reconstruct}\n \\psi=\\sum_a\\psi_a,\\qquad\n \\inner{\\psi}{\\phi}=\\sum_a\\inner{P_a\\psi}{P_a\\phi},\\qquad\n \\norm{\\psi}^2=\\sum_a\\norm{\\psi_a}^2.\n","context":1098,"kind":"use","source":21},{"line":29,"start":2168,"end":2181,"expression":"\\sum_aP_a=\\id","context":1099,"kind":"use","source":21},{"line":29,"start":2187,"end":2191,"expression":"\\psi","context":1099,"kind":"use","source":21},{"line":29,"start":2199,"end":2205,"expression":"a\\ne b","context":1099,"kind":"use","source":21},{"line":30,"start":2250,"end":2307,"expression":"\n \\inner{P_a\\psi}{P_b\\phi}\n =\\inner{\\psi}{P_aP_b\\phi}=0.\n","context":1099,"kind":"use","source":21},{"line":34,"start":2334,"end":2352,"expression":"\\inner{\\psi}{\\phi}","context":1099,"kind":"use","source":21},{"line":34,"start":2434,"end":2443,"expression":"\\phi=\\psi","context":1099,"kind":"use","source":21},{"line":40,"start":3002,"end":3010,"expression":"\\HH=\\C^4","context":1100,"kind":"use","source":21},{"line":41,"start":3052,"end":3178,"expression":"\n \\psi=\\frac1{\\sqrt7}(1,i,2,-i)^{\\mathsf T},\\qquad\n P_A=\\operatorname{diag}(1,1,0,0),\\quad\n P_B=\\operatorname{diag}(0,0,1,1).\n","context":1100,"kind":"use","source":21},{"line":46,"start":3187,"end":3222,"expression":"\\psi_A=(1,i,0,0)^{\\mathsf T}/\\sqrt7","context":1100,"kind":"use","source":21},{"line":47,"start":3229,"end":3265,"expression":"\\psi_B=(0,0,2,-i)^{\\mathsf T}/\\sqrt7","context":1100,"kind":"use","source":21},{"line":47,"start":3285,"end":3292,"expression":"p_A=2/7","context":1100,"kind":"use","source":21},{"line":47,"start":3299,"end":3306,"expression":"p_B=5/7","context":1100,"kind":"use","source":21},{"line":47,"start":3369,"end":3384,"expression":"1/7,1/7,4/7,1/7","context":1101,"kind":"use","source":21},{"line":56,"start":3974,"end":3993,"expression":"X\\in\\mathcal B(\\HH)","context":1102,"kind":"use","source":21},{"line":57,"start":4019,"end":4057,"expression":"\\label{f:corner-def}\n X_{ba}=P_bXP_a.\n","context":1102,"kind":"use","source":21},{"line":60,"start":4184,"end":4210,"expression":"X_{ba}:\\Ran P_a\\to\\Ran P_b","context":1102,"kind":"use","source":21},{"line":60,"start":4239,"end":4242,"expression":"\\HH","context":1102,"kind":"use","source":21},{"line":60,"start":4301,"end":4309,"expression":"\\Ran P_a","context":1103,"kind":"use","source":21},{"line":63,"start":4496,"end":4499,"expression":"X,Y","context":1104,"kind":"use","source":21},{"line":64,"start":4515,"end":4685,"expression":"\n X&=\\sum_{b,a}X_{ba}, & P_bX\\psi&=\\sum_aX_{ba}\\psi_a,\\label{f:block-reconstruction}\\\\\n (XY)_{ca}&=\\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.\n \\label{f:block-product}\n","context":1105,"kind":"use","source":21},{"line":69,"start":4731,"end":4734,"expression":"P_a","context":1106,"kind":"use","source":21},{"line":69,"start":4770,"end":4775,"expression":"X=X^*","context":1106,"kind":"use","source":21},{"line":73,"start":4853,"end":4866,"expression":"\\sum_bP_b=\\id","context":1107,"kind":"use","source":21},{"line":73,"start":4884,"end":4885,"expression":"X","context":1107,"kind":"use","source":21},{"line":73,"start":4892,"end":4905,"expression":"\\sum_aP_a=\\id","context":1107,"kind":"use","source":21},{"line":73,"start":4985,"end":4989,"expression":"\\psi","context":1107,"kind":"use","source":21},{"line":74,"start":5026,"end":5095,"expression":"\n P_cXYP_a=P_cX\\Bigl(\\sum_bP_b\\Bigr)YP_a\n =\\sum_b(P_cXP_b)(P_bYP_a),\n","context":1107,"kind":"use","source":21},{"line":78,"start":5105,"end":5114,"expression":"P_b^2=P_b","context":1107,"kind":"use","source":21},{"line":79,"start":5166,"end":5187,"expression":"(P_bXP_a)^*=P_aX^*P_b","context":1107,"kind":"use","source":21},{"line":86,"start":5798,"end":5872,"expression":"\n H=\\begin{pmatrix}\n 0&1&0&2\\\\1&0&-1&0\\\\0&-1&0&1\\\\2&0&1&0\n \\end{pmatrix}.\n","context":1108,"kind":"use","source":21},{"line":91,"start":5888,"end":5911,"expression":"\\C^4=\\C^2_A\\oplus\\C^2_B","context":1109,"kind":"use","source":21},{"line":92,"start":5936,"end":6067,"expression":"\n H_{AA}=H_{BB}=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n H_{BA}=\\begin{pmatrix}0&-1\\\\2&0\\end{pmatrix},\\qquad\n H_{AB}=H_{BA}^*.\n","context":1110,"kind":"use","source":21},{"line":97,"start":6086,"end":6087,"expression":"A","context":1111,"kind":"use","source":21},{"line":97,"start":6093,"end":6094,"expression":"B","context":1111,"kind":"use","source":21},{"line":100,"start":6429,"end":6442,"expression":"\\sum_aP_aXP_a","context":1112,"kind":"use","source":21},{"line":100,"start":6461,"end":6462,"expression":"X","context":1112,"kind":"use","source":21},{"line":106,"start":6883,"end":6890,"expression":"\\psi(t)","context":1113,"kind":"use","source":21},{"line":107,"start":6956,"end":7017,"expression":"\\label{f:flow}\n \\dot\\psi(t)=K(t)\\psi(t),\\qquad K(t)^*=-K(t).\n","context":1113,"kind":"use","source":21},{"line":110,"start":7038,"end":7042,"expression":"K(t)","context":1113,"kind":"use","source":21},{"line":113,"start":7466,"end":7618,"expression":"\\label{f:current}\n J_{b\\leftarrow a}(t)\n =2\\operatorname{Re}\\inner{\\psi_b(t)}{K(t)\\psi_a(t)}\n =2\\operatorname{Re}\\inner{\\psi_b(t)}{K_{ba}(t)\\psi_a(t)}.\n","context":1114,"kind":"use","source":21},{"line":118,"start":7670,"end":7686,"expression":"P_b\\psi_b=\\psi_b","context":1114,"kind":"use","source":21},{"line":118,"start":7780,"end":7781,"expression":"b","context":1114,"kind":"use","source":21},{"line":122,"start":8002,"end":8180,"expression":"\n \\dot p_b&=\\sum_aJ_{b\\leftarrow a},\\label{f:continuity}\\\\\n J_{b\\leftarrow a}&=-J_{a\\leftarrow b},\\qquad\n J_{a\\leftarrow a}=0,\\label{f:antisymmetry}\\\\\n \\frac{d}{dt}\\sum_ap_a&=0.\n","context":1115,"kind":"use","source":21},{"line":128,"start":8210,"end":8222,"expression":"C\\subseteq I","context":1116,"kind":"use","source":21},{"line":129,"start":8264,"end":8371,"expression":"\\label{f:boundary-current}\n \\frac{d}{dt}\\sum_{b\\in C}p_b\n =\\sum_{b\\in C}\\sum_{a\\notin C}J_{b\\leftarrow a}.\n","context":1116,"kind":"use","source":21},{"line":137,"start":8511,"end":8668,"expression":"\n \\dot p_b\n &=2\\operatorname{Re}\\inner{P_b\\psi}{P_b\\dot\\psi}\n =2\\operatorname{Re}\\inner{\\psi_b}{K\\psi}\\\\\n &=\\sum_a2\\operatorname{Re}\\inner{\\psi_b}{K\\psi_a}.\n","context":1117,"kind":"use","source":21},{"line":143,"start":8704,"end":8708,"expression":"\\psi","context":1118,"kind":"use","source":21},{"line":144,"start":8766,"end":8858,"expression":"\n \\inner{\\psi_a}{K\\psi_b}\n =-\\inner{K\\psi_a}{\\psi_b}\n =-\\overline{\\inner{\\psi_b}{K\\psi_a}}.\n","context":1118,"kind":"use","source":21},{"line":149,"start":8909,"end":8912,"expression":"a=b","context":1118,"kind":"use","source":21},{"line":149,"start":8975,"end":8984,"expression":"C\\times C","context":1118,"kind":"use","source":21},{"line":149,"start":9053,"end":9054,"expression":"C","context":1119,"kind":"use","source":21},{"line":149,"start":9118,"end":9121,"expression":"C=I","context":1120,"kind":"use","source":21},{"line":154,"start":9520,"end":9525,"expression":"K=-iH","context":1121,"kind":"use","source":21},{"line":154,"start":9533,"end":9538,"expression":"H=H^*","context":1121,"kind":"use","source":21},{"line":154,"start":9545,"end":9552,"expression":"\\hbar=1","context":1121,"kind":"use","source":21},{"line":155,"start":9607,"end":9701,"expression":"\\label{f:hamiltonian-current}\n J_{b\\leftarrow a}\n =2\\operatorname{Im}\\inner{\\psi_b}{H\\psi_a}.\n","context":1121,"kind":"use","source":21},{"line":159,"start":9725,"end":9766,"expression":"\\operatorname{Re}(-iz)=\\operatorname{Im}z","context":1121,"kind":"use","source":21},{"line":162,"start":10009,"end":10013,"expression":"\\C^2","context":1122,"kind":"use","source":21},{"line":163,"start":10045,"end":10148,"expression":"\n H=g\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\quad g>0,\n \\qquad \\psi(0)=\\begin{pmatrix}1\\\\0\\end{pmatrix}.\n","context":1123,"kind":"use","source":21},{"line":168,"start":10198,"end":10302,"expression":"\n \\psi(t)=\\begin{pmatrix}\\cos(gt)\\\\-i\\sin(gt)\\end{pmatrix},\n \\quad p_1=\\cos^2(gt),\\quad p_2=\\sin^2(gt).\n","context":1124,"kind":"use","source":21},{"line":173,"start":10322,"end":10416,"expression":"\n J_{2\\leftarrow1}\n =2\\operatorname{Im}\\bigl(i g\\sin(gt)\\cos(gt)\\bigr)\n =g\\sin(2gt)=\\dot p_2.\n","context":1125,"kind":"use","source":21},{"line":178,"start":10458,"end":10461,"expression":"t=0","context":1125,"kind":"use","source":21},{"line":181,"start":10743,"end":10765,"expression":"\\dot\\psi=K(\\psi,t)\\psi","context":1126,"kind":"use","source":21},{"line":181,"start":10873,"end":10882,"expression":"K(\\psi,t)","context":1126,"kind":"use","source":21},{"line":181,"start":10920,"end":10936,"expression":"\\norm{P_b\\psi}^2","context":1126,"kind":"use","source":21},{"line":189,"start":11464,"end":11474,"expression":"\\mathcal A","context":1127,"kind":"use","source":21},{"line":189,"start":11518,"end":11534,"expression":"f:I\\to\\mathcal A","context":1127,"kind":"use","source":21},{"line":189,"start":11597,"end":11610,"expression":"C_A=f^{-1}(A)","context":1127,"kind":"use","source":21},{"line":190,"start":11639,"end":11691,"expression":"\\label{f:coarse-projector}\n Q_A=\\sum_{a\\in C_A}P_a.\n","context":1127,"kind":"use","source":21},{"line":193,"start":11741,"end":11742,"expression":"I","context":1127,"kind":"use","source":21},{"line":193,"start":11749,"end":11754,"expression":"(Q_A)","context":1127,"kind":"use","source":21},{"line":197,"start":12041,"end":12228,"expression":"\n p_A^{\\mathrm c}&=\\norm{Q_A\\psi}^2=\\sum_{a\\in C_A}p_a,\\\\\n J_{B\\leftarrow A}^{\\mathrm c}\n &=2\\operatorname{Re}\\inner{Q_B\\psi}{KQ_A\\psi}\n =\\sum_{b\\in C_B}\\sum_{a\\in C_A}J_{b\\leftarrow a}.\n","context":1128,"kind":"use","source":21},{"line":203,"start":12254,"end":12310,"expression":"\\dot p_B^{\\mathrm c}=\\sum_AJ_{B\\leftarrow A}^{\\mathrm c}","context":1129,"kind":"use","source":21},{"line":204,"start":12317,"end":12377,"expression":"J_{B\\leftarrow A}^{\\mathrm c}=-J_{A\\leftarrow B}^{\\mathrm c}","context":1129,"kind":"use","source":21},{"line":211,"start":12844,"end":12852,"expression":"R_{Aa}=1","context":1130,"kind":"use","source":21},{"line":211,"start":12860,"end":12866,"expression":"f(a)=A","context":1130,"kind":"use","source":21},{"line":212,"start":12924,"end":13071,"expression":"\\label{f:incidence}\n p^{\\mathrm c}=Rp,\\qquad J^{\\mathrm c}=RJR^{\\mathsf T},\\qquad\n R^{\\mathsf T}\\boldsymbol1_{\\mathrm c}=\\boldsymbol1_{\\mathrm f}.\n","context":1130,"kind":"use","source":21},{"line":216,"start":13209,"end":13210,"expression":"S","context":1130,"kind":"use","source":21},{"line":217,"start":13247,"end":13325,"expression":"\n S(Rp)=(SR)p,\\qquad S(RJR^{\\mathsf T})S^{\\mathsf T}\n =(SR)J(SR)^{\\mathsf T}.\n","context":1131,"kind":"use","source":21},{"line":225,"start":13562,"end":13667,"expression":"\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n","context":1132,"kind":"use","source":21},{"line":230,"start":13864,"end":13878,"expression":"(-5/2,1,1,1/2)","context":1133,"kind":"use","source":21},{"line":230,"start":13886,"end":13895,"expression":"A=\\{1,2\\}","context":1134,"kind":"use","source":21},{"line":230,"start":13902,"end":13911,"expression":"B=\\{3,4\\}","context":1135,"kind":"use","source":21},{"line":230,"start":13937,"end":13938,"expression":"3","context":1136,"kind":"use","source":21},{"line":230,"start":13945,"end":13946,"expression":"1","context":1137,"kind":"use","source":21},{"line":231,"start":13965,"end":14004,"expression":"J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2","context":1138,"kind":"use","source":21},{"line":232,"start":14014,"end":14039,"expression":"\\dot p_A^{\\mathrm c}=-3/2","context":1139,"kind":"use","source":21},{"line":232,"start":14046,"end":14070,"expression":"\\dot p_B^{\\mathrm c}=3/2","context":1140,"kind":"use","source":21},{"line":235,"start":14180,"end":14193,"expression":"p^{\\mathrm c}","context":1141,"kind":"use","source":21},{"line":235,"start":14253,"end":14266,"expression":"J^{\\mathrm c}","context":1141,"kind":"use","source":21},{"line":236,"start":14462,"end":14515,"expression":"\n \\begin{pmatrix}0&c&-c\\\\-c&0&c\\\\c&-c&0\\end{pmatrix}\n","context":1142,"kind":"use","source":21},{"line":239,"start":14550,"end":14551,"expression":"c","context":1143,"kind":"use","source":21},{"line":241,"start":14942,"end":14945,"expression":"3/2","context":1144,"kind":"use","source":21},{"line":247,"start":15264,"end":15268,"expression":"U(t)","context":1145,"kind":"use","source":21},{"line":247,"start":15323,"end":15331,"expression":"U^*U=\\id","context":1145,"kind":"use","source":21},{"line":247,"start":15338,"end":15346,"expression":"UU^*=\\id","context":1145,"kind":"use","source":21},{"line":248,"start":15369,"end":15483,"expression":"\\label{f:moving-defs}\n P_a(t)=U(t)P_a^{(0)}U(t)^*,\\qquad\n \\phi(t)=U(t)^*\\psi(t),\\qquad \\Gamma(t)=U(t)^*\\dot U(t).\n","context":1145,"kind":"use","source":21},{"line":252,"start":15515,"end":15519,"expression":"\\phi","context":1145,"kind":"use","source":21},{"line":252,"start":15543,"end":15552,"expression":"P_a^{(0)}","context":1146,"kind":"use","source":21},{"line":253,"start":15618,"end":15672,"expression":"\n \\norm{P_a(t)\\psi(t)}^2\n =\\norm{P_a^{(0)}\\phi(t)}^2.\n","context":1147,"kind":"use","source":21},{"line":259,"start":15751,"end":15765,"expression":"\\dot\\psi=K\\psi","context":1148,"kind":"use","source":21},{"line":259,"start":15772,"end":15778,"expression":"K^*=-K","context":1148,"kind":"use","source":21},{"line":260,"start":15802,"end":15970,"expression":"\\label{f:effective-generator}\n \\dot\\phi=K_{\\mathrm{eff}}\\phi,\\qquad\n K_{\\mathrm{eff}}=U^*KU-\\Gamma,\\qquad\n \\Gamma^*=-\\Gamma,\\quad K_{\\mathrm{eff}}^*=-K_{\\mathrm{eff}}.\n","context":1148,"kind":"use","source":21},{"line":265,"start":16043,"end":16047,"expression":"\\phi","context":1148,"kind":"use","source":21},{"line":265,"start":16054,"end":16070,"expression":"K_{\\mathrm{eff}}","context":1148,"kind":"use","source":21},{"line":269,"start":16181,"end":16189,"expression":"U^*U=\\id","context":1149,"kind":"use","source":21},{"line":270,"start":16202,"end":16223,"expression":"\\dot U^*U+U^*\\dot U=0","context":1149,"kind":"use","source":21},{"line":270,"start":16239,"end":16255,"expression":"\\Gamma^*=-\\Gamma","context":1149,"kind":"use","source":21},{"line":271,"start":16290,"end":16293,"expression":"U^*","context":1149,"kind":"use","source":21},{"line":271,"start":16302,"end":16322,"expression":"\\dot U^*=-\\Gamma U^*","context":1149,"kind":"use","source":21},{"line":272,"start":16337,"end":16398,"expression":"\n \\dot\\phi=\\dot U^*\\psi+U^*\\dot\\psi\n =-\\Gamma\\phi+U^*KU\\phi.\n","context":1149,"kind":"use","source":21},{"line":279,"start":16610,"end":16616,"expression":"\\Gamma","context":1150,"kind":"use","source":21},{"line":280,"start":16835,"end":16865,"expression":"H_{\\mathrm{eff}}=U^*HU-i\\Gamma","context":1150,"kind":"use","source":21},{"line":280,"start":16899,"end":16905,"expression":"\\Gamma","context":1150,"kind":"use","source":21},{"line":283,"start":17023,"end":17034,"expression":"\\psi(t)=e_1","context":1151,"kind":"use","source":21},{"line":283,"start":17038,"end":17041,"expression":"K=0","context":1151,"kind":"use","source":21},{"line":284,"start":17050,"end":17146,"expression":"\n U(t)=\\begin{pmatrix}\\cos\\theta(t)&-\\sin\\theta(t)\\\\\n \\sin\\theta(t)&\\cos\\theta(t)\\end{pmatrix}.\n","context":1152,"kind":"use","source":21},{"line":288,"start":17184,"end":17193,"expression":"P_a^{(0)}","context":1153,"kind":"use","source":21},{"line":289,"start":17198,"end":17352,"expression":"\n \\phi=\\begin{pmatrix}\\cos\\theta\\\\-\\sin\\theta\\end{pmatrix},\\quad\n \\Gamma=\\dot\\theta\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix},\\quad\n K_{\\mathrm{eff}}=-\\Gamma.\n","context":1154,"kind":"use","source":21},{"line":294,"start":17379,"end":17395,"expression":"p_1=\\cos^2\\theta","context":1155,"kind":"use","source":21},{"line":294,"start":17402,"end":17418,"expression":"p_2=\\sin^2\\theta","context":1155,"kind":"use","source":21},{"line":295,"start":17427,"end":17476,"expression":"J_{2\\leftarrow1}=\\dot\\theta\\sin(2\\theta)=\\dot p_2","context":1155,"kind":"use","source":21},{"line":299,"start":17711,"end":17722,"expression":"V=\\dot UU^*","context":1156,"kind":"introduction-passage","source":21},{"line":300,"start":17781,"end":17870,"expression":"\n \\dot P_a=[V,P_a],\\qquad\n \\frac{d}{dt}(P_bXP_a)\n =[V,P_b]XP_a+P_b\\dot XP_a+P_bX[V,P_a].\n","context":1156,"kind":"introduction-passage","source":21},{"line":307,"start":18355,"end":18365,"expression":"-U^*\\dot U","context":1157,"kind":"use","source":21},{"line":313,"start":18478,"end":18486,"expression":"\\HH=\\C^n","context":1158,"kind":"use","source":21},{"line":313,"start":18551,"end":18555,"expression":"\\rho","context":1158,"kind":"use","source":21},{"line":313,"start":18563,"end":18586,"expression":"\\operatorname{Tr}\\rho=1","context":1158,"kind":"use","source":21},{"line":314,"start":18642,"end":18703,"expression":"\\label{f:mixed-probability}\n p_a=\\operatorname{Tr}(P_a\\rho).\n","context":1158,"kind":"use","source":21},{"line":317,"start":18807,"end":18816,"expression":"P_a^2=P_a","context":1158,"kind":"use","source":21},{"line":318,"start":18824,"end":18885,"expression":"\\operatorname{Tr}(P_a\\rho)=\\operatorname{Tr}(P_a\\rho P_a)\\ge0","context":1158,"kind":"use","source":21},{"line":319,"start":18908,"end":18941,"expression":"\\sum_ap_a=\\operatorname{Tr}\\rho=1","context":1159,"kind":"use","source":21},{"line":323,"start":19094,"end":19151,"expression":"\\label{f:von-neumann}\n \\dot\\rho=-i[H,\\rho],\\qquad H=H^*.\n","context":1160,"kind":"use","source":21},{"line":327,"start":19215,"end":19319,"expression":"\\label{f:mixed-current}\n J^{(\\rho)}_{b\\leftarrow a}\n =2\\operatorname{Im}\\operatorname{Tr}(P_bHP_a\\rho).\n","context":1160,"kind":"use","source":21},{"line":333,"start":19418,"end":19422,"expression":"\\rho","context":1161,"kind":"use","source":21},{"line":334,"start":19460,"end":19568,"expression":"\n \\dot p_b=\\sum_aJ^{(\\rho)}_{b\\leftarrow a},\\qquad\n J^{(\\rho)}_{b\\leftarrow a}=-J^{(\\rho)}_{a\\leftarrow b}.\n","context":1161,"kind":"use","source":21},{"line":342,"start":19686,"end":19723,"expression":"z_{ba}=\\operatorname{Tr}(P_bHP_a\\rho)","context":1162,"kind":"use","source":21},{"line":343,"start":19812,"end":19905,"expression":"\n \\overline{z_{ba}}=\\operatorname{Tr}(\\rho P_aHP_b)\n =\\operatorname{Tr}(P_aHP_b\\rho)=z_{ab}.\n","context":1162,"kind":"use","source":21},{"line":348,"start":19963,"end":20032,"expression":"\n \\dot p_b=-i\\sum_a(z_{ba}-z_{ab})\n =\\sum_a2\\operatorname{Im}z_{ba}.\n","context":1162,"kind":"use","source":21},{"line":355,"start":20191,"end":20220,"expression":"\\rho=|\\psi\\rangle\\langle\\psi|","context":1163,"kind":"use","source":21},{"line":355,"start":20260,"end":20261,"expression":"x","context":1163,"kind":"use","source":21},{"line":355,"start":20267,"end":20286,"expression":"\\psi\\inner{\\psi}{x}","context":1163,"kind":"use","source":21},{"line":356,"start":20298,"end":20342,"expression":"\\operatorname{Tr}(M\\rho)=\\inner{\\psi}{M\\psi}","context":1163,"kind":"use","source":21},{"line":356,"start":20359,"end":20368,"expression":"M=P_bHP_a","context":1163,"kind":"use","source":21},{"line":358,"start":20538,"end":20573,"expression":"2\\operatorname{Im}(H_{ba}\\rho_{ab})","context":1164,"kind":"use","source":21},{"line":359,"start":20644,"end":20818,"expression":"\n \\rho_+=\\frac12\\begin{pmatrix}1&i\\\\-i&1\\end{pmatrix},\\qquad\n \\rho_-=\\frac12\\begin{pmatrix}1&-i\\\\i&1\\end{pmatrix},\\qquad\n \\rho_0=\\frac12\\begin{pmatrix}1&0\\\\0&1\\end{pmatrix}.\n","context":1165,"kind":"use","source":21},{"line":364,"start":20879,"end":20888,"expression":"(1/2,1/2)","context":1166,"kind":"use","source":21},{"line":364,"start":20896,"end":20956,"expression":"H=g\\left(\\begin{smallmatrix}0&1\\\\1&0\\end{smallmatrix}\\right)","context":1167,"kind":"use","source":21},{"line":364,"start":20984,"end":21000,"expression":"J_{2\\leftarrow1}","context":1168,"kind":"use","source":21},{"line":364,"start":21020,"end":21026,"expression":"g,-g,0","context":1168,"kind":"use","source":21},{"line":372,"start":21821,"end":21826,"expression":"\\HH_A","context":1169,"kind":"use","source":21},{"line":372,"start":21833,"end":21838,"expression":"\\HH_B","context":1170,"kind":"use","source":21},{"line":372,"start":21909,"end":21914,"expression":"(P_a)","context":1171,"kind":"use","source":21},{"line":372,"start":21921,"end":21928,"expression":"(Q_\\mu)","context":1172,"kind":"use","source":21},{"line":373,"start":21955,"end":21984,"expression":"\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n","context":1173,"kind":"use","source":21},{"line":377,"start":22041,"end":22092,"expression":"R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}","context":1174,"kind":"use","source":21},{"line":378,"start":22107,"end":22124,"expression":"\\id_A\\otimes\\id_B","context":1175,"kind":"use","source":21},{"line":378,"start":22149,"end":22158,"expression":"\\rho_{AB}","context":1176,"kind":"use","source":21},{"line":379,"start":22184,"end":22319,"expression":"\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n","context":1177,"kind":"use","source":21},{"line":384,"start":22401,"end":22540,"expression":"\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n","context":1178,"kind":"use","source":21},{"line":388,"start":22606,"end":22613,"expression":"(e_k)_k","context":1179,"kind":"use","source":21},{"line":388,"start":22638,"end":22643,"expression":"\\HH_B","context":1180,"kind":"use","source":21},{"line":389,"start":22708,"end":22753,"expression":"(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}","context":1181,"kind":"use","source":21},{"line":390,"start":22881,"end":22882,"expression":"x","context":1182,"kind":"use","source":21},{"line":390,"start":22922,"end":22934,"expression":"x\\otimes e_k","context":1183,"kind":"use","source":21},{"line":392,"start":23011,"end":23042,"expression":"\\rho_{AB}=\\sigma_A\\otimes\\tau_B","context":1184,"kind":"use","source":21},{"line":392,"start":23071,"end":23092,"expression":"p_{a\\mu}=p_a^Ap_\\mu^B","context":1184,"kind":"use","source":21},{"line":393,"start":23162,"end":23205,"expression":"\\Psi=(e_0\\otimes e_0+e_1\\otimes e_1)/\\sqrt2","context":1184,"kind":"use","source":21},{"line":394,"start":23233,"end":23315,"expression":"\n (p_{a\\mu})=\\begin{pmatrix}1/2&0\\\\0&1/2\\end{pmatrix},\n \\qquad p^A=p^B=(1/2,1/2).\n","context":1185,"kind":"use","source":21},{"line":398,"start":23346,"end":23371,"expression":"(\\id_A/2)\\otimes(\\id_B/2)","context":1186,"kind":"use","source":21},{"line":398,"start":23422,"end":23425,"expression":"1/4","context":1186,"kind":"use","source":21},{"line":398,"start":23545,"end":23559,"expression":"e_0\\otimes e_0","context":1186,"kind":"use","source":21},{"line":398,"start":23566,"end":23580,"expression":"e_1\\otimes e_1","context":1186,"kind":"use","source":21},{"line":398,"start":23611,"end":23615,"expression":"\\Psi","context":1186,"kind":"use","source":21},{"line":406,"start":24427,"end":24441,"expression":"(e_a)_{a\\in I}","context":1187,"kind":"use","source":21},{"line":406,"start":24447,"end":24451,"expression":"\\C^n","context":1187,"kind":"use","source":21},{"line":406,"start":24501,"end":24516,"expression":"\\gamma_{ba}\\ge0","context":1187,"kind":"use","source":21},{"line":406,"start":24523,"end":24529,"expression":"b\\ne a","context":1187,"kind":"use","source":21},{"line":407,"start":24602,"end":24747,"expression":"\\label{f:jump-operators}\n L_{ba}=\\sqrt{\\gamma_{ba}}\\,|e_b\\rangle\\langle e_a|,\n \\qquad\n \\mathcal D_L(\\rho)=L\\rho L^*-\tfrac12(L^*L\\rho+\\rho L^*L).\n","context":1187,"kind":"use","source":21},{"line":413,"start":24842,"end":24911,"expression":"\\label{f:lindblad}\n \\dot\\rho=\\sum_{b\\ne a}\\mathcal D_{L_{ba}}(\\rho).\n","context":1188,"kind":"use","source":21},{"line":419,"start":25189,"end":25202,"expression":"p_b=\\rho_{bb}","context":1189,"kind":"use","source":21},{"line":420,"start":25252,"end":25412,"expression":"\\label{f:jump-current}\n \\dot p_b=\\sum_aF_{b\\leftarrow a},\\qquad\n F_{b\\leftarrow a}=\\gamma_{ba}p_a-\\gamma_{ab}p_b,\n \\qquad F_{b\\leftarrow a}=-F_{a\\leftarrow b}.\n","context":1189,"kind":"use","source":21},{"line":429,"start":25607,"end":25613,"expression":"a\\to b","context":1190,"kind":"use","source":21},{"line":430,"start":25618,"end":25697,"expression":"\n L_{ba}\\rho L_{ba}^*=\\gamma_{ba}p_aP_b,\\qquad\n L_{ba}^*L_{ba}=\\gamma_{ba}P_a.\n","context":1190,"kind":"use","source":21},{"line":434,"start":25722,"end":25736,"expression":"\\gamma_{ba}p_a","context":1190,"kind":"use","source":21},{"line":434,"start":25742,"end":25743,"expression":"b","context":1190,"kind":"use","source":21},{"line":434,"start":25815,"end":25816,"expression":"a","context":1190,"kind":"use","source":21},{"line":437,"start":26155,"end":26161,"expression":"i\\ne j","context":1191,"kind":"use","source":21},{"line":437,"start":26169,"end":26197,"expression":"r_i=\\sum_{b\\ne i}\\gamma_{bi}","context":1191,"kind":"use","source":21},{"line":437,"start":26226,"end":26227,"expression":"i","context":1191,"kind":"use","source":21},{"line":438,"start":26352,"end":26422,"expression":"\\label{f:coherence-decay}\n \\dot\\rho_{ij}=-\\tfrac12(r_i+r_j)\\rho_{ij}.\n","context":1191,"kind":"use","source":21},{"line":441,"start":26455,"end":26497,"expression":"\\rho_{ij}(t)=e^{-(r_i+r_j)t/2}\\rho_{ij}(0)","context":1192,"kind":"use","source":21},{"line":444,"start":26677,"end":26704,"expression":"\\gamma_{21}=\\gamma_{12}=k>0","context":1193,"kind":"use","source":21},{"line":444,"start":26711,"end":26720,"expression":"p_1+p_2=1","context":1193,"kind":"use","source":21},{"line":445,"start":26730,"end":26854,"expression":"\n \\dot p_1=k(1-2p_1),\\qquad\n p_1(t)=\\tfrac12+\\bigl(p_1(0)-\\tfrac12\\bigr)e^{-2kt},\n \\qquad \\rho_{12}(t)=e^{-kt}\\rho_{12}(0).\n","context":1193,"kind":"use","source":21},{"line":450,"start":27131,"end":27141,"expression":"-i[H,\\rho]","context":1194,"kind":"use","source":21},{"line":453,"start":27496,"end":27499,"expression":"A,B","context":1195,"kind":"use","source":21},{"line":454,"start":27512,"end":27581,"expression":"\n \\sum_{b\\in C_B}\\gamma_{ba}=q_{BA}\n \\quad\\text{for every }a\\in C_A.\n","context":1195,"kind":"use","source":21},{"line":458,"start":27619,"end":27640,"expression":"q_{BA}p_A^{\\mathrm c}","context":1195,"kind":"use","source":21},{"line":459,"start":27647,"end":27726,"expression":"\\dot p_B^{\\mathrm c}=\\sum_{A\\ne B}(q_{BA}p_A^{\\mathrm c}-q_{AB}p_B^{\\mathrm c})","context":1195,"kind":"use","source":21},{"line":462,"start":28168,"end":28174,"expression":"\\rho_+","context":1196,"kind":"use","source":21},{"line":462,"start":28181,"end":28187,"expression":"\\rho_-","context":1196,"kind":"use","source":21},{"line":462,"start":28227,"end":28230,"expression":"p_2","context":1196,"kind":"use","source":21},{"line":468,"start":28491,"end":28499,"expression":"p_a(t)>0","context":1197,"kind":"use","source":21},{"line":468,"start":28542,"end":28574,"expression":"\\dot p_b=\\sum_aJ_{b\\leftarrow a}","context":1197,"kind":"use","source":21},{"line":468,"start":28582,"end":28583,"expression":"J","context":1197,"kind":"use","source":21},{"line":468,"start":28636,"end":28637,"expression":"f","context":1197,"kind":"use","source":21},{"line":469,"start":28682,"end":28811,"expression":"\\label{f:functional-current}\n \\frac{d}{dt}\\sum_b f(p_b)\n =\\frac12\\sum_{b,a}J_{b\\leftarrow a}\n       \\bigl(f'(p_b)-f'(p_a)\\bigr).\n","context":1197,"kind":"use","source":21},{"line":476,"start":29070,"end":29093,"expression":"S(p)=-\\sum_ap_a\\log p_a","context":1198,"kind":"use","source":21},{"line":477,"start":29123,"end":29213,"expression":"\\label{f:entropy-current}\n \\dot S=-\\frac12\\sum_{b,a}J_{b\\leftarrow a}\\log\\frac{p_b}{p_a}.\n","context":1199,"kind":"use","source":21},{"line":480,"start":29312,"end":29313,"expression":"q","context":1199,"kind":"use","source":21},{"line":480,"start":29331,"end":29346,"expression":"M_q=\\sum_ap_a^q","context":1199,"kind":"use","source":21},{"line":481,"start":29365,"end":29436,"expression":"\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n","context":1199,"kind":"use","source":21},{"line":485,"start":29444,"end":29449,"expression":"q\\ne1","context":1199,"kind":"use","source":21},{"line":485,"start":29473,"end":29491,"expression":"H_q=\\log M_q/(1-q)","context":1200,"kind":"use","source":21},{"line":486,"start":29514,"end":29542,"expression":"\\dot H_q=\\dot M_q/((1-q)M_q)","context":1201,"kind":"use","source":21},{"line":486,"start":29620,"end":29625,"expression":"M_q>0","context":1202,"kind":"use","source":21},{"line":490,"start":30248,"end":30255,"expression":"\\pi_a>0","context":1203,"kind":"use","source":21},{"line":491,"start":30321,"end":30384,"expression":"\\label{f:detailed-balance}\n \\gamma_{ba}\\pi_a=\\gamma_{ab}\\pi_b.\n","context":1203,"kind":"use","source":21},{"line":494,"start":30426,"end":30439,"expression":"u_a=p_a/\\pi_a","context":1203,"kind":"use","source":21},{"line":494,"start":30459,"end":30482,"expression":"c_{ba}=\\gamma_{ba}\\pi_a","context":1203,"kind":"use","source":21},{"line":494,"start":30491,"end":30508,"expression":"c_{ba}=c_{ab}\\ge0","context":1203,"kind":"use","source":21},{"line":495,"start":30515,"end":30548,"expression":"F_{b\\leftarrow a}=c_{ba}(u_a-u_b)","context":1203,"kind":"use","source":21},{"line":500,"start":30795,"end":30961,"expression":"\\label{f:dissipation}\n D(p\\Vert\\pi)&=\\sum_ap_a\\log\\frac{p_a}{\\pi_a},\\\\\n \\frac{d}{dt}D(p\\Vert\\pi)\n &=-\\frac12\\sum_{b,a}c_{ba}(u_b-u_a)(\\log u_b-\\log u_a)\\le0.\n \\notag\n","context":1204,"kind":"use","source":21},{"line":509,"start":31022,"end":31025,"expression":"\\pi","context":1205,"kind":"use","source":21},{"line":510,"start":31110,"end":31164,"expression":"\\tfrac12\\sum_{b,a}F_{b\\leftarrow a}(\\log u_b-\\log u_a)","context":1205,"kind":"use","source":21},{"line":511,"start":31206,"end":31207,"expression":"F","context":1205,"kind":"use","source":21},{"line":511,"start":31242,"end":31252,"expression":"(0,\\infty)","context":1205,"kind":"use","source":21},{"line":512,"start":31272,"end":31300,"expression":"(u_b-u_a)(\\log u_b-\\log u_a)","context":1205,"kind":"use","source":21},{"line":515,"start":31487,"end":31495,"expression":"c_{ba}=0","context":1206,"kind":"use","source":21},{"line":515,"start":31537,"end":31545,"expression":"c_{ba}>0","context":1206,"kind":"use","source":21},{"line":515,"start":31581,"end":31588,"expression":"u_b=u_a","context":1206,"kind":"use","source":21},{"line":515,"start":31811,"end":31816,"expression":"p=\\pi","context":1207,"kind":"use","source":21},{"line":6,"start":607,"end":608,"expression":"K","context":1208,"kind":"use","source":22},{"line":6,"start":617,"end":623,"expression":"K^*=-K","context":1208,"kind":"use","source":22},{"line":6,"start":658,"end":663,"expression":"K=-iH","context":1208,"kind":"use","source":22},{"line":6,"start":707,"end":713,"expression":"J_{ba}","context":1208,"kind":"use","source":22},{"line":9,"start":762,"end":765,"expression":"\\HH","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":9,"start":803,"end":804,"expression":"A","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":10,"start":891,"end":935,"expression":"\n F_a=F_a^*\\geq0,\\qquad \\sum_{a\\in A}F_a=I.\n","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":13,"start":961,"end":965,"expression":"\\psi","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":13,"start":971,"end":996,"expression":"p_a=\\inner{\\psi}{F_a\\psi}","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":13,"start":1031,"end":1046,"expression":"0\\leq F_a\\leq I","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":13,"start":1053,"end":1061,"expression":"p_a\\geq0","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":13,"start":1068,"end":1090,"expression":"\\sum_a p_a=\\norm\\psi^2","context":1209,"kind":"definition-passage","definition_title":"Finite effect atlas","source":22},{"line":16,"start":1131,"end":1138,"expression":"F_a=P_a","context":1210,"kind":"use","source":22},{"line":16,"start":1145,"end":1165,"expression":"p_a=\\norm{P_a\\psi}^2","context":1210,"kind":"use","source":22},{"line":16,"start":1181,"end":1184,"expression":"p_a","context":1210,"kind":"use","source":22},{"line":16,"start":1194,"end":1210,"expression":"\\norm{F_a\\psi}^2","context":1210,"kind":"use","source":22},{"line":16,"start":1237,"end":1260,"expression":"\\inner{\\psi}{F_a^2\\psi}","context":1210,"kind":"use","source":22},{"line":16,"start":1313,"end":1322,"expression":"F_a-F_a^2","context":1210,"kind":"use","source":22},{"line":19,"start":1379,"end":1383,"expression":"\\C^2","context":1211,"kind":"use","source":22},{"line":20,"start":1393,"end":1515,"expression":"\n F_+=\\begin{pmatrix}4/5&0\\\\0&1/5\\end{pmatrix},\\qquad\n F_-=\\begin{pmatrix}1/5&0\\\\0&4/5\\end{pmatrix},\\qquad \\psi=\\binom10.\n","context":1212,"kind":"use","source":22},{"line":24,"start":1540,"end":1543,"expression":"4/5","context":1213,"kind":"use","source":22},{"line":24,"start":1550,"end":1553,"expression":"1/5","context":1213,"kind":"use","source":22},{"line":24,"start":1604,"end":1609,"expression":"16/25","context":1213,"kind":"use","source":22},{"line":24,"start":1616,"end":1620,"expression":"1/25","context":1213,"kind":"use","source":22},{"line":24,"start":1637,"end":1642,"expression":"17/25","context":1213,"kind":"use","source":22},{"line":24,"start":1672,"end":1692,"expression":"F_+\\psi+F_-\\psi=\\psi","context":1213,"kind":"use","source":22},{"line":29,"start":1924,"end":1938,"expression":"\\dot\\psi=K\\psi","context":1214,"kind":"use","source":22},{"line":30,"start":1964,"end":2116,"expression":"\n J^{F}_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}\n             -\\operatorname{Re}\\inner{F_a\\psi}{KF_b\\psi}.\n \\label{eq:foundation-effect-current}\n","context":1214,"kind":"use","source":22},{"line":36,"start":2221,"end":2277,"expression":"\n J^F_{ba}=2\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}.\n","context":1214,"kind":"use","source":22},{"line":43,"start":2458,"end":2510,"expression":"\n \\dot p_b=\\sum_a J^F_{ba},\\qquad \\sum_b\\dot p_b=0.\n","context":1215,"kind":"use","source":22},{"line":46,"start":2517,"end":2527,"expression":"F_b=F_b(t)","context":1216,"kind":"use","source":22},{"line":47,"start":2594,"end":2685,"expression":"\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n","context":1217,"kind":"use","source":22},{"line":53,"start":2741,"end":2744,"expression":"F_b","context":1217,"kind":"use","source":22},{"line":54,"start":2773,"end":2825,"expression":"\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n","context":1217,"kind":"use","source":22},{"line":57,"start":2916,"end":2917,"expression":"a","context":1218,"kind":"use","source":22},{"line":57,"start":2930,"end":2941,"expression":"\\sum_aF_a=I","context":1219,"kind":"use","source":22},{"line":58,"start":2951,"end":3057,"expression":"\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n","context":1220,"kind":"use","source":22},{"line":62,"start":3155,"end":3158,"expression":"s_b","context":1221,"kind":"use","source":22},{"line":62,"start":3188,"end":3204,"expression":"\\sum_b\\dot F_b=0","context":1222,"kind":"use","source":22},{"line":65,"start":3286,"end":3294,"expression":"r:A\\to B","context":1223,"kind":"use","source":22},{"line":65,"start":3337,"end":3365,"expression":"G_\\beta=\\sum_{r(a)=\\beta}F_a","context":1223,"kind":"use","source":22},{"line":66,"start":3389,"end":3513,"expression":"\n p^{\\rm c}_\\beta=\\sum_{r(a)=\\beta}p_a,\\qquad\n J^{\\rm c}_{\\beta\\alpha}\n =\\sum_{\\substack{r(b)=\\beta\\\\r(a)=\\alpha}}J^F_{ba}.\n","context":1224,"kind":"use","source":22},{"line":71,"start":3636,"end":3639,"expression":"J^F","context":1225,"kind":"use","source":22},{"line":75,"start":3898,"end":3903,"expression":"\\KK_a","context":1226,"kind":"use","source":22},{"line":75,"start":3923,"end":3938,"expression":"V_a:\\HH\\to\\KK_a","context":1227,"kind":"use","source":22},{"line":76,"start":3952,"end":3972,"expression":"\n \\sum_aV_a^*V_a=I.\n","context":1228,"kind":"use","source":22},{"line":79,"start":4002,"end":4030,"expression":"V:\\HH\\to\\KK=\\bigoplus_a\\KK_a","context":1228,"kind":"use","source":22},{"line":79,"start":4036,"end":4053,"expression":"V\\psi=(V_a\\psi)_a","context":1228,"kind":"use","source":22},{"line":79,"start":4083,"end":4108,"expression":"V^*(y_a)_a=\\sum_aV_a^*y_a","context":1228,"kind":"use","source":22},{"line":80,"start":4126,"end":4167,"expression":"\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n","context":1228,"kind":"use","source":22},{"line":83,"start":4176,"end":4177,"expression":"R","context":1228,"kind":"use","source":22},{"line":83,"start":4227,"end":4228,"expression":"V","context":1228,"kind":"use","source":22},{"line":83,"start":4256,"end":4259,"expression":"\\KK","context":1228,"kind":"use","source":22},{"line":83,"start":4266,"end":4271,"expression":"\\Pi_a","context":1228,"kind":"use","source":22},{"line":84,"start":4329,"end":4404,"expression":"\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n","context":1229,"kind":"use","source":22},{"line":88,"start":4469,"end":4478,"expression":"\\KK_a=\\HH","context":1230,"kind":"use","source":22},{"line":88,"start":4485,"end":4498,"expression":"V_a=F_a^{1/2}","context":1231,"kind":"use","source":22},{"line":90,"start":4664,"end":4683,"expression":"X\\in\\mathcal B(\\HH)","context":1232,"kind":"use","source":22},{"line":90,"start":4708,"end":4728,"expression":"X_{ba}^{V}=V_bXV_a^*","context":1233,"kind":"use","source":22},{"line":91,"start":4792,"end":4872,"expression":"\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n","context":1234,"kind":"use","source":22},{"line":95,"start":4908,"end":4926,"expression":"\\widetilde X=VXV^*","context":1235,"kind":"use","source":22},{"line":95,"start":4974,"end":4980,"expression":"V^*V=I","context":1235,"kind":"use","source":22},{"line":95,"start":5000,"end":5001,"expression":"R","context":1235,"kind":"use","source":22},{"line":95,"start":5009,"end":5016,"expression":"I_{\\KK}","context":1235,"kind":"use","source":22},{"line":96,"start":5121,"end":5200,"expression":"\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n","context":1236,"kind":"use","source":22},{"line":100,"start":5262,"end":5271,"expression":"A=B=\\Pi_a","context":1237,"kind":"use","source":22},{"line":101,"start":5295,"end":5390,"expression":"\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n","context":1238,"kind":"use","source":22},{"line":106,"start":5554,"end":5568,"expression":"F_a-F_a^2\\geq0","context":1239,"kind":"use","source":22},{"line":108,"start":5631,"end":5649,"expression":"\\widetilde K=VKV^*","context":1240,"kind":"use","source":22},{"line":108,"start":5677,"end":5713,"expression":"\\frac{d}{dt}V\\psi=\\widetilde K V\\psi","context":1240,"kind":"use","source":22},{"line":109,"start":5765,"end":5884,"expression":"\n 2\\operatorname{Re}\\inner{\\Pi_bV\\psi}{\\widetilde K\\Pi_aV\\psi}\n =2\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}=J^F_{ba}.\n","context":1240,"kind":"use","source":22},{"line":115,"start":6165,"end":6197,"expression":"F_+=\\operatorname{diag}(4/5,1/5)","context":1241,"kind":"use","source":22},{"line":115,"start":6204,"end":6213,"expression":"F_-=I-F_+","context":1241,"kind":"use","source":22},{"line":125,"start":6722,"end":6736,"expression":"(P_a)_{a\\in A}","context":1242,"kind":"introduction-passage","source":22},{"line":125,"start":6811,"end":6812,"expression":"I","context":1242,"kind":"introduction-passage","source":22},{"line":125,"start":6827,"end":6837,"expression":"F\\subset A","context":1242,"kind":"introduction-passage","source":22},{"line":125,"start":6847,"end":6867,"expression":"S_F=\\sum_{a\\in F}P_a","context":1242,"kind":"introduction-passage","source":22},{"line":125,"start":6897,"end":6906,"expression":"S_Fx\\to x","context":1242,"kind":"introduction-passage","source":22},{"line":125,"start":6924,"end":6931,"expression":"x\\in\\HH","context":1242,"kind":"introduction-passage","source":22},{"line":126,"start":6997,"end":7079,"expression":"\n \\norm{S_Fx}^2=\\sum_{a\\in F}\\norm{P_ax}^2,\\qquad\n \\sum_a\\norm{P_ax}^2=\\norm x^2.\n","context":1242,"kind":"introduction-passage","source":22},{"line":132,"start":7164,"end":7186,"expression":"H:D(H)\\subset\\HH\\to\\HH","context":1243,"kind":"introduction-passage","source":22},{"line":132,"start":7239,"end":7247,"expression":"HP_a\\psi","context":1243,"kind":"introduction-passage","source":22},{"line":133,"start":7297,"end":7359,"expression":"\n P_aD(H)\\subset D(H).\n \\label{eq:foundation-domain-preserve}\n","context":1243,"kind":"introduction-passage","source":22},{"line":137,"start":7382,"end":7415,"expression":"\\norm x_H^2=\\norm x^2+\\norm{Hx}^2","context":1243,"kind":"introduction-passage","source":22},{"line":138,"start":7485,"end":7599,"expression":"\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n","context":1243,"kind":"introduction-passage","source":22},{"line":142,"start":7634,"end":7643,"expression":"S_Fx\\to x","context":1244,"kind":"use","source":22},{"line":142,"start":7650,"end":7661,"expression":"HS_Fx\\to Hx","context":1245,"kind":"use","source":22},{"line":142,"start":7672,"end":7675,"expression":"\\HH","context":1246,"kind":"use","source":22},{"line":142,"start":7750,"end":7751,"expression":"H","context":1247,"kind":"use","source":22},{"line":145,"start":7823,"end":7841,"expression":"\\ell^2(\\N_{\\geq1})","context":1248,"kind":"use","source":22},{"line":145,"start":7849,"end":7858,"expression":"He_n=ne_n","context":1248,"kind":"use","source":22},{"line":145,"start":7915,"end":7924,"expression":"x_N=e_N/N","context":1248,"kind":"use","source":22},{"line":145,"start":7935,"end":7953,"expression":"\\norm{x_N}=1/N\\to0","context":1248,"kind":"use","source":22},{"line":145,"start":7963,"end":7976,"expression":"\\norm{Hx_N}=1","context":1248,"kind":"use","source":22},{"line":145,"start":8109,"end":8117,"expression":"Hx_N=e_N","context":1248,"kind":"use","source":22},{"line":150,"start":8198,"end":8209,"expression":"(\\lambda_n)","context":1249,"kind":"use","source":22},{"line":151,"start":8245,"end":8364,"expression":"\n D(H_\\lambda)=\\left\\{c\\in\\ell^2:\n       \\sum_n\\lambda_n^2|c_n|^2<\\infty\\right\\},\\qquad\n (H_\\lambda c)_n=\\lambda_nc_n.\n","context":1249,"kind":"use","source":22},{"line":159,"start":8626,"end":8635,"expression":"H_\\lambda","context":1250,"kind":"use","source":22},{"line":159,"start":8704,"end":8716,"expression":"D(H_\\lambda)","context":1250,"kind":"use","source":22},{"line":162,"start":8755,"end":8774,"expression":"y\\in D(H_\\lambda^*)","context":1250,"kind":"use","source":22},{"line":162,"start":8781,"end":8795,"expression":"H_\\lambda^*y=z","context":1250,"kind":"use","source":22},{"line":162,"start":8836,"end":8839,"expression":"e_n","context":1250,"kind":"use","source":22},{"line":162,"start":8848,"end":8864,"expression":"z_n=\\lambda_ny_n","context":1250,"kind":"use","source":22},{"line":162,"start":8874,"end":8884,"expression":"z\\in\\ell^2","context":1250,"kind":"use","source":22},{"line":162,"start":8901,"end":8902,"expression":"y","context":1250,"kind":"use","source":22},{"line":162,"start":8916,"end":8928,"expression":"D(H_\\lambda)","context":1250,"kind":"use","source":22},{"line":162,"start":9015,"end":9040,"expression":"S_Nc=\\sum_{n\\leq N}c_ne_n","context":1251,"kind":"use","source":22},{"line":163,"start":9045,"end":9127,"expression":"\n \\norm{c-S_Nc}_{H_\\lambda}^2\n =\\sum_{n>N}(1+\\lambda_n^2)|c_n|^2\\longrightarrow0.\n","context":1252,"kind":"use","source":22},{"line":169,"start":9157,"end":9166,"expression":"\\lambda_n","context":1253,"kind":"use","source":22},{"line":169,"start":9209,"end":9238,"expression":"\\sum_n\\lambda_n|c_n|^2<\\infty","context":1253,"kind":"use","source":22},{"line":169,"start":9264,"end":9279,"expression":"H_\\lambda^{1/2}","context":1253,"kind":"use","source":22},{"line":169,"start":9311,"end":9320,"expression":"H_\\lambda","context":1253,"kind":"use","source":22},{"line":169,"start":9329,"end":9340,"expression":"\\lambda_n=n","context":1253,"kind":"use","source":22},{"line":169,"start":9347,"end":9357,"expression":"c_n=n^{-s}","context":1253,"kind":"use","source":22},{"line":170,"start":9387,"end":9529,"expression":"\n \\begin{array}{c|c}\n \\text{membership}&\\text{condition}\\\\ \\hline\n \\ell^2&s>1/2\\\\\n D(H_\\lambda^{1/2})&s>1\\\\\n D(H_\\lambda)&s>3/2.\n \\end{array}\n","context":1254,"kind":"use","source":22},{"line":178,"start":9741,"end":9756,"expression":"\\lambda_n=n+1/2","context":1255,"kind":"use","source":22},{"line":178,"start":9763,"end":9769,"expression":"n\\geq0","context":1255,"kind":"use","source":22},{"line":178,"start":9823,"end":9824,"expression":"n","context":1255,"kind":"use","source":22},{"line":178,"start":9830,"end":9833,"expression":"n+1","context":1255,"kind":"use","source":22},{"line":180,"start":9904,"end":9913,"expression":"He_n=ne_n","context":1256,"kind":"use","source":22},{"line":180,"start":9917,"end":9923,"expression":"n\\geq1","context":1256,"kind":"use","source":22},{"line":180,"start":9940,"end":9950,"expression":"a_n=n^{-s}","context":1256,"kind":"use","source":22},{"line":180,"start":10101,"end":10102,"expression":"p","context":1256,"kind":"use","source":22},{"line":184,"start":10221,"end":10236,"expression":"\\psi(t)\\in D(H)","context":1257,"kind":"use","source":22},{"line":184,"start":10271,"end":10274,"expression":"\\HH","context":1258,"kind":"use","source":22},{"line":184,"start":10291,"end":10307,"expression":"\\dot\\psi=-iH\\psi","context":1259,"kind":"use","source":22},{"line":185,"start":10347,"end":10430,"expression":"\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n","context":1260,"kind":"use","source":22},{"line":189,"start":10449,"end":10463,"expression":"J_{ba}=-J_{ab}","context":1260,"kind":"use","source":22},{"line":190,"start":10517,"end":10570,"expression":"\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n","context":1260,"kind":"use","source":22},{"line":193,"start":10622,"end":10639,"expression":"HS_F\\psi\\to H\\psi","context":1261,"kind":"use","source":22},{"line":193,"start":10740,"end":10741,"expression":"H","context":1262,"kind":"use","source":22},{"line":194,"start":10811,"end":10866,"expression":"\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n","context":1263,"kind":"use","source":22},{"line":197,"start":10887,"end":10888,"expression":"a","context":1264,"kind":"use","source":22},{"line":197,"start":10904,"end":10949,"expression":"\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}","context":1265,"kind":"use","source":22},{"line":198,"start":11012,"end":11241,"expression":"\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|\\\\\n &\\leq\\left(\\sum_a\\norm{P_aHP_b\\psi}^2\\right)^{1/2}\n       \\left(\\sum_a\\norm{P_a\\psi}^2\\right)^{1/2}\n =\\norm{HP_b\\psi}\\norm\\psi.\n","context":1266,"kind":"use","source":22},{"line":205,"start":11268,"end":11289,"expression":"\\dot p_b=\\sum_aJ_{ba}","context":1267,"kind":"use","source":22},{"line":206,"start":11398,"end":11637,"expression":"\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{P_b\\psi}\\norm{HS_F\\psi-H\\psi},\\label{eq:foundation-graph-row}\\\\\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{HP_b\\psi}\\norm{S_F\\psi-\\psi}.\\label{eq:foundation-strong-row}\n","context":1268,"kind":"use","source":22},{"line":212,"start":11792,"end":11800,"expression":"(J_{ba})","context":1269,"kind":"use","source":22},{"line":216,"start":11880,"end":11908,"expression":"A=\\bigsqcup_{\\alpha}C_\\alpha","context":1270,"kind":"use","source":22},{"line":216,"start":11927,"end":11959,"expression":"Q_\\alpha=\\sum_{a\\in C_\\alpha}P_a","context":1270,"kind":"use","source":22},{"line":217,"start":12020,"end":12082,"expression":"\n \\norm{Q_\\alpha\\psi}^2=\\sum_{a\\in C_\\alpha}\\norm{P_a\\psi}^2.\n","context":1270,"kind":"use","source":22},{"line":220,"start":12118,"end":12131,"expression":"HQ_\\alpha\\psi","context":1270,"kind":"use","source":22},{"line":220,"start":12156,"end":12181,"expression":"Q_\\alpha D(H)\\subset D(H)","context":1270,"kind":"use","source":22},{"line":223,"start":12384,"end":12592,"expression":"\n J^{\\rm c}_{\\beta\\alpha}\n &=2\\operatorname{Im}\\inner{Q_\\beta\\psi}{HQ_\\alpha\\psi}\\\\\n &=\\sum_{b\\in C_\\beta}\\left(\\sum_{a\\in C_\\alpha}J_{ba}\\right)\n =\\sum_{a\\in C_\\alpha}\\left(\\sum_{b\\in C_\\beta}J_{ba}\\right).\n","context":1271,"kind":"use","source":22},{"line":229,"start":12633,"end":12634,"expression":"a","context":1272,"kind":"use","source":22},{"line":229,"start":12643,"end":12644,"expression":"H","context":1272,"kind":"use","source":22},{"line":229,"start":12652,"end":12659,"expression":"P_b\\psi","context":1272,"kind":"use","source":22},{"line":229,"start":12688,"end":12701,"expression":"\\sum_bP_b\\psi","context":1272,"kind":"use","source":22},{"line":229,"start":12726,"end":12739,"expression":"HQ_\\alpha\\psi","context":1272,"kind":"use","source":22},{"line":229,"start":12851,"end":12873,"expression":"C_\\beta\\times C_\\alpha","context":1272,"kind":"use","source":22},{"line":231,"start":13036,"end":13051,"expression":"a_{2n,2n+1}=1/n","context":1273,"kind":"use","source":22},{"line":231,"start":13055,"end":13071,"expression":"a_{2n+1,2n}=-1/n","context":1273,"kind":"use","source":22},{"line":231,"start":13078,"end":13084,"expression":"n\\geq1","context":1273,"kind":"use","source":22},{"line":233,"start":13404,"end":13427,"expression":"\\norm{Q_\\beta\\psi(t)}^2","context":1274,"kind":"use","source":22},{"line":233,"start":13582,"end":13598,"expression":"\\norm{\\psi(t)}^2","context":1274,"kind":"use","source":22},{"line":242,"start":14148,"end":14159,"expression":"(\\lambda_n)","context":1275,"kind":"use","source":22},{"line":243,"start":14184,"end":14272,"expression":"\n (U(t)c)_n=e^{-it\\lambda_n}c_n,\\qquad c\\in\\ell^2.\n \\label{eq:foundation-spectral-flow}\n","context":1275,"kind":"use","source":22},{"line":247,"start":14334,"end":14338,"expression":"U(t)","context":1275,"kind":"use","source":22},{"line":247,"start":14364,"end":14379,"expression":"U(t+s)=U(t)U(s)","context":1275,"kind":"use","source":22},{"line":247,"start":14387,"end":14399,"expression":"U(-t)=U(t)^*","context":1275,"kind":"use","source":22},{"line":247,"start":14484,"end":14485,"expression":"c","context":1275,"kind":"use","source":22},{"line":247,"start":14519,"end":14525,"expression":"\\ell^2","context":1275,"kind":"use","source":22},{"line":250,"start":14727,"end":14741,"expression":"t\\mapsto U(t)c","context":1276,"kind":"use","source":22},{"line":250,"start":14765,"end":14771,"expression":"\\ell^2","context":1276,"kind":"use","source":22},{"line":250,"start":14822,"end":14839,"expression":"c\\in D(H_\\lambda)","context":1276,"kind":"use","source":22},{"line":251,"start":14860,"end":14940,"expression":"\n \\frac{d}{dt}U(t)c=-iH_\\lambda U(t)c,\n \\qquad H_\\lambda U(t)c=U(t)H_\\lambda c.\n","context":1276,"kind":"use","source":22},{"line":257,"start":14979,"end":14996,"expression":"c\\in D(H_\\lambda)","context":1276,"kind":"use","source":22},{"line":257,"start":15045,"end":15075,"expression":"-i\\lambda_ne^{-it\\lambda_n}c_n","context":1276,"kind":"use","source":22},{"line":257,"start":15100,"end":15118,"expression":"|e^{-iu}-1|\\leq|u|","context":1277,"kind":"use","source":22},{"line":257,"start":15154,"end":15168,"expression":"|\\lambda_nc_n|","context":1278,"kind":"use","source":22},{"line":257,"start":15224,"end":15243,"expression":"4\\lambda_n^2|c_n|^2","context":1279,"kind":"use","source":22},{"line":257,"start":15321,"end":15327,"expression":"\\ell^2","context":1280,"kind":"use","source":22},{"line":259,"start":15350,"end":15356,"expression":"\\ell^2","context":1281,"kind":"use","source":22},{"line":259,"start":15370,"end":15371,"expression":"d","context":1281,"kind":"use","source":22},{"line":260,"start":15419,"end":15457,"expression":"\n d_n=-i\\lambda_ne^{-it\\lambda_n}c_n.\n","context":1281,"kind":"use","source":22},{"line":263,"start":15467,"end":15477,"expression":"d\\in\\ell^2","context":1281,"kind":"use","source":22},{"line":263,"start":15489,"end":15520,"expression":"\\sum_n\\lambda_n^2|c_n|^2<\\infty","context":1281,"kind":"use","source":22},{"line":266,"start":15787,"end":15788,"expression":"w","context":1282,"kind":"use","source":22},{"line":266,"start":15801,"end":15821,"expression":"\\dot w=-iH_\\lambda w","context":1282,"kind":"use","source":22},{"line":267,"start":15841,"end":15913,"expression":"\n \\frac{d}{dt}\\norm w^2\n =2\\operatorname{Re}\\inner{w}{-iH_\\lambda w}=0.\n","context":1282,"kind":"use","source":22},{"line":274,"start":16178,"end":16227,"expression":"\n \\norm{U(t)c-U(s)c}\\leq|t-s|\\norm{H_\\lambda c}.\n","context":1283,"kind":"use","source":22},{"line":279,"start":16612,"end":16618,"expression":"n^{-2}","context":1284,"kind":"use","source":22},{"line":279,"start":16625,"end":16634,"expression":"He_n=ne_n","context":1284,"kind":"use","source":22},{"line":283,"start":16825,"end":16829,"expression":"U(t)","context":1285,"kind":"use","source":22},{"line":283,"start":16848,"end":16858,"expression":"c\\in\\ell^2","context":1285,"kind":"use","source":22},{"line":284,"start":16877,"end":16949,"expression":"\n \\norm{U(t)c-S_NU(t)c}=\\norm{c-S_Nc}.\n \\label{eq:foundation-time-tail}\n","context":1285,"kind":"use","source":22},{"line":288,"start":17011,"end":17017,"expression":"t\\in\\R","context":1285,"kind":"use","source":22},{"line":288,"start":17101,"end":17118,"expression":"c\\in D(H_\\lambda)","context":1285,"kind":"use","source":22},{"line":289,"start":17227,"end":17357,"expression":"\n \\norm{\\frac{d}{dt}U(t)c-\n       \\frac{d}{dt}S_NU(t)c}\n =\\norm{H_\\lambda c-S_NH_\\lambda c}.\n \\label{eq:foundation-velocity-tail}\n","context":1286,"kind":"use","source":22},{"line":298,"start":17566,"end":17577,"expression":"\\lambda_n=n","context":1287,"kind":"use","source":22},{"line":298,"start":17586,"end":17593,"expression":"c_n=C/n","context":1287,"kind":"use","source":22},{"line":298,"start":17601,"end":17631,"expression":"C=(\\sum_{n\\geq1}n^{-2})^{-1/2}","context":1287,"kind":"use","source":22},{"line":298,"start":17656,"end":17662,"expression":"\\ell^2","context":1287,"kind":"use","source":22},{"line":298,"start":17679,"end":17691,"expression":"D(H_\\lambda)","context":1287,"kind":"use","source":22},{"line":298,"start":17869,"end":17875,"expression":"\\ell^2","context":1287,"kind":"use","source":22},{"line":298,"start":17952,"end":17960,"expression":"C\\sqrt N","context":1288,"kind":"use","source":22},{"line":301,"start":18026,"end":18035,"expression":"c_n=C/n^2","context":1289,"kind":"use","source":22},{"line":302,"start":18068,"end":18171,"expression":"\n \\norm{c-S_Nc}^2=C^2\\sum_{n>N}n^{-4},\\qquad\n \\norm{H_\\lambda c-S_NH_\\lambda c}^2=C^2\\sum_{n>N}n^{-2}.\n","context":1289,"kind":"use","source":22},{"line":306,"start":18193,"end":18204,"expression":"O(N^{-3/2})","context":1289,"kind":"use","source":22},{"line":306,"start":18243,"end":18254,"expression":"O(N^{-1/2})","context":1289,"kind":"use","source":22},{"line":313,"start":18472,"end":18475,"expression":"S_N","context":1290,"kind":"use","source":22},{"line":313,"start":18524,"end":18525,"expression":"I","context":1290,"kind":"use","source":22},{"line":313,"start":18584,"end":18585,"expression":"X","context":1290,"kind":"use","source":22},{"line":313,"start":18621,"end":18628,"expression":"S_NXS_N","context":1290,"kind":"use","source":22},{"line":313,"start":18677,"end":18678,"expression":"X","context":1290,"kind":"use","source":22},{"line":318,"start":18819,"end":18820,"expression":"X","context":1291,"kind":"use","source":22},{"line":318,"start":18835,"end":18847,"expression":"S_NXS_N\\to X","context":1291,"kind":"use","source":22},{"line":319,"start":18869,"end":18870,"expression":"X","context":1291,"kind":"use","source":22},{"line":319,"start":18885,"end":18905,"expression":"\\norm{S_NXS_N-X}\\to0","context":1291,"kind":"use","source":22},{"line":320,"start":18935,"end":18936,"expression":"X","context":1291,"kind":"use","source":22},{"line":320,"start":18960,"end":18982,"expression":"\\norm{S_NXS_N-X}_2\\to0","context":1291,"kind":"use","source":22},{"line":325,"start":19073,"end":19107,"expression":"\n S_NXS_N-X=S_NX(S_N-I)+(S_N-I)X.\n","context":1291,"kind":"use","source":22},{"line":328,"start":19117,"end":19132,"expression":"\\norm{S_N}\\leq1","context":1291,"kind":"use","source":22},{"line":328,"start":19159,"end":19160,"expression":"x","context":1292,"kind":"use","source":22},{"line":329,"start":19184,"end":19227,"expression":"\n \\norm X\\norm{(S_N-I)x}+\\norm{(S_N-I)Xx},\n","context":1293,"kind":"use","source":22},{"line":334,"start":19265,"end":19266,"expression":"X","context":1294,"kind":"use","source":22},{"line":334,"start":19289,"end":19290,"expression":"Y","context":1294,"kind":"use","source":22},{"line":334,"start":19298,"end":19320,"expression":"\\norm{X-Y}<\\varepsilon","context":1294,"kind":"use","source":22},{"line":334,"start":19421,"end":19424,"expression":"S_N","context":1294,"kind":"use","source":22},{"line":334,"start":19488,"end":19491,"expression":"Y^*","context":1294,"kind":"use","source":22},{"line":334,"start":19500,"end":19520,"expression":"\\norm{S_NYS_N-Y}\\to0","context":1294,"kind":"use","source":22},{"line":334,"start":19588,"end":19600,"expression":"2\\varepsilon","context":1294,"kind":"use","source":22},{"line":336,"start":19673,"end":19678,"expression":"(e_j)","context":1295,"kind":"use","source":22},{"line":337,"start":19693,"end":19735,"expression":"\n \\norm X_2^2=\\sum_j\\norm{Xe_j}^2<\\infty.\n","context":1295,"kind":"use","source":22},{"line":340,"start":19751,"end":19762,"expression":"(S_N-I)Xe_j","context":1295,"kind":"use","source":22},{"line":340,"start":19815,"end":19829,"expression":"4\\norm{Xe_j}^2","context":1295,"kind":"use","source":22},{"line":340,"start":19861,"end":19882,"expression":"\\norm{(S_N-I)X}_2\\to0","context":1295,"kind":"use","source":22},{"line":340,"start":19936,"end":19945,"expression":"S_N^*=S_N","context":1295,"kind":"use","source":22},{"line":340,"start":19952,"end":19974,"expression":"\\norm{X^*}_2=\\norm X_2","context":1295,"kind":"use","source":22},{"line":340,"start":19999,"end":20030,"expression":"\\norm{AY}_2\\leq\\norm A\\norm Y_2","context":1295,"kind":"use","source":22},{"line":346,"start":20680,"end":20683,"expression":"S_N","context":1296,"kind":"use","source":22},{"line":346,"start":20703,"end":20704,"expression":"N","context":1296,"kind":"use","source":22},{"line":346,"start":20724,"end":20727,"expression":"X=I","context":1296,"kind":"use","source":22},{"line":347,"start":20732,"end":20767,"expression":"\n \\norm{S_NIS_N-I}=\\norm{S_N-I}=1.\n","context":1296,"kind":"use","source":22},{"line":350,"start":20785,"end":20792,"expression":"e_{N+1}","context":1296,"kind":"use","source":22},{"line":350,"start":20836,"end":20845,"expression":"S_Nx\\to x","context":1296,"kind":"use","source":22},{"line":350,"start":20864,"end":20874,"expression":"x\\in\\ell^2","context":1296,"kind":"use","source":22},{"line":355,"start":21092,"end":21104,"expression":"f(H_\\lambda)","context":1297,"kind":"use","source":22},{"line":356,"start":21146,"end":21215,"expression":"\n \\norm{f(H_\\lambda)-S_Nf(H_\\lambda)S_N}\n =\\sup_{n>N}|f(\\lambda_n)|.\n","context":1297,"kind":"use","source":22},{"line":360,"start":21246,"end":21271,"expression":"f(\\lambda)=e^{-it\\lambda}","context":1297,"kind":"use","source":22},{"line":362,"start":21481,"end":21492,"expression":"\\lambda_n=n","context":1298,"kind":"use","source":22},{"line":362,"start":21511,"end":21516,"expression":"i\\eta","context":1298,"kind":"use","source":22},{"line":362,"start":21520,"end":21528,"expression":"\\eta\\ne0","context":1298,"kind":"use","source":22},{"line":363,"start":21555,"end":21642,"expression":"\n \\sup_{n>N}\\frac1{\\sqrt{n^2+\\eta^2}}\n =\\frac1{\\sqrt{(N+1)^2+\\eta^2}}\\longrightarrow0.\n","context":1298,"kind":"use","source":22},{"line":367,"start":21666,"end":21673,"expression":"e^{-tn}","context":1298,"kind":"use","source":22},{"line":367,"start":21686,"end":21697,"expression":"e^{-t(N+1)}","context":1298,"kind":"use","source":22},{"line":367,"start":21710,"end":21713,"expression":"t>0","context":1298,"kind":"use","source":22},{"line":367,"start":21773,"end":21776,"expression":"t=0","context":1298,"kind":"use","source":22},{"line":370,"start":21938,"end":21998,"expression":"\n b(n)=\\frac{n}{\\sqrt{1+n^2}},\\qquad\n q(n)=\\frac{n-i}{n+i}.\n","context":1299,"kind":"use","source":22},{"line":375,"start":22036,"end":22073,"expression":"\n A_N(f)=S_Nf(H_\\lambda)S_N+(I-S_N).\n","context":1299,"kind":"use","source":22},{"line":379,"start":22098,"end":22194,"expression":"\n \\norm{b(H_\\lambda)-A_N(b)}\n =\\sup_{n>N}\\left|\\frac{n}{\\sqrt{1+n^2}}-1\\right|\\longrightarrow0,\n","context":1299,"kind":"use","source":22},{"line":384,"start":22203,"end":22282,"expression":"\n \\norm{q(H_\\lambda)-A_N(q)}\n =\\sup_{n>N}\\frac2{\\sqrt{n^2+1}}\\longrightarrow0.\n","context":1299,"kind":"use","source":22},{"line":391,"start":22542,"end":22635,"expression":"\n S_NHS_N\\psi-H\\psi\n =S_N(HS_N\\psi-H\\psi)+(S_N-I)H\\psi\\longrightarrow0,\n \\qquad\\psi\\in D(H).\n","context":1300,"kind":"use","source":22},{"line":396,"start":22757,"end":22758,"expression":"H","context":1300,"kind":"use","source":22},{"line":398,"start":23044,"end":23050,"expression":"n\\geq1","context":1301,"kind":"use","source":22},{"line":402,"start":23257,"end":23258,"expression":"H","context":1302,"kind":"use","source":22},{"line":402,"start":23278,"end":23279,"expression":"P","context":1303,"kind":"use","source":22},{"line":402,"start":23328,"end":23332,"expression":"D(H)","context":1304,"kind":"use","source":22},{"line":402,"start":23340,"end":23345,"expression":"[H,P]","context":1305,"kind":"use","source":22},{"line":402,"start":23351,"end":23355,"expression":"D(H)","context":1306,"kind":"use","source":22},{"line":402,"start":23388,"end":23389,"expression":"B","context":1307,"kind":"use","source":22},{"line":402,"start":23405,"end":23406,"expression":"z","context":1308,"kind":"use","source":22},{"line":402,"start":23414,"end":23428,"expression":"R_z=(H-z)^{-1}","context":1309,"kind":"use","source":22},{"line":402,"start":23440,"end":23443,"expression":"\\HH","context":1309,"kind":"use","source":22},{"line":402,"start":23451,"end":23455,"expression":"D(H)","context":1309,"kind":"use","source":22},{"line":403,"start":23588,"end":23608,"expression":"\n [R_z,P]=-R_zBR_z.\n","context":1310,"kind":"use","source":22},{"line":406,"start":23624,"end":23631,"expression":"x\\in\\HH","context":1311,"kind":"use","source":22},{"line":406,"start":23641,"end":23644,"expression":"H-z","context":1312,"kind":"use","source":22},{"line":406,"start":23650,"end":23655,"expression":"PR_zx","context":1313,"kind":"use","source":22},{"line":406,"start":23668,"end":23676,"expression":"Px+BR_zx","context":1314,"kind":"use","source":22},{"line":406,"start":23691,"end":23694,"expression":"R_z","context":1315,"kind":"use","source":22},{"line":407,"start":23705,"end":23820,"expression":"\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n","context":1316,"kind":"use","source":22},{"line":421,"start":24386,"end":24390,"expression":"\\C^N","context":1317,"kind":"use","source":22},{"line":424,"start":24614,"end":24627,"expression":"D:\\C^N\\to\\C^N","context":1318,"kind":"use","source":22},{"line":425,"start":24661,"end":24681,"expression":"\n D(fg)=D(f)g+fD(g)\n","context":1318,"kind":"use","source":22},{"line":428,"start":24757,"end":24760,"expression":"D=0","context":1318,"kind":"use","source":22},{"line":431,"start":24822,"end":24825,"expression":"e_a","context":1318,"kind":"use","source":22},{"line":431,"start":24842,"end":24851,"expression":"e_a^2=e_a","context":1318,"kind":"use","source":22},{"line":431,"start":24860,"end":24877,"expression":"D(e_a)=2e_aD(e_a)","context":1318,"kind":"use","source":22},{"line":431,"start":24897,"end":24903,"expression":"b\\ne a","context":1318,"kind":"use","source":22},{"line":431,"start":24934,"end":24937,"expression":"b=a","context":1318,"kind":"use","source":22},{"line":431,"start":24960,"end":24982,"expression":"(D(e_a))_a=2(D(e_a))_a","context":1319,"kind":"use","source":22},{"line":431,"start":25011,"end":25019,"expression":"D(e_a)=0","context":1320,"kind":"use","source":22},{"line":431,"start":25066,"end":25069,"expression":"D=0","context":1321,"kind":"use","source":22},{"line":434,"start":25140,"end":25144,"expression":"\\C^N","context":1322,"kind":"use","source":22},{"line":434,"start":25271,"end":25302,"expression":"\\C[\\varepsilon]/(\\varepsilon^2)","context":1322,"kind":"use","source":22},{"line":434,"start":25314,"end":25344,"expression":"D(a+b\\varepsilon)=b\\varepsilon","context":1322,"kind":"use","source":22},{"line":437,"start":25511,"end":25533,"expression":"D_hf_j=(f_{j+1}-f_j)/h","context":1323,"kind":"use","source":22},{"line":437,"start":25542,"end":25547,"expression":"h\\ne0","context":1323,"kind":"use","source":22},{"line":438,"start":25580,"end":25637,"expression":"\n D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j\n =h(D_hf)_j(D_hg)_j.\n","context":1323,"kind":"use","source":22},{"line":447,"start":25999,"end":26006,"expression":"M_N(\\C)","context":1324,"kind":"introduction-passage","source":22},{"line":447,"start":26090,"end":26091,"expression":"K","context":1324,"kind":"introduction-passage","source":22},{"line":447,"start":26124,"end":26129,"expression":"B=B^*","context":1324,"kind":"introduction-passage","source":22},{"line":448,"start":26144,"end":26173,"expression":"\n \\delta_BX=i[B,X]=i(BX-XB).\n","context":1324,"kind":"introduction-passage","source":22},{"line":452,"start":26223,"end":26468,"expression":"\n \\delta_B(XY)&=(\\delta_BX)Y+X\\delta_BY,\\label{eq:foundation-matrix-leibniz}\\\\\n (\\delta_BX)^*&=\\delta_B(X^*),\\label{eq:foundation-matrix-star}\\\\\n \\operatorname{Tr}((\\delta_BX)Y)&=-\\operatorname{Tr}(X\\delta_BY).\n \\label{eq:foundation-matrix-ibp}\n","context":1325,"kind":"introduction-passage","source":22},{"line":458,"start":26522,"end":26527,"expression":"-iXBY","context":1326,"kind":"use","source":22},{"line":458,"start":26534,"end":26538,"expression":"iXBY","context":1326,"kind":"use","source":22},{"line":458,"start":26626,"end":26627,"expression":"i","context":1326,"kind":"use","source":22},{"line":458,"start":26636,"end":26641,"expression":"B^*=B","context":1326,"kind":"use","source":22},{"line":458,"start":26714,"end":26715,"expression":"B","context":1326,"kind":"use","source":22},{"line":458,"start":26721,"end":26743,"expression":"\\operatorname{Tr}(BXY)","context":1326,"kind":"use","source":22},{"line":459,"start":26917,"end":27035,"expression":"\n \\inner{\\delta_BX}{Y}_{\\rm HS}=-\\inner{X}{\\delta_BY}_{\\rm HS},\n \\qquad\\inner{X}{Y}_{\\rm HS}=\\operatorname{Tr}(X^*Y).\n","context":1327,"kind":"use","source":22},{"line":464,"start":27105,"end":27113,"expression":"\\delta_B","context":1328,"kind":"use","source":22},{"line":468,"start":27302,"end":27391,"expression":"\n B=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n X=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\n","context":1329,"kind":"use","source":22},{"line":473,"start":27408,"end":27463,"expression":"\n \\delta_BX=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}\\ne0.\n","context":1330,"kind":"use","source":22},{"line":479,"start":27644,"end":27651,"expression":"B_i,B_j","context":1331,"kind":"use","source":22},{"line":479,"start":27659,"end":27685,"expression":"\\mathcal F_{ij}=i[B_i,B_j]","context":1331,"kind":"use","source":22},{"line":480,"start":27770,"end":27908,"expression":"\n [\\delta_{B_i},\\delta_{B_j}]X\n =-[[B_i,B_j],X]\n =i[\\mathcal F_{ij},X]=\\delta_{\\mathcal F_{ij}}X.\n \\label{eq:foundation-corner-curvature}\n","context":1331,"kind":"use","source":22},{"line":486,"start":28077,"end":28078,"expression":"X","context":1332,"kind":"use","source":22},{"line":486,"start":28110,"end":28111,"expression":"X","context":1333,"kind":"use","source":22},{"line":492,"start":28612,"end":28617,"expression":"(P_a)","context":1334,"kind":"use","source":22},{"line":492,"start":28653,"end":28667,"expression":"X_{ba}=P_bXP_a","context":1334,"kind":"use","source":22},{"line":492,"start":28674,"end":28688,"expression":"B_{ba}=P_bBP_a","context":1334,"kind":"use","source":22},{"line":492,"start":28705,"end":28716,"expression":"I=\\sum_cP_c","context":1334,"kind":"use","source":22},{"line":493,"start":28740,"end":28844,"expression":"\n P_b\\delta_B(X)P_a\n =i\\sum_c\\left(B_{bc}X_{ca}-X_{bc}B_{ca}\\right).\n \\label{eq:foundation-corner-path}\n","context":1334,"kind":"use","source":22},{"line":498,"start":28904,"end":28905,"expression":"c","context":1334,"kind":"use","source":22},{"line":498,"start":28930,"end":28931,"expression":"X","context":1334,"kind":"use","source":22},{"line":498,"start":28939,"end":28940,"expression":"B","context":1334,"kind":"use","source":22},{"line":498,"start":28968,"end":28969,"expression":"B","context":1334,"kind":"use","source":22},{"line":498,"start":28977,"end":28978,"expression":"X","context":1334,"kind":"use","source":22},{"line":500,"start":29164,"end":29173,"expression":"\\delta_BX","context":1335,"kind":"use","source":22},{"line":500,"start":29205,"end":29237,"expression":"Q_\\alpha=\\sum_{a\\in C_\\alpha}P_a","context":1335,"kind":"use","source":22},{"line":501,"start":29242,"end":29337,"expression":"\n Q_\\beta\\delta_B(X)Q_\\alpha\n =\\sum_{\\substack{b\\in C_\\beta\\\\a\\in C_\\alpha}}P_b\\delta_B(X)P_a.\n","context":1336,"kind":"use","source":22},{"line":23,"start":3239,"end":3243,"expression":"\\C^N","context":1337,"kind":"use","source":23},{"line":28,"start":3514,"end":3526,"expression":"\\inner{x}{y}","context":1338,"kind":"notation-table","source":23},{"line":28,"start":3552,"end":3553,"expression":"x","context":1338,"kind":"notation-table","source":23},{"line":28,"start":3570,"end":3571,"expression":"y","context":1338,"kind":"notation-table","source":23},{"line":29,"start":3589,"end":3604,"expression":"P_a=P_a^*=P_a^2","context":1339,"kind":"notation-table","source":23},{"line":30,"start":3678,"end":3692,"expression":"L_{ba}=P_bLP_a","context":1340,"kind":"notation-table","source":23},{"line":30,"start":3723,"end":3724,"expression":"a","context":1340,"kind":"notation-table","source":23},{"line":30,"start":3737,"end":3738,"expression":"b","context":1340,"kind":"notation-table","source":23},{"line":31,"start":3761,"end":3775,"expression":"\\dot\\psi=K\\psi","context":1341,"kind":"notation-table","source":23},{"line":31,"start":3784,"end":3790,"expression":"K^*=-K","context":1341,"kind":"notation-table","source":23},{"line":31,"start":3824,"end":3829,"expression":"K=-iH","context":1341,"kind":"notation-table","source":23},{"line":31,"start":3841,"end":3848,"expression":"\\hbar=1","context":1341,"kind":"notation-table","source":23},{"line":32,"start":3870,"end":3890,"expression":"p_a=\\norm{P_a\\psi}^2","context":1342,"kind":"notation-table","source":23},{"line":33,"start":4046,"end":4063,"expression":"J_{b\\leftarrow a}","context":1343,"kind":"notation-table","source":23},{"line":33,"start":4088,"end":4089,"expression":"b","context":1343,"kind":"notation-table","source":23},{"line":33,"start":4104,"end":4105,"expression":"a","context":1343,"kind":"notation-table","source":23},{"line":33,"start":4112,"end":4144,"expression":"\\dot p_b=\\sum_aJ_{b\\leftarrow a}","context":1343,"kind":"notation-table","source":23},{"line":34,"start":4193,"end":4202,"expression":"\\rho\\geq0","context":1344,"kind":"notation-table","source":23},{"line":34,"start":4206,"end":4229,"expression":"\\operatorname{tr}\\rho=1","context":1344,"kind":"notation-table","source":23},{"line":36,"start":4473,"end":4478,"expression":"H\\psi","context":1345,"kind":"notation-table","source":23},{"line":36,"start":4504,"end":4518,"expression":"\\psi\\in\\Dom(H)","context":1345,"kind":"notation-table","source":23},{"line":37,"start":4541,"end":4542,"expression":"K","context":1346,"kind":"notation-table","source":23},{"line":37,"start":4579,"end":4586,"expression":"\\hbar=1","context":1346,"kind":"notation-table","source":23},{"line":37,"start":4606,"end":4611,"expression":"K=-iH","context":1346,"kind":"notation-table","source":23},{"line":5,"start":474,"end":475,"expression":"E","context":1347,"kind":"use","source":25},{"line":5,"start":516,"end":525,"expression":"f(h)\\in E","context":1347,"kind":"use","source":25},{"line":5,"start":543,"end":555,"expression":"0<|h|<\\delta","context":1347,"kind":"use","source":25},{"line":5,"start":582,"end":585,"expression":"b>1","context":1347,"kind":"use","source":25},{"line":5,"start":592,"end":596,"expression":"f(h)","context":1347,"kind":"use","source":25},{"line":10,"start":988,"end":1000,"expression":"T_0f(h)=f(h)","context":1348,"kind":"definition-passage","definition_title":"The actual refinement tower","source":25},{"line":10,"start":1012,"end":1018,"expression":"m\\geq0","context":1348,"kind":"definition-passage","definition_title":"The actual refinement tower","source":25},{"line":11,"start":1030,"end":1088,"expression":"\n T_{m+1}f(h)=\\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.\n","context":1348,"kind":"definition-passage","definition_title":"The actual refinement tower","source":25},{"line":17,"start":1164,"end":1170,"expression":"r\\in\\N","context":1349,"kind":"use","source":25},{"line":17,"start":1177,"end":1183,"expression":"v\\in E","context":1349,"kind":"use","source":25},{"line":18,"start":1188,"end":1268,"expression":"\n T_m(h^rv)=c_{m,r}h^rv,\\qquad\n c_{m,r}=\\prod_{k=1}^{m}\\frac{b^{k-r}-1}{b^k-1}.\n","context":1349,"kind":"use","source":25},{"line":22,"start":1317,"end":1326,"expression":"c_{m,0}=1","context":1349,"kind":"use","source":25},{"line":22,"start":1334,"end":1343,"expression":"c_{m,r}=0","context":1349,"kind":"use","source":25},{"line":22,"start":1350,"end":1363,"expression":"1\\leq r\\leq m","context":1349,"kind":"use","source":25},{"line":25,"start":1467,"end":1468,"expression":"m","context":1349,"kind":"use","source":25},{"line":26,"start":1522,"end":1559,"expression":"\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n","context":1350,"kind":"use","source":25},{"line":29,"start":1612,"end":1615,"expression":"r=0","context":1351,"kind":"use","source":25},{"line":29,"start":1647,"end":1660,"expression":"1\\leq r\\leq m","context":1352,"kind":"use","source":25},{"line":29,"start":1678,"end":1681,"expression":"k=r","context":1353,"kind":"use","source":25},{"line":33,"start":1754,"end":1790,"expression":"\n f(h)=F+\\sum_{r=1}^{m}h^rv_r+R(h).\n","context":1354,"kind":"use","source":25},{"line":37,"start":1849,"end":1870,"expression":"\n T_mf(h)-F=T_mR(h).\n","context":1354,"kind":"use","source":25},{"line":44,"start":2120,"end":2121,"expression":"p","context":1355,"kind":"use","source":25},{"line":45,"start":2162,"end":2249,"expression":"\n \\norm{R(h)}\\leq C|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n","context":1356,"kind":"use","source":25},{"line":50,"start":2259,"end":2277,"expression":"C,\\varepsilon\\geq0","context":1357,"kind":"use","source":25},{"line":51,"start":2289,"end":2339,"expression":"\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n","context":1357,"kind":"use","source":25},{"line":55,"start":2390,"end":2447,"expression":"\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n","context":1357,"kind":"use","source":25},{"line":60,"start":2508,"end":2520,"expression":"h\\mapsto h/b","context":1358,"kind":"use","source":25},{"line":61,"start":2581,"end":2665,"expression":"\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n","context":1359,"kind":"use","source":25},{"line":65,"start":2770,"end":2773,"expression":"m+1","context":1360,"kind":"use","source":25},{"line":65,"start":2779,"end":2804,"expression":"(b^{m+1-p}+1)/(b^{m+1}-1)","context":1361,"kind":"use","source":25},{"line":65,"start":2814,"end":2846,"expression":"\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p","context":1362,"kind":"use","source":25},{"line":65,"start":2879,"end":2882,"expression":"p=0","context":1363,"kind":"use","source":25},{"line":65,"start":2888,"end":2891,"expression":"g-f","context":1364,"kind":"use","source":25},{"line":68,"start":2983,"end":2988,"expression":"p=m+1","context":1365,"kind":"use","source":25},{"line":68,"start":3135,"end":3142,"expression":"A_{m,0}","context":1365,"kind":"use","source":25},{"line":68,"start":3264,"end":3265,"expression":"h","context":1365,"kind":"use","source":25},{"line":71,"start":3318,"end":3321,"expression":"b=2","context":1366,"kind":"use","source":25},{"line":72,"start":3364,"end":3489,"expression":"\n T_1f(h)&=2f(h/2)-f(h),\\\\\n T_2f(h)&=\\frac{8f(h/4)-6f(h/2)+f(h)}{3},\\\\\n T_3f(h)&=\\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.\n","context":1367,"kind":"use","source":25},{"line":77,"start":3536,"end":3555,"expression":"f(h)=1+2h-3h^2+4h^3","context":1368,"kind":"use","source":25},{"line":78,"start":3578,"end":3653,"expression":"\n T_1f(h)=1+\\frac32h^2-3h^3,\\qquad\n T_2f(h)=1+\\frac12h^3,\\qquad T_3f(h)=1.\n","context":1368,"kind":"use","source":25},{"line":82,"start":3704,"end":3705,"expression":"h","context":1368,"kind":"use","source":25},{"line":82,"start":3746,"end":3749,"expression":"T_2","context":1368,"kind":"use","source":25},{"line":82,"start":3779,"end":3790,"expression":"\\varepsilon","context":1368,"kind":"use","source":25},{"line":83,"start":3823,"end":3866,"expression":"\n \\frac{8+6+1}{3}\\varepsilon=5\\varepsilon.\n","context":1368,"kind":"use","source":25},{"line":86,"start":4142,"end":4162,"expression":"|h|^3/2+5\\varepsilon","context":1369,"kind":"use","source":25},{"line":6,"start":752,"end":755,"expression":"L^2","context":1370,"kind":"use","source":26},{"line":3,"start":191,"end":218,"expression":"\\psi_{x+\\widehat\\mu}-\\psi_x","context":1371,"kind":"use","source":27},{"line":5,"start":416,"end":417,"expression":"V","context":1372,"kind":"use","source":27},{"line":5,"start":482,"end":486,"expression":"\\C^r","context":1373,"kind":"use","source":27},{"line":6,"start":625,"end":754,"expression":"\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n","context":1374,"kind":"use","source":27},{"line":10,"start":780,"end":796,"expression":"g=(g_x)_{x\\in V}","context":1374,"kind":"use","source":27},{"line":10,"start":822,"end":840,"expression":"\\psi_x^g=g_x\\psi_x","context":1375,"kind":"use","source":27},{"line":11,"start":935,"end":1087,"expression":"\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n","context":1376,"kind":"use","source":27},{"line":16,"start":1184,"end":1201,"expression":"T_{y\\leftarrow x}","context":1377,"kind":"use","source":27},{"line":16,"start":1227,"end":1255,"expression":"g_yT_{y\\leftarrow x}g_x^{-1}","context":1378,"kind":"use","source":27},{"line":19,"start":1360,"end":1506,"expression":"\\label{g:forward-link}\n u_{x,\\mu}=U_{x,\\mu}^{-1}:\\HH_x\\longrightarrow\\HH_{x+\\widehat\\mu},\n \\qquad u_{x,\\mu}^g=g_{x+\\widehat\\mu}u_{x,\\mu}g_x^{-1}.\n","context":1379,"kind":"use","source":27},{"line":23,"start":1588,"end":1591,"expression":"L_j","context":1379,"kind":"use","source":27},{"line":23,"start":1632,"end":1633,"expression":"u","context":1379,"kind":"use","source":27},{"line":23,"start":1711,"end":1712,"expression":"U","context":1379,"kind":"use","source":27},{"line":26,"start":1994,"end":1997,"expression":"h>0","context":1380,"kind":"use","source":27},{"line":27,"start":2020,"end":2287,"expression":"\n (D^+_{\\mu,U}\\psi)_x\n &=\\frac{U_{x,\\mu}\\psi_{x+\\widehat\\mu}-\\psi_x}{h},\\label{g:forward-difference}\\\\\n (\\Delta_U\\psi)_x\n &=\\frac1{h^2}\\sum_\\mu\\left(\n U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n +U_{x-\\widehat\\mu,\\mu}^{-1}\\psi_{x-\\widehat\\mu}-2\\psi_x\\right).\n \\label{g:laplacian}\n","context":1381,"kind":"use","source":27},{"line":36,"start":2305,"end":2342,"expression":"D^+_{\\mu,U^g}(g\\psi)=gD^+_{\\mu,U}\\psi","context":1382,"kind":"use","source":27},{"line":37,"start":2349,"end":2382,"expression":"\\Delta_{U^g}(g\\psi)=g\\Delta_U\\psi","context":1382,"kind":"use","source":27},{"line":41,"start":2504,"end":2517,"expression":"x+\\widehat\\mu","context":1383,"kind":"use","source":27},{"line":41,"start":2536,"end":2539,"expression":"g_x","context":1383,"kind":"use","source":27},{"line":42,"start":2579,"end":2662,"expression":"\n (U_{x-\\widehat\\mu,\\mu}^g)^{-1}g_{x-\\widehat\\mu}\n =g_xU_{x-\\widehat\\mu,\\mu}^{-1}.\n","context":1383,"kind":"use","source":27},{"line":46,"start":2730,"end":2733,"expression":"g_x","context":1383,"kind":"use","source":27},{"line":49,"start":2779,"end":2820,"expression":"\\inner{\\phi}{\\psi}_V=\\sum_x\\phi_x^*\\psi_x","context":1384,"kind":"use","source":27},{"line":50,"start":2940,"end":3054,"expression":"\\label{g:laplace-form}\n -\\inner{\\psi}{\\Delta_U\\psi}_V\n =\\sum_{x,\\mu}\\left\\lVert(D^+_{\\mu,U}\\psi)_x\\right\\rVert^2.\n","context":1385,"kind":"use","source":27},{"line":55,"start":3117,"end":3229,"expression":"h^{-2}(\\norm{\\psi_{x+\\widehat\\mu}}^2+\\norm{\\psi_x}^2\n-2\\operatorname{Re}(\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}))","context":1385,"kind":"use","source":27},{"line":57,"start":3355,"end":3363,"expression":"\\Delta_U","context":1386,"kind":"use","source":27},{"line":59,"start":3583,"end":3584,"expression":"U","context":1387,"kind":"use","source":27},{"line":59,"start":3611,"end":3612,"expression":"u","context":1387,"kind":"use","source":27},{"line":65,"start":3951,"end":3956,"expression":"N\\ge1","context":1388,"kind":"use","source":27},{"line":65,"start":3978,"end":3988,"expression":"j\\in\\Z/N\\Z","context":1388,"kind":"use","source":27},{"line":65,"start":3996,"end":4011,"expression":"\\psi_j^n\\in\\C^2","context":1388,"kind":"use","source":27},{"line":65,"start":4023,"end":4036,"expression":"U_j^n\\in U(1)","context":1388,"kind":"use","source":27},{"line":65,"start":4070,"end":4083,"expression":"T_j^n\\in U(1)","context":1388,"kind":"use","source":27},{"line":65,"start":4101,"end":4106,"expression":"(j,n)","context":1388,"kind":"use","source":27},{"line":65,"start":4112,"end":4119,"expression":"(j,n+1)","context":1388,"kind":"use","source":27},{"line":65,"start":4169,"end":4196,"expression":"C_{1,j}^n,C_{2,j}^n\\in U(2)","context":1388,"kind":"use","source":27},{"line":66,"start":4222,"end":4420,"expression":"\n \\eta_j&=C_{1,j}^n\\psi_j^n,\\label{g:walk-coin}\\\\\n \\xi_j^+&=(U_{j-1}^n)^{-1}\\eta_{j-1}^+,\n &\\xi_j^-&=U_j^n\\eta_{j+1}^-,\\label{g:walk-shift}\\\\\n \\psi_j^{n+1}&=T_j^n C_{2,j}^n\\xi_j.\\label{g:walk-step}\n","context":1389,"kind":"use","source":27},{"line":72,"start":4442,"end":4443,"expression":"+","context":1390,"kind":"use","source":27},{"line":72,"start":4478,"end":4479,"expression":"j","context":1390,"kind":"use","source":27},{"line":72,"start":4493,"end":4494,"expression":"-","context":1390,"kind":"use","source":27},{"line":72,"start":4529,"end":4530,"expression":"j","context":1390,"kind":"use","source":27},{"line":74,"start":4739,"end":4744,"expression":"g_j^n","context":1391,"kind":"use","source":27},{"line":75,"start":4767,"end":4876,"expression":"\\label{g:temporal-link}\n (U_j^n)^g=g_j^nU_j^n(g_{j+1}^n)^{-1},\n \\qquad (T_j^n)^g=g_j^{n+1}T_j^n(g_j^n)^{-1}.\n","context":1391,"kind":"use","source":27},{"line":79,"start":4948,"end":4949,"expression":"j","context":1391,"kind":"use","source":27},{"line":79,"start":5045,"end":5057,"expression":"\\mathcal W_n","context":1391,"kind":"use","source":27},{"line":80,"start":5085,"end":5196,"expression":"\\label{g:walk-covariance}\n \\mathcal W_n[U^g,T^g]G^n=G^{n+1}\\mathcal W_n[U,T],\n \\qquad (G^n\\psi)_j=g_j^n\\psi_j.\n","context":1391,"kind":"use","source":27},{"line":87,"start":5504,"end":5530,"expression":"\\rho_j^n=\\norm{\\psi_j^n}^2","context":1392,"kind":"use","source":27},{"line":88,"start":5552,"end":5619,"expression":"\\label{g:walk-current}\n J_{j+1/2}^n=|\\eta_j^+|^2-|\\eta_{j+1}^-|^2.\n","context":1392,"kind":"use","source":27},{"line":92,"start":5672,"end":5743,"expression":"\\label{g:walk-balance}\n \\rho_j^{n+1}-\\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.\n","context":1392,"kind":"use","source":27},{"line":95,"start":5872,"end":5873,"expression":"A","context":1393,"kind":"use","source":27},{"line":96,"start":5922,"end":5973,"expression":"Q_A^{n+1}-Q_A^n=J_{\\partial_-A}^n-J_{\\partial_+A}^n","context":1394,"kind":"use","source":27},{"line":101,"start":6054,"end":6088,"expression":"\\rho_j^n=|\\eta_j^+|^2+|\\eta_j^-|^2","context":1395,"kind":"use","source":27},{"line":102,"start":6169,"end":6215,"expression":"\\rho_j^{n+1}=|\\eta_{j-1}^+|^2+|\\eta_{j+1}^-|^2","context":1395,"kind":"use","source":27},{"line":103,"start":6354,"end":6360,"expression":"\\eta_j","context":1395,"kind":"use","source":27},{"line":103,"start":6366,"end":6377,"expression":"g_j^n\\eta_j","context":1395,"kind":"use","source":27},{"line":114,"start":7294,"end":7299,"expression":"A=A^*","context":1396,"kind":"use","source":27},{"line":114,"start":7306,"end":7311,"expression":"A^2=I","context":1396,"kind":"use","source":27},{"line":115,"start":7360,"end":7439,"expression":"\\label{g:clifford-projections}\n P^+_A=\\tfrac12(I+A),\\qquad P^-_A=\\tfrac12(I-A)\n","context":1396,"kind":"use","source":27},{"line":118,"start":7526,"end":7538,"expression":"P_A^+P_A^-=0","context":1396,"kind":"use","source":27},{"line":118,"start":7584,"end":7589,"expression":"A^2=I","context":1397,"kind":"use","source":27},{"line":118,"start":7623,"end":7628,"expression":"A=A^*","context":1398,"kind":"use","source":27},{"line":120,"start":7868,"end":7879,"expression":"\\tau:V\\to V","context":1399,"kind":"use","source":27},{"line":120,"start":7908,"end":7919,"expression":"u:V\\to U(1)","context":1400,"kind":"use","source":27},{"line":121,"start":7945,"end":8060,"expression":"\\label{g:clifford-shift}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_A^+\\psi(\\tau^{-1}x)\n +u(x)^{-1}P_A^-\\psi(\\tau x).\n","context":1401,"kind":"use","source":27},{"line":127,"start":8174,"end":8296,"expression":"\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n","context":1402,"kind":"use","source":27},{"line":132,"start":8479,"end":8480,"expression":"x","context":1403,"kind":"use","source":27},{"line":132,"start":8491,"end":8495,"expression":"\\tau","context":1404,"kind":"use","source":27},{"line":133,"start":8536,"end":8629,"expression":"\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n","context":1405,"kind":"use","source":27},{"line":137,"start":8649,"end":8678,"expression":"u^g(x)=g(\\tau x)u(x)g(x)^{-1}","context":1406,"kind":"use","source":27},{"line":137,"start":8717,"end":8750,"expression":"S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}","context":1407,"kind":"use","source":27},{"line":139,"start":9000,"end":9001,"expression":"2","context":1408,"kind":"use","source":27},{"line":139,"start":9031,"end":9032,"expression":"4","context":1408,"kind":"use","source":27},{"line":139,"start":9099,"end":9108,"expression":"\\C^\\times","context":1408,"kind":"use","source":27},{"line":141,"start":9318,"end":9322,"expression":"U(1)","context":1409,"kind":"use","source":27},{"line":141,"start":9337,"end":9342,"expression":"SU(2)","context":1409,"kind":"use","source":27},{"line":141,"start":9514,"end":9529,"expression":"\\C^4\\otimes\\C^2","context":1409,"kind":"use","source":27},{"line":148,"start":10000,"end":10110,"expression":"\\label{g:plaquette}\n W_{x,\\mu\\nu}\n =U_{x,\\mu}U_{x+\\widehat\\mu,\\nu}\n U_{x+\\widehat\\nu,\\mu}^{-1}U_{x,\\nu}^{-1}.\n","context":1410,"kind":"use","source":27},{"line":153,"start":10152,"end":10153,"expression":"x","context":1410,"kind":"use","source":27},{"line":153,"start":10190,"end":10203,"expression":"x+\\widehat\\nu","context":1410,"kind":"use","source":27},{"line":153,"start":10274,"end":10275,"expression":"x","context":1410,"kind":"use","source":27},{"line":154,"start":10429,"end":10501,"expression":"\\label{g:plaquette-covariance}\n W_{x,\\mu\\nu}^g=g_xW_{x,\\mu\\nu}g_x^{-1}.\n","context":1411,"kind":"use","source":27},{"line":157,"start":10521,"end":10525,"expression":"U(1)","context":1412,"kind":"use","source":27},{"line":157,"start":10677,"end":10678,"expression":"x","context":1413,"kind":"use","source":27},{"line":159,"start":10687,"end":10692,"expression":"SU(2)","context":1414,"kind":"use","source":27},{"line":160,"start":10772,"end":10962,"expression":"\\label{g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n","context":1415,"kind":"use","source":27},{"line":165,"start":11039,"end":11042,"expression":"g_x","context":1415,"kind":"use","source":27},{"line":165,"start":11072,"end":11087,"expression":"S_W[U^g]=S_W[U]","context":1415,"kind":"use","source":27},{"line":165,"start":11100,"end":11105,"expression":"SU(2)","context":1416,"kind":"use","source":27},{"line":165,"start":11131,"end":11155,"expression":"e^{i\\theta},e^{-i\\theta}","context":1417,"kind":"use","source":27},{"line":165,"start":11179,"end":11195,"expression":"1-\\cos\\theta\\ge0","context":1418,"kind":"use","source":27},{"line":165,"start":11241,"end":11261,"expression":"1-\\operatorname{Re}W","context":1419,"kind":"use","source":27},{"line":170,"start":11658,"end":11740,"expression":"\\label{g:abelian-cube}\n \\prod_{f\\subset\\partial c}W_f^{\\operatorname{or}(f,c)}=1.\n","context":1420,"kind":"use","source":27},{"line":173,"start":11912,"end":11929,"expression":"W_f=e^{i\\theta_f}","context":1421,"kind":"use","source":27},{"line":173,"start":11942,"end":11950,"expression":"\\theta_f","context":1422,"kind":"use","source":27},{"line":173,"start":12032,"end":12038,"expression":"2\\pi\\Z","context":1423,"kind":"use","source":27},{"line":179,"start":12555,"end":12568,"expression":"W=e^{i\\theta}","context":1424,"kind":"use","source":27},{"line":179,"start":12602,"end":12639,"expression":"\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS","context":1425,"kind":"use","source":27},{"line":179,"start":12648,"end":12649,"expression":"q","context":1426,"kind":"use","source":27},{"line":179,"start":12670,"end":12680,"expression":"F_{\\mu\\nu}","context":1427,"kind":"use","source":27},{"line":179,"start":12768,"end":12784,"expression":"a_\\mu,a_\\nu=O(a)","context":1428,"kind":"use","source":27},{"line":179,"start":12827,"end":12828,"expression":"F","context":1429,"kind":"use","source":27},{"line":179,"start":12864,"end":12867,"expression":"x_c","context":1430,"kind":"use","source":27},{"line":179,"start":12874,"end":12875,"expression":"F","context":1431,"kind":"use","source":27},{"line":179,"start":12881,"end":12884,"expression":"C^2","context":1432,"kind":"use","source":27},{"line":180,"start":12958,"end":13033,"expression":"\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n","context":1433,"kind":"use","source":27},{"line":184,"start":13078,"end":13181,"expression":"\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n","context":1434,"kind":"use","source":27},{"line":188,"start":13223,"end":13258,"expression":"1-\\cos\\theta=\\theta^2/2+O(\\theta^4)","context":1435,"kind":"use","source":27},{"line":188,"start":13265,"end":13278,"expression":"\\theta=O(a^2)","context":1436,"kind":"use","source":27},{"line":188,"start":13306,"end":13312,"expression":"O(a^4)","context":1437,"kind":"use","source":27},{"line":188,"start":13330,"end":13336,"expression":"O(a^6)","context":1438,"kind":"use","source":27},{"line":188,"start":13371,"end":13377,"expression":"O(a^8)","context":1439,"kind":"use","source":27},{"line":188,"start":13446,"end":13452,"expression":"O(a^3)","context":1440,"kind":"use","source":27},{"line":188,"start":13480,"end":13486,"expression":"O(a^5)","context":1441,"kind":"use","source":27},{"line":194,"start":13955,"end":13960,"expression":"N\\ge3","context":1442,"kind":"use","source":27},{"line":196,"start":14095,"end":14098,"expression":"h>0","context":1443,"kind":"use","source":27},{"line":196,"start":14102,"end":14108,"expression":"m\\in\\R","context":1443,"kind":"use","source":27},{"line":196,"start":14140,"end":14155,"expression":"\\alpha=\\alpha^*","context":1443,"kind":"use","source":27},{"line":196,"start":14162,"end":14175,"expression":"\\beta=\\beta^*","context":1443,"kind":"use","source":27},{"line":196,"start":14181,"end":14185,"expression":"\\C^s","context":1443,"kind":"use","source":27},{"line":196,"start":14193,"end":14206,"expression":"\\psi_j\\in\\C^s","context":1443,"kind":"use","source":27},{"line":196,"start":14213,"end":14225,"expression":"A_j,E_j\\in\\R","context":1443,"kind":"use","source":27},{"line":196,"start":14237,"end":14251,"expression":"U_j=e^{-ihA_j}","context":1443,"kind":"use","source":27},{"line":196,"start":14290,"end":14291,"expression":"A","context":1443,"kind":"use","source":27},{"line":197,"start":14331,"end":14540,"expression":"\n H_h&=H_{\\mathrm m}+H_E,\\label{g:hamiltonian}\\\\\n H_{\\mathrm m}\n &=hm\\sum_j\\psi_j^*\\beta\\psi_j\n   +\\sum_j\\operatorname{Im}(\\psi_j^*\\alpha U_j\\psi_{j+1}),\n &H_E&=\\frac h2\\sum_jE_j^2.\\label{g:hamiltonian-parts}\n","context":1444,"kind":"use","source":27},{"line":205,"start":14814,"end":14950,"expression":"\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n","context":1445,"kind":"use","source":27},{"line":210,"start":15008,"end":15057,"expression":"ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j","context":1446,"kind":"use","source":27},{"line":210,"start":15061,"end":15096,"expression":"h\\dot A_j=\\partial H_h/\\partial E_j","context":1447,"kind":"use","source":27},{"line":210,"start":15104,"end":15140,"expression":"h\\dot E_j=-\\partial H_h/\\partial A_j","context":1448,"kind":"use","source":27},{"line":211,"start":15179,"end":15412,"expression":"\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}\\\\\n \\dot A_j&=E_j,\n &\\dot E_j&=J_j,\n &J_j&=\\operatorname{Re}(\\psi_j^*\\alpha U_j\\psi_{j+1}).\n \\label{g:field-ode}\n","context":1449,"kind":"use","source":27},{"line":220,"start":15441,"end":15473,"expression":"z_j=\\psi_j^*\\alpha U_j\\psi_{j+1}","context":1450,"kind":"use","source":27},{"line":221,"start":15482,"end":15566,"expression":"\\partial_{A_j}\\operatorname{Im}z_j\n=\\operatorname{Im}(-ihz_j)=-h\\operatorname{Re}z_j","context":1450,"kind":"use","source":27},{"line":223,"start":15602,"end":15614,"expression":"\\dot E_j=J_j","context":1450,"kind":"use","source":27},{"line":227,"start":15824,"end":15962,"expression":"\\label{g:gauss-def}\n \\rho_j=\\psi_j^*\\psi_j,\\qquad\n \\bar\\rho=\\frac1N\\sum_j\\rho_j,\\qquad\n \\mathcal G_j=\\frac{E_j-E_{j-1}}h+\\rho_j-\\bar\\rho.\n","context":1451,"kind":"use","source":27},{"line":233,"start":15998,"end":16111,"expression":"\\label{g:gauss-balance}\n \\dot\\rho_j+\\frac{J_j-J_{j-1}}h=0,\n \\qquad\\dot{\\bar\\rho}=0,\n \\qquad\\dot{\\mathcal G}_j=0.\n","context":1451,"kind":"use","source":27},{"line":238,"start":16166,"end":16180,"expression":"\\mathcal G_j=0","context":1452,"kind":"use","source":27},{"line":242,"start":16281,"end":16287,"expression":"\\rho_j","context":1453,"kind":"use","source":27},{"line":242,"start":16298,"end":16347,"expression":"\\dot\\rho_j=2\\operatorname{Re}(\\psi_j^*\\dot\\psi_j)","context":1453,"kind":"use","source":27},{"line":242,"start":16389,"end":16408,"expression":"\\psi_j^*\\beta\\psi_j","context":1453,"kind":"use","source":27},{"line":242,"start":16457,"end":16463,"expression":"-J_j/h","context":1453,"kind":"use","source":27},{"line":242,"start":16507,"end":16539,"expression":"\\psi_{j-1}^*\\alpha U_{j-1}\\psi_j","context":1453,"kind":"use","source":27},{"line":242,"start":16549,"end":16564,"expression":"\\alpha=\\alpha^*","context":1453,"kind":"use","source":27},{"line":242,"start":16592,"end":16593,"expression":"U","context":1453,"kind":"use","source":27},{"line":242,"start":16617,"end":16626,"expression":"J_{j-1}/h","context":1453,"kind":"use","source":27},{"line":242,"start":16679,"end":16695,"expression":"\\dot{\\bar\\rho}=0","context":1454,"kind":"use","source":27},{"line":243,"start":16709,"end":16833,"expression":"\n \\dot{\\mathcal G}_j\n =\\frac{\\dot E_j-\\dot E_{j-1}}h+\\dot\\rho_j-\\dot{\\bar\\rho}\n =\\frac{J_j-J_{j-1}}h-\\frac{J_j-J_{j-1}}h=0.\n","context":1455,"kind":"use","source":27},{"line":250,"start":16869,"end":16877,"expression":"\\bar\\rho","context":1456,"kind":"use","source":27},{"line":250,"start":17136,"end":17148,"expression":"\\mathcal G_j","context":1456,"kind":"use","source":27},{"line":250,"start":17333,"end":17344,"expression":"a\\le j\\le b","context":1457,"kind":"use","source":27},{"line":251,"start":17398,"end":17460,"expression":"\n \\frac{d}{dt}\\left(h\\sum_{j=a}^{b}\\rho_j\\right)=J_{a-1}-J_b.\n","context":1458,"kind":"use","source":27},{"line":257,"start":17572,"end":17681,"expression":"\\label{g:work-balance}\n \\dot H_E=h\\sum_jE_jJ_j,\\qquad\n \\dot H_{\\mathrm m}=-h\\sum_jE_jJ_j,\\qquad\n \\dot H_h=0.\n","context":1459,"kind":"use","source":27},{"line":265,"start":17761,"end":17773,"expression":"\\dot E_j=J_j","context":1460,"kind":"use","source":27},{"line":265,"start":17810,"end":17873,"expression":"q_j=\\partial H_{\\mathrm m}/\\partial\\overline\\psi_j=ih\\dot\\psi_j","context":1460,"kind":"use","source":27},{"line":266,"start":17928,"end":17969,"expression":"2\\operatorname{Re}\\sum_jq_j^*\\dot\\psi_j=0","context":1460,"kind":"use","source":27},{"line":266,"start":17981,"end":18019,"expression":"q_j^*\\dot\\psi_j=-ih\\norm{\\dot\\psi_j}^2","context":1460,"kind":"use","source":27},{"line":267,"start":18086,"end":18164,"expression":"\n \\sum_j\\frac{\\partial H_{\\mathrm m}}{\\partial A_j}\\dot A_j\n =-h\\sum_jJ_jE_j.\n","context":1460,"kind":"use","source":27},{"line":276,"start":18322,"end":18328,"expression":"\\chi_j","context":1461,"kind":"use","source":27},{"line":277,"start":18366,"end":18500,"expression":"\\label{g:residual-gauge}\n \\psi_j\\mapsto e^{i\\chi_j}\\psi_j,\n \\qquad A_j\\mapsto A_j+\\frac{\\chi_{j+1}-\\chi_j}{h},\n \\qquad E_j\\mapsto E_j\n","context":1461,"kind":"use","source":27},{"line":282,"start":18567,"end":18570,"expression":"U_j","context":1461,"kind":"use","source":27},{"line":282,"start":18661,"end":18664,"expression":"z_j","context":1461,"kind":"use","source":27},{"line":284,"start":18808,"end":18814,"expression":"\\chi_j","context":1462,"kind":"use","source":27},{"line":284,"start":18870,"end":18887,"expression":"i\\dot\\chi_j\\psi_j","context":1462,"kind":"use","source":27},{"line":284,"start":18897,"end":18905,"expression":"\\dot A_j","context":1462,"kind":"use","source":27},{"line":284,"start":18917,"end":18946,"expression":"(\\dot\\chi_{j+1}-\\dot\\chi_j)/h","context":1462,"kind":"use","source":27},{"line":284,"start":18955,"end":18958,"expression":"E_j","context":1462,"kind":"use","source":27},{"line":284,"start":19154,"end":19155,"expression":"T","context":1463,"kind":"use","source":27},{"line":292,"start":19774,"end":19775,"expression":"A","context":1464,"kind":"use","source":27},{"line":292,"start":19788,"end":19791,"expression":"C^2","context":1464,"kind":"use","source":27},{"line":292,"start":19808,"end":19809,"expression":"x","context":1464,"kind":"use","source":27},{"line":292,"start":19817,"end":19821,"expression":"\\psi","context":1464,"kind":"use","source":27},{"line":292,"start":19827,"end":19830,"expression":"C^3","context":1464,"kind":"use","source":27},{"line":293,"start":19863,"end":20013,"expression":"\\label{g:integrated-link}\n U_h(x)=\\exp\\left(-i\\int_x^{x+h}A(s)\\,ds\\right),\\qquad\n (D_h^A\\psi)(x)\n =\\frac{U_h(x)\\psi(x+h)-U_h(x-h)^{-1}\\psi(x-h)}{2h}.\n","context":1464,"kind":"use","source":27},{"line":298,"start":20040,"end":20041,"expression":"A","context":1464,"kind":"use","source":27},{"line":298,"start":20152,"end":20155,"expression":"A_j","context":1465,"kind":"use","source":27},{"line":298,"start":20181,"end":20213,"expression":"h^{-1}\\int_{x_j}^{x_j+h}A(s)\\,ds","context":1466,"kind":"use","source":27},{"line":301,"start":20360,"end":20361,"expression":"A","context":1467,"kind":"use","source":27},{"line":301,"start":20409,"end":20413,"expression":"\\psi","context":1467,"kind":"use","source":27},{"line":302,"start":20496,"end":20585,"expression":"\\label{g:stencil-limit}\n (D_h^A\\psi)(x)=(\\partial_x-iA(x))\\psi(x)+O(h^2)\n \\qquad(h\\to0).\n","context":1467,"kind":"use","source":27},{"line":310,"start":20734,"end":20735,"expression":"x","context":1468,"kind":"use","source":27},{"line":311,"start":20743,"end":20787,"expression":"f_x(s)=\\exp(-i\\int_x^{x+s}A(r)\\,dr)\\psi(x+s)","context":1468,"kind":"use","source":27},{"line":312,"start":20853,"end":20859,"expression":"f_x(h)","context":1468,"kind":"use","source":27},{"line":312,"start":20866,"end":20873,"expression":"f_x(-h)","context":1468,"kind":"use","source":27},{"line":313,"start":20903,"end":20932,"expression":"f_x'(0)=\\psi'(x)-iA(x)\\psi(x)","context":1468,"kind":"use","source":27},{"line":314,"start":20956,"end":20959,"expression":"f_x","context":1468,"kind":"use","source":27},{"line":315,"start":21022,"end":21053,"expression":"f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)","context":1468,"kind":"use","source":27},{"line":316,"start":21069,"end":21071,"expression":"2h","context":1468,"kind":"use","source":27},{"line":316,"start":21122,"end":21123,"expression":"A","context":1469,"kind":"use","source":27},{"line":316,"start":21130,"end":21134,"expression":"\\psi","context":1470,"kind":"use","source":27},{"line":316,"start":21172,"end":21178,"expression":"f_x'''","context":1471,"kind":"use","source":27},{"line":319,"start":21211,"end":21244,"expression":"\\widetilde U_h(x)=e^{-ihA(x+h/2)}","context":1472,"kind":"use","source":27},{"line":319,"start":21340,"end":21346,"expression":"O(h^3)","context":1472,"kind":"use","source":27},{"line":319,"start":21354,"end":21377,"expression":"|e^{ia}-e^{ib}|\\le|a-b|","context":1472,"kind":"use","source":27},{"line":319,"start":21389,"end":21392,"expression":"a,b","context":1472,"kind":"use","source":27},{"line":319,"start":21470,"end":21476,"expression":"O(h^3)","context":1472,"kind":"use","source":27},{"line":319,"start":21491,"end":21495,"expression":"\\psi","context":1472,"kind":"use","source":27},{"line":319,"start":21533,"end":21539,"expression":"O(h^2)","context":1472,"kind":"use","source":27},{"line":322,"start":21640,"end":21646,"expression":"A(x)=x","context":1473,"kind":"use","source":27},{"line":322,"start":21669,"end":21678,"expression":"\\psi(x)=1","context":1473,"kind":"use","source":27},{"line":322,"start":21718,"end":21745,"expression":"\\widehat U_h(x)=e^{-ihA(x)}","context":1473,"kind":"use","source":27},{"line":323,"start":21773,"end":21870,"expression":"\n \\frac{\\widehat U_h(0)-\\widehat U_h(-h)^{-1}}{2h}\n =\\frac{1-e^{-ih^2}}{2h}=\\frac{i}{2}h+O(h^3).\n","context":1473,"kind":"use","source":27},{"line":327,"start":21894,"end":21919,"expression":"(\\partial_x-iA(0))\\psi(0)","context":1473,"kind":"use","source":27},{"line":327,"start":22001,"end":22015,"expression":"U_j=e^{-ihA_j}","context":1474,"kind":"use","source":27},{"line":336,"start":22584,"end":22597,"expression":"\\psi_x\\in\\C^2","context":1475,"kind":"use","source":27},{"line":336,"start":22601,"end":22619,"expression":"U_{x,\\mu}\\in SU(2)","context":1475,"kind":"use","source":27},{"line":336,"start":22640,"end":22652,"expression":"g_x\\in SU(2)","context":1475,"kind":"use","source":27},{"line":337,"start":22727,"end":22800,"expression":"\\label{g:linked-scalar}\n z_{x,\\mu}=\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n","context":1475,"kind":"use","source":27},{"line":340,"start":22855,"end":22860,"expression":"g_x^*","context":1475,"kind":"use","source":27},{"line":340,"start":22871,"end":22874,"expression":"g_x","context":1475,"kind":"use","source":27},{"line":340,"start":22897,"end":22910,"expression":"x+\\widehat\\mu","context":1475,"kind":"use","source":27},{"line":341,"start":22993,"end":23054,"expression":"\\label{g:su2-current}\n j_{x,\\mu}=2\\operatorname{Im}z_{x,\\mu}\n","context":1476,"kind":"use","source":27},{"line":346,"start":23286,"end":23312,"expression":"\\sigma^1,\\sigma^2,\\sigma^3","context":1477,"kind":"use","source":27},{"line":346,"start":23368,"end":23396,"expression":"q_x^a=\\psi_x^*\\sigma^a\\psi_x","context":1478,"kind":"use","source":27},{"line":347,"start":23452,"end":23565,"expression":"\\label{g:adjoint-color}\n g_x^*\\sigma^a g_x=\\sum_b R_x{}^a{}_b\\sigma^b,\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n","context":1478,"kind":"use","source":27},{"line":351,"start":23691,"end":23720,"expression":"\\tfrac12\\operatorname{Tr}(AB)","context":1479,"kind":"use","source":27},{"line":351,"start":23801,"end":23821,"expression":"R_xR_x^{\\mathsf T}=I","context":1480,"kind":"use","source":27},{"line":351,"start":23921,"end":23935,"expression":"\\psi_x^*\\psi_x","context":1481,"kind":"use","source":27},{"line":353,"start":24112,"end":24113,"expression":"W","context":1482,"kind":"use","source":27},{"line":353,"start":24259,"end":24264,"expression":"SU(2)","context":1482,"kind":"use","source":27},{"line":355,"start":24436,"end":24441,"expression":"SU(2)","context":1483,"kind":"use","source":27},{"line":361,"start":24913,"end":24921,"expression":"2\\times2","context":1484,"kind":"use","source":27},{"line":363,"start":24997,"end":25003,"expression":"X_{ij}","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25044,"end":25051,"expression":"(i+1,j)","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25057,"end":25062,"expression":"(i,j)","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25072,"end":25077,"expression":"i=0,1","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25084,"end":25091,"expression":"j=0,1,2","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25101,"end":25107,"expression":"Y_{ij}","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25146,"end":25153,"expression":"(i,j+1)","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25159,"end":25164,"expression":"(i,j)","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25174,"end":25181,"expression":"i=0,1,2","context":1485,"kind":"introduction-passage","source":27},{"line":363,"start":25188,"end":25193,"expression":"j=0,1","context":1485,"kind":"introduction-passage","source":27},{"line":364,"start":25247,"end":25345,"expression":"\\label{g:fine-plaquette}\n W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},\n \\qquad i,j\\in\\{0,1\\}.\n","context":1485,"kind":"introduction-passage","source":27},{"line":368,"start":25366,"end":25372,"expression":"W_{ij}","context":1485,"kind":"introduction-passage","source":27},{"line":368,"start":25412,"end":25417,"expression":"(i,j)","context":1486,"kind":"introduction-passage","source":27},{"line":368,"start":25454,"end":25459,"expression":"(0,j)","context":1487,"kind":"introduction-passage","source":27},{"line":369,"start":25482,"end":25560,"expression":"\\label{g:row-holonomies}\n R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},\n \\qquad j=0,1.\n","context":1488,"kind":"introduction-passage","source":27},{"line":379,"start":26075,"end":26241,"expression":"\n W^{(2)}\n &:=X_{00}X_{10}Y_{20}Y_{21}\n X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}\n \\label{g:coarse-boundary}\\\\\n &=R_0Y_{00}R_1Y_{00}^{-1}.\\label{g:patch-identity}\n","context":1489,"kind":"use","source":27},{"line":392,"start":26444,"end":26611,"expression":"\n R_j\n &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}\n X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\\\\n &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.\n","context":1490,"kind":"use","source":27},{"line":398,"start":26644,"end":26661,"expression":"X_{0j}^{-1}X_{0j}","context":1491,"kind":"use","source":27},{"line":398,"start":26668,"end":26685,"expression":"Y_{1j}^{-1}Y_{1j}","context":1491,"kind":"use","source":27},{"line":399,"start":26750,"end":26937,"expression":"\n R_0Y_{00}R_1Y_{00}^{-1}\n ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}\n \\underbrace{Y_{00}^{-1}Y_{00}}_{I}\\\\\n &\\quad\\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.\n","context":1492,"kind":"use","source":27},{"line":405,"start":26955,"end":26972,"expression":"X_{01}^{-1}X_{01}","context":1493,"kind":"use","source":27},{"line":405,"start":26996,"end":27013,"expression":"X_{11}^{-1}X_{11}","context":1493,"kind":"use","source":27},{"line":408,"start":27157,"end":27162,"expression":"(0,0)","context":1494,"kind":"use","source":27},{"line":416,"start":28151,"end":28173,"expression":"A,B\\in\\mathfrak{su}(2)","context":1495,"kind":"use","source":27},{"line":417,"start":28218,"end":28277,"expression":"\\label{g:commutator-loop}\n W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.\n","context":1495,"kind":"use","source":27},{"line":420,"start":28298,"end":28299,"expression":"A","context":1495,"kind":"use","source":27},{"line":420,"start":28306,"end":28307,"expression":"B","context":1495,"kind":"use","source":27},{"line":420,"start":28362,"end":28363,"expression":"h","context":1495,"kind":"use","source":27},{"line":420,"start":28402,"end":28407,"expression":"W_0=I","context":1495,"kind":"use","source":27},{"line":420,"start":28503,"end":28506,"expression":"|h|","context":1496,"kind":"use","source":27},{"line":421,"start":28532,"end":28590,"expression":"\\label{g:log-curvature}\n F_h=h^{-2}\\log W_h,\\qquad h\\ne0.\n","context":1497,"kind":"use","source":27},{"line":430,"start":29270,"end":29454,"expression":"\n F_h&=[A,B]+hC_3(A,B)+O(h^2),\\label{g:log-series}\\\\\n C_3(A,B)&=\\tfrac12[A+B,[A,B]],\\label{g:cubic-coefficient}\\\\\n F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).\n \\label{g:richardson-formula}\n","context":1498,"kind":"use","source":27},{"line":436,"start":29480,"end":29490,"expression":"G\\in SU(2)","context":1499,"kind":"use","source":27},{"line":436,"start":29522,"end":29525,"expression":"A,B","context":1499,"kind":"use","source":27},{"line":436,"start":29544,"end":29547,"expression":"F_h","context":1499,"kind":"use","source":27},{"line":436,"start":29554,"end":29563,"expression":"F_h^{[1]}","context":1499,"kind":"use","source":27},{"line":436,"start":29569,"end":29570,"expression":"G","context":1499,"kind":"use","source":27},{"line":441,"start":29681,"end":29731,"expression":"\n W_h=I+h^2[A,B]+\\frac{h^3}{2}[A+B,[A,B]]+O(h^4).\n","context":1500,"kind":"use","source":27},{"line":444,"start":29762,"end":29786,"expression":"\\log(I+Z)=Z+O(\\norm Z^2)","context":1500,"kind":"use","source":27},{"line":444,"start":29806,"end":29820,"expression":"Z=W_h-I=O(h^2)","context":1500,"kind":"use","source":27},{"line":444,"start":29855,"end":29858,"expression":"h^4","context":1500,"kind":"use","source":27},{"line":444,"start":29874,"end":29877,"expression":"h^2","context":1500,"kind":"use","source":27},{"line":444,"start":29944,"end":29950,"expression":"hC_3/2","context":1500,"kind":"use","source":27},{"line":444,"start":29957,"end":29969,"expression":"2F_{h/2}-F_h","context":1500,"kind":"use","source":27},{"line":444,"start":30005,"end":30011,"expression":"O(h^2)","context":1501,"kind":"use","source":27},{"line":447,"start":30288,"end":30294,"expression":"h^{-2}","context":1502,"kind":"use","source":27},{"line":24,"start":3025,"end":3037,"expression":"\\inner{v}{w}","context":1503,"kind":"use","source":28},{"line":24,"start":3063,"end":3064,"expression":"v","context":1503,"kind":"use","source":28},{"line":24,"start":3081,"end":3082,"expression":"w","context":1503,"kind":"use","source":28},{"line":24,"start":3114,"end":3117,"expression":"T^*","context":1503,"kind":"use","source":28},{"line":25,"start":3144,"end":3161,"expression":"T_{y\\leftarrow x}","context":1504,"kind":"use","source":28},{"line":25,"start":3182,"end":3183,"expression":"x","context":1504,"kind":"use","source":28},{"line":25,"start":3202,"end":3203,"expression":"y","context":1504,"kind":"use","source":28},{"line":25,"start":3235,"end":3236,"expression":"g","context":1504,"kind":"use","source":28},{"line":25,"start":3251,"end":3279,"expression":"g_yT_{y\\leftarrow x}g_x^{-1}","context":1504,"kind":"use","source":28},{"line":26,"start":3300,"end":3334,"expression":"T_{z\\leftarrow y}T_{y\\leftarrow x}","context":1505,"kind":"use","source":28},{"line":26,"start":3359,"end":3360,"expression":"x","context":1505,"kind":"use","source":28},{"line":26,"start":3366,"end":3367,"expression":"y","context":1505,"kind":"use","source":28},{"line":26,"start":3381,"end":3382,"expression":"y","context":1505,"kind":"use","source":28},{"line":26,"start":3388,"end":3389,"expression":"z","context":1505,"kind":"use","source":28},{"line":27,"start":3458,"end":3461,"expression":"U_j","context":1506,"kind":"use","source":28},{"line":27,"start":3467,"end":3480,"expression":"U_j\\psi_{j+1}","context":1506,"kind":"use","source":28},{"line":27,"start":3499,"end":3502,"expression":"j+1","context":1506,"kind":"use","source":28},{"line":27,"start":3513,"end":3514,"expression":"j","context":1506,"kind":"use","source":28},{"line":27,"start":3523,"end":3546,"expression":"U_j'=g_jU_jg_{j+1}^{-1}","context":1506,"kind":"use","source":28},{"line":28,"start":3622,"end":3626,"expression":"U(1)","context":1507,"kind":"use","source":28},{"line":28,"start":3725,"end":3730,"expression":"SU(2)","context":1507,"kind":"use","source":28},{"line":29,"start":3941,"end":3948,"expression":"\\hbar=1","context":1508,"kind":"use","source":28},{"line":5,"start":494,"end":498,"expression":"u(x)","context":1509,"kind":"introduction-passage","source":30},{"line":5,"start":517,"end":518,"expression":"x","context":1510,"kind":"introduction-passage","source":30},{"line":5,"start":524,"end":530,"expression":"\\tau x","context":1511,"kind":"introduction-passage","source":30},{"line":6,"start":538,"end":571,"expression":"\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n","context":1511,"kind":"introduction-passage","source":30},{"line":9,"start":580,"end":594,"expression":"u(x)=U(x)^{-1}","context":1511,"kind":"introduction-passage","source":30},{"line":9,"start":602,"end":606,"expression":"U(x)","context":1511,"kind":"introduction-passage","source":30},{"line":9,"start":743,"end":747,"expression":"U(1)","context":1511,"kind":"introduction-passage","source":30},{"line":9,"start":824,"end":829,"expression":"A=A^*","context":1512,"kind":"introduction-passage","source":30},{"line":9,"start":833,"end":838,"expression":"A^2=1","context":1513,"kind":"introduction-passage","source":30},{"line":9,"start":849,"end":853,"expression":"\\C^4","context":1514,"kind":"introduction-passage","source":30},{"line":9,"start":865,"end":881,"expression":"P_\\pm=(1\\pm A)/2","context":1515,"kind":"introduction-passage","source":30},{"line":10,"start":929,"end":1041,"expression":"\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n","context":1516,"kind":"introduction-passage","source":30},{"line":19,"start":1288,"end":1291,"expression":"e_x","context":1517,"kind":"introduction-passage","source":30},{"line":19,"start":1324,"end":1327,"expression":"m_x","context":1517,"kind":"introduction-passage","source":30},{"line":19,"start":1359,"end":1369,"expression":"x\\to\\tau x","context":1517,"kind":"introduction-passage","source":30},{"line":20,"start":1412,"end":1459,"expression":"\n \\sigma e_x=m_x,\\qquad \\sigma m_x=e_{\\tau x}.\n","context":1517,"kind":"introduction-passage","source":30},{"line":23,"start":1476,"end":1498,"expression":"\\sigma^2e_x=e_{\\tau x}","context":1517,"kind":"introduction-passage","source":30},{"line":23,"start":1505,"end":1527,"expression":"\\sigma^2m_x=m_{\\tau x}","context":1517,"kind":"introduction-passage","source":30},{"line":23,"start":1655,"end":1661,"expression":"\\sigma","context":1518,"kind":"introduction-passage","source":30},{"line":25,"start":1778,"end":1782,"expression":"a(x)","context":1519,"kind":"use","source":30},{"line":25,"start":1789,"end":1793,"expression":"b(x)","context":1519,"kind":"use","source":30},{"line":26,"start":1850,"end":1894,"expression":"\n b(x)a(x)=u(x).\n \\label{g:ordered-product}\n","context":1519,"kind":"use","source":30},{"line":30,"start":1938,"end":1949,"expression":"v(e_x)=a(x)","context":1519,"kind":"use","source":30},{"line":30,"start":1953,"end":1964,"expression":"v(m_x)=b(x)","context":1519,"kind":"use","source":30},{"line":31,"start":2026,"end":2207,"expression":"\n (I_a\\psi)(e_x)&=\\psi(x),&\n (I_a\\psi)(m_x)&=a(x)\\psi(x),\\label{g:edge-interpolation}\\\\\n (R_a\\Psi)(x)&=\\tfrac12\\bigl(\\Psi(e_x)+a(x)^{-1}\\Psi(m_x)\\bigr).\n \\label{g:edge-restriction}\n","context":1520,"kind":"use","source":30},{"line":39,"start":2398,"end":2402,"expression":"g(x)","context":1521,"kind":"use","source":30},{"line":39,"start":2425,"end":2429,"expression":"q(x)","context":1521,"kind":"use","source":30},{"line":40,"start":2492,"end":2558,"expression":"\n a^g(x)=q(x)a(x)g(x)^{-1},\\qquad\n b^g(x)=g(\\tau x)b(x)q(x)^{-1}.\n","context":1521,"kind":"use","source":30},{"line":44,"start":2593,"end":2599,"expression":"b^ga^g","context":1521,"kind":"use","source":30},{"line":45,"start":2665,"end":2702,"expression":"\n a^g(x)g(x)\\psi(x)=q(x)a(x)\\psi(x),\n","context":1521,"kind":"use","source":30},{"line":51,"start":2969,"end":2973,"expression":"\\psi","context":1522,"kind":"use","source":30},{"line":52,"start":3045,"end":3125,"expression":"\n S_{A,\\sigma,v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n","context":1523,"kind":"use","source":30},{"line":59,"start":3282,"end":3297,"expression":"P_+P_-=P_-P_+=0","context":1524,"kind":"use","source":30},{"line":59,"start":3424,"end":3427,"expression":"e_x","context":1525,"kind":"use","source":30},{"line":59,"start":3459,"end":3499,"expression":"b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)","context":1526,"kind":"use","source":30},{"line":59,"start":3531,"end":3559,"expression":"a(x)^{-1}b(x)^{-1}=u(x)^{-1}","context":1527,"kind":"use","source":30},{"line":59,"start":3612,"end":3615,"expression":"m_x","context":1528,"kind":"use","source":30},{"line":60,"start":3648,"end":3717,"expression":"\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n","context":1529,"kind":"use","source":30},{"line":64,"start":3727,"end":3750,"expression":"b(x)^{-1}=a(x)u(x)^{-1}","context":1530,"kind":"use","source":30},{"line":64,"start":3762,"end":3766,"expression":"a(x)","context":1531,"kind":"use","source":30},{"line":69,"start":3974,"end":4097,"expression":"\n \\inner{\\Psi}{\\Phi}_{1/2}\n =\\tfrac12\\sum_x\\bigl(\n    \\inner{\\Psi(e_x)}{\\Phi(e_x)}\n   +\\inner{\\Psi(m_x)}{\\Phi(m_x)}\\bigr).\n","context":1532,"kind":"use","source":30},{"line":75,"start":4114,"end":4118,"expression":"a(x)","context":1532,"kind":"use","source":30},{"line":76,"start":4147,"end":4284,"expression":"\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n","context":1532,"kind":"use","source":30},{"line":82,"start":4338,"end":4342,"expression":"a(x)","context":1533,"kind":"use","source":30},{"line":82,"start":4407,"end":4416,"expression":"a(x)^{-1}","context":1534,"kind":"use","source":30},{"line":82,"start":4476,"end":4479,"expression":"1/2","context":1535,"kind":"use","source":30},{"line":82,"start":4497,"end":4505,"expression":"R_aI_a=1","context":1536,"kind":"use","source":30},{"line":82,"start":4512,"end":4518,"expression":"I_aR_a","context":1537,"kind":"use","source":30},{"line":84,"start":4758,"end":4759,"expression":"N","context":1538,"kind":"use","source":30},{"line":84,"start":4772,"end":4777,"expression":"N^2=M","context":1538,"kind":"use","source":30},{"line":84,"start":4786,"end":4787,"expression":"N","context":1538,"kind":"use","source":30},{"line":85,"start":4844,"end":4899,"expression":"\n N^2S_{A,\\sigma,v}^{\\,2}N^2I_a\n   =I_aMS_{A,\\tau,u}M.\n","context":1538,"kind":"use","source":30},{"line":89,"start":4955,"end":4956,"expression":"N","context":1538,"kind":"use","source":30},{"line":89,"start":5095,"end":5112,"expression":"N S_{A,\\sigma,v}N","context":1539,"kind":"use","source":30},{"line":91,"start":5319,"end":5330,"expression":"a(s)=\\sin s","context":1540,"kind":"use","source":30},{"line":97,"start":5737,"end":5752,"expression":"2\\times2\\times2","context":1541,"kind":"use","source":30},{"line":97,"start":5788,"end":5807,"expression":"(x,y,z)\\in\\{0,1\\}^3","context":1541,"kind":"use","source":30},{"line":97,"start":5831,"end":5842,"expression":"\\ell_x(y,z)","context":1541,"kind":"use","source":30},{"line":97,"start":5846,"end":5857,"expression":"\\ell_y(x,z)","context":1541,"kind":"use","source":30},{"line":97,"start":5864,"end":5875,"expression":"\\ell_z(x,y)","context":1541,"kind":"use","source":30},{"line":99,"start":6072,"end":6074,"expression":"xy","context":1542,"kind":"use","source":30},{"line":99,"start":6092,"end":6093,"expression":"z","context":1542,"kind":"use","source":30},{"line":100,"start":6101,"end":6178,"expression":"\n p_{xy}(z)=\\ell_y(1,z)\\ell_x(0,z),\\qquad\n p_{yx}(z)=\\ell_x(1,z)\\ell_y(0,z).\n","context":1542,"kind":"use","source":30},{"line":104,"start":6189,"end":6227,"expression":"\\Omega_{xy}(z)=p_{xy}(z)p_{yx}(z)^{-1}","context":1542,"kind":"use","source":30},{"line":104,"start":6246,"end":6248,"expression":"xz","context":1542,"kind":"use","source":30},{"line":104,"start":6255,"end":6257,"expression":"yz","context":1542,"kind":"use","source":30},{"line":104,"start":6384,"end":6390,"expression":"\\Omega","context":1542,"kind":"use","source":30},{"line":104,"start":6417,"end":6425,"expression":"\\Omega=1","context":1542,"kind":"use","source":30},{"line":107,"start":6591,"end":6598,"expression":"p_{XYZ}","context":1543,"kind":"use","source":30},{"line":107,"start":6602,"end":6609,"expression":"p_{XZY}","context":1543,"kind":"use","source":30},{"line":107,"start":6613,"end":6620,"expression":"p_{YXZ}","context":1543,"kind":"use","source":30},{"line":107,"start":6624,"end":6631,"expression":"p_{YZX}","context":1543,"kind":"use","source":30},{"line":107,"start":6635,"end":6642,"expression":"p_{ZXY}","context":1543,"kind":"use","source":30},{"line":107,"start":6649,"end":6656,"expression":"p_{ZYX}","context":1543,"kind":"use","source":30},{"line":109,"start":6822,"end":7028,"expression":"\n p_{XYZ}&=\\Omega_{xy}(0)p_{YXZ},&\n p_{XYZ}&=\\Omega_{yz}(1)p_{XZY},\\\\\n p_{YXZ}&=\\Omega_{xz}(1)p_{YZX},&\n p_{XZY}&=\\Omega_{xz}(0)p_{ZXY},\\\\\n p_{YZX}&=\\Omega_{yz}(0)p_{ZYX},&\n p_{ZXY}&=\\Omega_{xy}(1)p_{ZYX}.\n","context":1544,"kind":"use","source":30},{"line":121,"start":7342,"end":7452,"expression":"\n \\Omega_{yz}(1)\\Omega_{xz}(0)\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n","context":1545,"kind":"use","source":30},{"line":126,"start":7567,"end":7572,"expression":"xy(1)","context":1545,"kind":"use","source":30},{"line":129,"start":7694,"end":7697,"expression":"XYZ","context":1546,"kind":"use","source":30},{"line":129,"start":7708,"end":7715,"expression":"YXZ,YZX","context":1547,"kind":"use","source":30},{"line":129,"start":7721,"end":7724,"expression":"ZYX","context":1548,"kind":"use","source":30},{"line":129,"start":7771,"end":7778,"expression":"XZY,ZXY","context":1549,"kind":"use","source":30},{"line":129,"start":7795,"end":7802,"expression":"p_{XYZ}","context":1550,"kind":"use","source":30},{"line":129,"start":7859,"end":7866,"expression":"p_{ZYX}","context":1551,"kind":"use","source":30},{"line":129,"start":7978,"end":7994,"expression":"\\Omega_{xy}(1)=1","context":1552,"kind":"use","source":30},{"line":135,"start":8584,"end":8588,"expression":"r(v)","context":1553,"kind":"use","source":30},{"line":136,"start":8615,"end":8738,"expression":"\n (I_\\ell\\psi)(v)=r(v)\\psi,\n \\qquad\n \\inner{\\Psi}{\\Phi}_{\\rm cube}\n =\\tfrac18\\sum_{v\\in\\{0,1\\}^3}\\inner{\\Psi(v)}{\\Phi(v)}.\n","context":1553,"kind":"use","source":30},{"line":142,"start":8748,"end":8752,"expression":"r(v)","context":1553,"kind":"use","source":30},{"line":144,"start":9059,"end":9060,"expression":"p","context":1554,"kind":"use","source":30},{"line":144,"start":9067,"end":9068,"expression":"q","context":1554,"kind":"use","source":30},{"line":144,"start":9076,"end":9086,"expression":"p=\\Omega q","context":1554,"kind":"use","source":30},{"line":145,"start":9142,"end":9270,"expression":"\n D(p,q;\\psi)=p\\psi-q\\psi=q(\\Omega-1)\\psi,\n \\qquad\n \\norm{D(p,q;\\psi)}^2=|\\Omega-1|^2\\norm{\\psi}^2.\n \\label{g:curvature-defect}\n","context":1554,"kind":"use","source":30},{"line":151,"start":9368,"end":9369,"expression":"q","context":1554,"kind":"use","source":30},{"line":151,"start":9382,"end":9387,"expression":"|q|=1","context":1554,"kind":"use","source":30},{"line":151,"start":9451,"end":9459,"expression":"\\Omega=1","context":1555,"kind":"use","source":30},{"line":151,"start":9491,"end":9501,"expression":"p_{XYZ}=-1","context":1556,"kind":"use","source":30},{"line":151,"start":9508,"end":9517,"expression":"p_{YXZ}=1","context":1557,"kind":"use","source":30},{"line":154,"start":9758,"end":9906,"expression":"\n \\kappa(\\ell)={}&|\\Omega_{xy}(0)-1|+|\\Omega_{xz}(1)-1|\n  +|\\Omega_{yz}(0)-1|\\notag\\\\\n &+|\\Omega_{yz}(1)-1|+|\\Omega_{xz}(0)-1|.\n \\label{g:fiveface}\n","context":1558,"kind":"use","source":30},{"line":160,"start":9939,"end":9944,"expression":"xy(1)","context":1559,"kind":"use","source":30},{"line":168,"start":10518,"end":10546,"expression":"\\mathrm{Site}\\times\\{0,1\\}^3","context":1560,"kind":"use","source":30},{"line":168,"start":10571,"end":10579,"expression":"\\sigma_i","context":1560,"kind":"use","source":30},{"line":168,"start":10693,"end":10699,"expression":"\\tau_i","context":1560,"kind":"use","source":30},{"line":168,"start":10720,"end":10730,"expression":"\\sigma_i^2","context":1560,"kind":"use","source":30},{"line":168,"start":10748,"end":10754,"expression":"\\tau_i","context":1561,"kind":"use","source":30},{"line":168,"start":10772,"end":10786,"expression":"r(s,c)\\in U(1)","context":1562,"kind":"use","source":30},{"line":169,"start":10798,"end":10830,"expression":"\n (I_r\\psi)(s,c)=r(s,c)\\psi(s).\n","context":1563,"kind":"use","source":30},{"line":176,"start":11000,"end":11048,"expression":"\n v_i(p)=r(\\sigma_i p)\\,\\gamma_i(p)\\,r(p)^{-1},\n","context":1564,"kind":"use","source":30},{"line":179,"start":11058,"end":11071,"expression":"\\gamma_i(p)=1","context":1565,"kind":"use","source":30},{"line":179,"start":11105,"end":11123,"expression":"\\gamma_i(p)=u_i(s)","context":1566,"kind":"use","source":30},{"line":180,"start":11211,"end":11297,"expression":"\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n","context":1567,"kind":"use","source":30},{"line":186,"start":11556,"end":11610,"expression":"\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n","context":1568,"kind":"use","source":30},{"line":189,"start":11710,"end":11723,"expression":"M-X-Y-Z-Y-X-M","context":1569,"kind":"use","source":30},{"line":191,"start":12027,"end":12028,"expression":"x","context":1570,"kind":"use","source":30},{"line":191,"start":12035,"end":12036,"expression":"y","context":1570,"kind":"use","source":30},{"line":194,"start":12415,"end":12420,"expression":"(a,b)","context":1571,"kind":"use","source":30},{"line":194,"start":12451,"end":12455,"expression":"v=ba","context":1571,"kind":"use","source":30},{"line":194,"start":12489,"end":12490,"expression":"v","context":1571,"kind":"use","source":30},{"line":194,"start":12545,"end":12551,"expression":"u\\ne v","context":1571,"kind":"use","source":30},{"line":195,"start":12599,"end":12745,"expression":"\n D_{\\rm fine}\\psi\n  &:=S_{A,\\sigma,(a,b)}^2I_a\\psi-I_aS_{A,\\tau,u}\\psi\\notag\\\\\n  &=I_a(S_{A,\\tau,v}-S_{A,\\tau,u})\\psi.\n \\label{g:exact-mismatch}\n","context":1572,"kind":"use","source":30},{"line":201,"start":12824,"end":12825,"expression":"v","context":1573,"kind":"use","source":30},{"line":201,"start":12851,"end":12854,"expression":"I_a","context":1573,"kind":"use","source":30},{"line":201,"start":12885,"end":12888,"expression":"I_a","context":1573,"kind":"use","source":30},{"line":203,"start":13188,"end":13189,"expression":"i","context":1574,"kind":"use","source":30},{"line":203,"start":13207,"end":13210,"expression":"u=1","context":1574,"kind":"use","source":30},{"line":203,"start":13231,"end":13235,"expression":"v=-1","context":1574,"kind":"use","source":30},{"line":203,"start":13282,"end":13283,"expression":"i","context":1574,"kind":"use","source":30},{"line":203,"start":13366,"end":13372,"expression":"\\tau=1","context":1574,"kind":"use","source":30},{"line":204,"start":13380,"end":13434,"expression":"\n S_{A,1,1}=P_++P_-=1,\\qquad\n S_{A,1,-1}=-P_+-P_-=-1.\n","context":1574,"kind":"use","source":30},{"line":208,"start":13466,"end":13472,"expression":"-2\\psi","context":1574,"kind":"use","source":30},{"line":208,"start":13516,"end":13528,"expression":"2\\norm{\\psi}","context":1575,"kind":"use","source":30},{"line":216,"start":14214,"end":14220,"expression":"\\ell^2","context":1576,"kind":"use","source":30},{"line":217,"start":14390,"end":14488,"expression":"\n \\HH_S=\\ell^2(S\\times\\{0,1,2,3\\};\\C),\\qquad\n \\norm{\\psi}^2=\\sum_{x\\in S}\\norm{\\psi(x)}_{\\C^4}^2.\n","context":1576,"kind":"use","source":30},{"line":221,"start":14711,"end":14712,"expression":"S","context":1577,"kind":"use","source":30},{"line":224,"start":14834,"end":14835,"expression":"A","context":1578,"kind":"use","source":30},{"line":224,"start":14874,"end":14878,"expression":"\\tau","context":1579,"kind":"use","source":30},{"line":224,"start":14903,"end":14916,"expression":"u,v:S\\to U(1)","context":1579,"kind":"use","source":30},{"line":225,"start":14940,"end":15076,"expression":"\n \\norm{S_{A,\\tau,v}-S_{A,\\tau,u}}\n \\leq\\delta(u,v):=\\max_{x\\in S}|v(x)-u(x)|\n =\\max_{x\\in S}|v(x)u(x)^{-1}-1|.\n \\label{g:constant-one}\n","context":1579,"kind":"use","source":30},{"line":235,"start":15448,"end":15648,"expression":"\n \\norm{(S_v-S_u)\\psi}^2\n =\\sum_x\\bigl[&|v(\\tau^{-1}x)-u(\\tau^{-1}x)|^2\n                   \\norm{P_+\\psi(\\tau^{-1}x)}^2\\\\\n &+|v(x)^{-1}-u(x)^{-1}|^2\n                   \\norm{P_-\\psi(\\tau x)}^2\\bigr].\n","context":1580,"kind":"use","source":30},{"line":242,"start":15807,"end":15828,"expression":"|v^{-1}-u^{-1}|=|v-u|","context":1581,"kind":"use","source":30},{"line":242,"start":15859,"end":15867,"expression":"\\delta^2","context":1581,"kind":"use","source":30},{"line":242,"start":15900,"end":15904,"expression":"\\tau","context":1581,"kind":"use","source":30},{"line":242,"start":15916,"end":15963,"expression":"\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2","context":1581,"kind":"use","source":30},{"line":242,"start":15989,"end":16010,"expression":"\\delta^2\\norm{\\psi}^2","context":1582,"kind":"use","source":30},{"line":242,"start":16112,"end":16121,"expression":"u(x)^{-1}","context":1583,"kind":"use","source":30},{"line":242,"start":16215,"end":16218,"expression":"I_a","context":1584,"kind":"use","source":30},{"line":245,"start":16450,"end":16458,"expression":"1/\\sqrt2","context":1585,"kind":"use","source":30},{"line":245,"start":16526,"end":16532,"expression":"\\sqrt2","context":1585,"kind":"use","source":30},{"line":248,"start":16749,"end":16757,"expression":"I:X\\to Y","context":1586,"kind":"use","source":30},{"line":248,"start":16803,"end":16813,"expression":"F_j:Y\\to Y","context":1587,"kind":"use","source":30},{"line":248,"start":16817,"end":16827,"expression":"C_j:X\\to X","context":1587,"kind":"use","source":30},{"line":248,"start":16864,"end":16877,"expression":"D_j=F_jI-IC_j","context":1587,"kind":"use","source":30},{"line":249,"start":16921,"end":16954,"expression":"\n F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.\n","context":1587,"kind":"use","source":30},{"line":252,"start":17003,"end":17013,"expression":"1,\\ldots,N","context":1587,"kind":"use","source":30},{"line":253,"start":17032,"end":17141,"expression":"\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n","context":1587,"kind":"use","source":30},{"line":258,"start":17197,"end":17204,"expression":"F_j,C_j","context":1588,"kind":"use","source":30},{"line":259,"start":17278,"end":17346,"expression":"\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n","context":1589,"kind":"use","source":30},{"line":263,"start":17371,"end":17372,"expression":"I","context":1590,"kind":"use","source":30},{"line":263,"start":17454,"end":17457,"expression":"D_j","context":1591,"kind":"use","source":30},{"line":265,"start":17602,"end":17605,"expression":"I=1","context":1592,"kind":"use","source":30},{"line":265,"start":17642,"end":17653,"expression":"v_x,v_y,v_z","context":1593,"kind":"use","source":30},{"line":265,"start":17688,"end":17699,"expression":"u_x,u_y,u_z","context":1594,"kind":"use","source":30},{"line":265,"start":17754,"end":17761,"expression":"M_0,M_1","context":1595,"kind":"use","source":30},{"line":265,"start":17786,"end":17803,"expression":"M_0-X-Y-Z-Y-X-M_1","context":1596,"kind":"use","source":30},{"line":265,"start":17814,"end":17838,"expression":"\\delta_i=\\delta(u_i,v_i)","context":1596,"kind":"use","source":30},{"line":266,"start":17946,"end":18020,"expression":"\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n","context":1597,"kind":"use","source":30},{"line":270,"start":18049,"end":18050,"expression":"X","context":1598,"kind":"use","source":30},{"line":270,"start":18057,"end":18058,"expression":"Y","context":1599,"kind":"use","source":30},{"line":279,"start":18927,"end":18936,"expression":"z\\in U(1)","context":1600,"kind":"use","source":30},{"line":279,"start":18944,"end":18957,"expression":"P_z\\psi=z\\psi","context":1600,"kind":"use","source":30},{"line":279,"start":18988,"end":18998,"expression":"p=\\Omega q","context":1600,"kind":"use","source":30},{"line":280,"start":19008,"end":19068,"expression":"\n P_p-P_q=q(\\Omega-1)\\id,\\qquad\n \\norm{P_p-P_q}=|\\Omega-1|.\n","context":1600,"kind":"use","source":30},{"line":287,"start":19250,"end":19367,"expression":"\n (p_{XYZ},p_{YXZ}),\\quad(p_{YXZ},p_{YZX}),\\quad\n (p_{YZX},p_{ZYX}),\\quad(p_{XYZ},p_{XZY}),\\quad\n (p_{XZY},p_{ZXY}).\n","context":1601,"kind":"use","source":30},{"line":292,"start":19375,"end":19378,"expression":"Q_f","context":1601,"kind":"use","source":30},{"line":292,"start":19461,"end":19464,"expression":"Q_c","context":1601,"kind":"use","source":30},{"line":293,"start":19616,"end":19650,"expression":"\n \\norm{Q_f-Q_c}\\leq\\kappa(\\ell).\n","context":1602,"kind":"use","source":30},{"line":300,"start":20038,"end":20069,"expression":"\n D_{\\rm total}=Q_fW_v-Q_cW_u.\n","context":1603,"kind":"use","source":30},{"line":304,"start":20093,"end":20192,"expression":"\n \\norm{D_{\\rm total}}\n \\leq 2\\delta_x+2\\delta_y+\\delta_z+\\kappa(\\ell).\n \\label{g:combined-budget}\n","context":1603,"kind":"use","source":30},{"line":309,"start":20211,"end":20218,"expression":"v_i=u_i","context":1603,"kind":"use","source":30},{"line":312,"start":20344,"end":20350,"expression":"Q_fW_u","context":1604,"kind":"use","source":30},{"line":313,"start":20355,"end":20398,"expression":"\n D_{\\rm total}=Q_f(W_v-W_u)+(Q_f-Q_c)W_u.\n","context":1605,"kind":"use","source":30},{"line":316,"start":20437,"end":20440,"expression":"W_u","context":1606,"kind":"use","source":30},{"line":326,"start":21289,"end":21300,"expression":"A_x,A_y,A_z","context":1607,"kind":"use","source":30},{"line":327,"start":21397,"end":21505,"expression":"\n \\theta_i(p,h)=\\int_0^h A_i(p+t e_i)\\,dt,\n \\qquad u_i(p,h)=e^{i\\theta_i(p,h)}.\n \\label{g:integrated-links}\n","context":1607,"kind":"use","source":30},{"line":332,"start":21551,"end":21554,"expression":"A_i","context":1608,"kind":"use","source":30},{"line":332,"start":21640,"end":21643,"expression":"h<0","context":1609,"kind":"use","source":30},{"line":335,"start":21717,"end":21720,"expression":"A_i","context":1610,"kind":"use","source":30},{"line":336,"start":21830,"end":21960,"expression":"\n \\theta_i(p,h)\n &=\\int_0^{h/2}A_i(p+t e_i)\\,dt\n   +\\int_{h/2}^{h}A_i(p+t e_i)\\,dt\\\\\n &=\\theta_i(p,h/2)+\\theta_i(p+(h/2)e_i,h/2).\n","context":1611,"kind":"use","source":30},{"line":343,"start":22031,"end":22105,"expression":"\n u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h).\n \\label{g:integral-composition}\n","context":1612,"kind":"use","source":30},{"line":349,"start":22456,"end":22492,"expression":"A_i(p+t e_i)=A_i(p)+t\\,\\partial_iA_i","context":1613,"kind":"use","source":30},{"line":350,"start":22500,"end":22550,"expression":"\n \\theta_i(p,h)=hA_i(p)+\\tfrac12h^2\\partial_iA_i.\n","context":1613,"kind":"use","source":30},{"line":353,"start":22580,"end":22589,"expression":"h^2F_{ij}","context":1613,"kind":"use","source":30},{"line":353,"start":22599,"end":22633,"expression":"F_{ij}=\\partial_iA_j-\\partial_jA_i","context":1613,"kind":"use","source":30},{"line":353,"start":22643,"end":22669,"expression":"\\Omega_{ij}=e^{ih^2F_{ij}}","context":1613,"kind":"use","source":30},{"line":353,"start":22679,"end":22696,"expression":"|e^{it}-1|\\leq|t|","context":1613,"kind":"use","source":30},{"line":354,"start":22749,"end":22850,"expression":"\n \\norm{D_{\\rm total}(h)}\n \\leq h^2\\bigl(|F_{xy}|+2|F_{xz}|+2|F_{yz}|\\bigr).\n \\label{g:affine-bound}\n","context":1613,"kind":"use","source":30},{"line":362,"start":23166,"end":23179,"expression":"\\partial_iA_j","context":1614,"kind":"use","source":30},{"line":362,"start":23186,"end":23192,"expression":"i\\ne j","context":1614,"kind":"use","source":30},{"line":363,"start":23480,"end":23532,"expression":"\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n","context":1615,"kind":"use","source":30},{"line":366,"start":23549,"end":23560,"expression":"B_{y\\mid x}","context":1616,"kind":"use","source":30},{"line":366,"start":23588,"end":23591,"expression":"A_y","context":1617,"kind":"use","source":30},{"line":366,"start":23601,"end":23602,"expression":"x","context":1618,"kind":"use","source":30},{"line":369,"start":23823,"end":23867,"expression":"\n |A_j(p+h e_i)-A_j(p)|\\leq B_{j\\mid i}|h|.\n","context":1619,"kind":"use","source":30},{"line":372,"start":23904,"end":23905,"expression":"j","context":1619,"kind":"use","source":30},{"line":373,"start":23939,"end":24028,"expression":"\n |\\theta_j(p+h e_i,h)-\\theta_j(p,h)|\n \\leq B_{j\\mid i}|h|^2.\n \\label{g:angle-variation}\n","context":1619,"kind":"use","source":30},{"line":378,"start":24090,"end":24104,"expression":"B_{j\\mid i}|h|","context":1619,"kind":"use","source":30},{"line":378,"start":24147,"end":24150,"expression":"|h|","context":1619,"kind":"use","source":30},{"line":380,"start":24230,"end":24232,"expression":"xy","context":1620,"kind":"use","source":30},{"line":381,"start":24287,"end":24400,"expression":"\n \\Theta_{xy}(p,h)\n ={}&\\theta_y(p+h e_x,h)-\\theta_y(p,h)\\\\\n    &-\\bigl(\\theta_x(p+h e_y,h)-\\theta_x(p,h)\\bigr).\n","context":1621,"kind":"use","source":30},{"line":387,"start":24425,"end":24550,"expression":"\n |\\Theta_{xy}(p,h)|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2,\n \\qquad\n |\\Omega_{xy}(p,h)-1|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2.\n","context":1622,"kind":"use","source":30},{"line":394,"start":24580,"end":24610,"expression":"e^{it}-1=\\int_0^t i e^{is}\\,ds","context":1622,"kind":"use","source":30},{"line":394,"start":24640,"end":24643,"expression":"|t|","context":1622,"kind":"use","source":30},{"line":394,"start":24874,"end":24880,"expression":"F_{ij}","context":1623,"kind":"use","source":30},{"line":398,"start":25014,"end":25165,"expression":"\n \\mathcal A_{\\partial}={}&B_{y\\mid x}+B_{x\\mid y}\n +2(B_{z\\mid x}+B_{x\\mid z})\\notag\\\\\n &+2(B_{z\\mid y}+B_{y\\mid z}).\n \\label{g:derivative-amplitude}\n","context":1624,"kind":"use","source":30},{"line":404,"start":25386,"end":25387,"expression":"p","context":1625,"kind":"use","source":30},{"line":405,"start":25532,"end":25613,"expression":"\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n","context":1626,"kind":"use","source":30},{"line":411,"start":25703,"end":25731,"expression":"\\delta_x=\\delta_y=\\delta_z=0","context":1627,"kind":"use","source":30},{"line":411,"start":25820,"end":25822,"expression":"xy","context":1628,"kind":"use","source":30},{"line":411,"start":25835,"end":25837,"expression":"xz","context":1629,"kind":"use","source":30},{"line":411,"start":25854,"end":25856,"expression":"yz","context":1630,"kind":"use","source":30},{"line":411,"start":25873,"end":25908,"expression":"\\kappa\\leq h^2\\mathcal A_{\\partial}","context":1631,"kind":"use","source":30},{"line":417,"start":26458,"end":26467,"expression":"A_x=A_z=0","context":1632,"kind":"use","source":30},{"line":417,"start":26474,"end":26491,"expression":"A_y(x,y,z)=\\sin x","context":1632,"kind":"use","source":30},{"line":417,"start":26509,"end":26522,"expression":"F_{xy}=\\cos x","context":1632,"kind":"use","source":30},{"line":417,"start":26581,"end":26594,"expression":"B_{y\\mid x}=1","context":1632,"kind":"use","source":30},{"line":417,"start":26601,"end":26624,"expression":"\\mathcal A_{\\partial}=1","context":1632,"kind":"use","source":30},{"line":418,"start":26634,"end":26723,"expression":"\n \\theta_y(p,h)=h\\sin x,\\qquad\n \\Omega_{xy}(p,h)=\\exp\\!\\bigl(ih[\\sin(x+h)-\\sin x]\\bigr).\n","context":1632,"kind":"use","source":30},{"line":422,"start":26802,"end":26805,"expression":"h^2","context":1632,"kind":"use","source":30},{"line":424,"start":27084,"end":27094,"expression":"A_y=\\sin x","context":1633,"kind":"use","source":30},{"line":424,"start":27186,"end":27206,"expression":"|\\partial_xA_y|\\leq1","context":1633,"kind":"use","source":30},{"line":430,"start":27398,"end":27412,"expression":"h_n=a_0 2^{-n}","context":1634,"kind":"use","source":30},{"line":431,"start":27472,"end":27533,"expression":"\n \\varepsilon_n=C4^{-n},\\qquad C=a_0^2\\mathcal A_{\\partial}.\n","context":1634,"kind":"use","source":30},{"line":434,"start":27663,"end":27680,"expression":"\\varepsilon_n\\to0","context":1634,"kind":"use","source":30},{"line":438,"start":27893,"end":27896,"expression":"f_n","context":1635,"kind":"use","source":30},{"line":439,"start":27953,"end":28010,"expression":"\n d(f_n,f_{n+1})\\leq C4^{-n}\\qquad(n\\geq0),\\quad C\\geq0.\n","context":1635,"kind":"use","source":30},{"line":442,"start":28019,"end":28022,"expression":"f_n","context":1635,"kind":"use","source":30},{"line":442,"start":28043,"end":28051,"expression":"f_\\infty","context":1635,"kind":"use","source":30},{"line":443,"start":28074,"end":28164,"expression":"\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n","context":1635,"kind":"use","source":30},{"line":450,"start":28216,"end":28219,"expression":"m>n","context":1636,"kind":"use","source":30},{"line":451,"start":28254,"end":28316,"expression":"\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n","context":1637,"kind":"use","source":30},{"line":455,"start":28355,"end":28356,"expression":"n","context":1638,"kind":"use","source":30},{"line":455,"start":28426,"end":28436,"expression":"m\\to\\infty","context":1639,"kind":"use","source":30},{"line":458,"start":28957,"end":28960,"expression":"f_n","context":1640,"kind":"use","source":30},{"line":462,"start":29704,"end":29718,"expression":"\\frac43C4^{-n}","context":1641,"kind":"use","source":30},{"line":469,"start":30104,"end":30107,"expression":"L^2","context":1642,"kind":"use","source":30},{"line":472,"start":30388,"end":30391,"expression":"v>0","context":1643,"kind":"use","source":30},{"line":472,"start":30435,"end":30445,"expression":"q_a\\in\\C^4","context":1643,"kind":"use","source":30},{"line":472,"start":30456,"end":30459,"expression":"C_a","context":1643,"kind":"use","source":30},{"line":472,"start":30494,"end":30513,"expression":"\\psi_a=\\sqrt v\\,q_a","context":1643,"kind":"use","source":30},{"line":473,"start":30544,"end":30588,"expression":"\n (J\\psi)(x)=v^{-1/2}\\psi_a\\quad(x\\in C_a),\n","context":1643,"kind":"use","source":30},{"line":477,"start":30662,"end":30750,"expression":"\n \\norm{J\\psi}_{L^2}^2\n =\\sum_a\\int_{C_a}v^{-1}\\norm{\\psi_a}^2\n =\\sum_a\\norm{\\psi_a}^2.\n","context":1643,"kind":"use","source":30},{"line":482,"start":30813,"end":30832,"expression":"v\\sum_a\\norm{q_a}^2","context":1644,"kind":"use","source":30},{"line":482,"start":30985,"end":30988,"expression":"L^2","context":1645,"kind":"use","source":30},{"line":484,"start":31009,"end":31010,"expression":"n","context":1646,"kind":"use","source":30},{"line":484,"start":31037,"end":31040,"expression":"8^n","context":1646,"kind":"use","source":30},{"line":484,"start":31064,"end":31070,"expression":"2^{-n}","context":1646,"kind":"use","source":30},{"line":484,"start":31093,"end":31099,"expression":"8^{-n}","context":1646,"kind":"use","source":30},{"line":484,"start":31480,"end":31483,"expression":"d+1","context":1647,"kind":"use","source":30},{"line":484,"start":31507,"end":31508,"expression":"n","context":1648,"kind":"use","source":30},{"line":489,"start":32328,"end":32331,"expression":"L^2","context":1649,"kind":"use","source":30},{"line":491,"start":32873,"end":32876,"expression":"L^2","context":1650,"kind":"use","source":30},{"line":493,"start":33049,"end":33052,"expression":"P_n","context":1651,"kind":"use","source":30},{"line":494,"start":33064,"end":33150,"expression":"\n \\norm{P_nf-f}_{L^2}\\longrightarrow0\n \\qquad\\text{for every }f\\in L^2([0,1]^3;\\C^4).\n","context":1651,"kind":"use","source":30},{"line":7,"start":892,"end":901,"expression":"(Z^A,P_A)","context":1652,"kind":"use","source":32},{"line":8,"start":963,"end":1105,"expression":"\n \\{F,G\\}=\\sum_A\\operatorname{Tr}\\left(\n \\nabla_{Z^A}F\\nabla_{P_A}G-\n \\nabla_{P_A}F\\nabla_{Z^A}G\\right).\n \\label{c:constraints:trace-bracket}\n","context":1652,"kind":"use","source":32},{"line":14,"start":1199,"end":1206,"expression":"\\Lambda","context":1653,"kind":"use","source":32},{"line":15,"start":1229,"end":1296,"expression":"\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n","context":1654,"kind":"use","source":32},{"line":19,"start":1384,"end":1463,"expression":"\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n","context":1655,"kind":"use","source":32},{"line":24,"start":1510,"end":1541,"expression":"\\nabla_PD[\\Lambda]=i[\\Lambda,Z]","context":1656,"kind":"use","source":32},{"line":25,"start":1548,"end":1579,"expression":"\\nabla_ZD[\\Lambda]=i[P,\\Lambda]","context":1657,"kind":"use","source":32},{"line":29,"start":1687,"end":1773,"expression":"\n \\{D[\\Lambda],D[\\Gamma]\\}=D[i[\\Lambda,\\Gamma]].\n \\label{c:constraints:inner-closure}\n","context":1658,"kind":"use","source":32},{"line":33,"start":1809,"end":1819,"expression":"\\mathcal E","context":1658,"kind":"use","source":32},{"line":34,"start":1828,"end":1875,"expression":"\\{\\mathcal E,D[\\Lambda]\\}=i[\\Lambda,\\mathcal E]","context":1658,"kind":"use","source":32},{"line":35,"start":1884,"end":1919,"expression":"H[N]=\\operatorname{Tr}(N\\mathcal E)","context":1658,"kind":"use","source":32},{"line":36,"start":1947,"end":1986,"expression":"\n \\{D[\\Lambda],H[N]\\}=H[i[\\Lambda,N]].\n","context":1658,"kind":"use","source":32},{"line":42,"start":2108,"end":2395,"expression":"\n \\{D[\\Lambda],D[\\Gamma]\\}\n &=\\operatorname{Tr}\\left(\n   -[P,\\Lambda][\\Gamma,Z]+[\\Lambda,Z][P,\\Gamma]\\right)\\\\\n &=\\operatorname{Tr}\\left(P\\bigl(\n   -[\\Lambda,[\\Gamma,Z]]+[\\Gamma,[\\Lambda,Z]]\\bigr)\\right)\\\\\n &=-\\operatorname{Tr}\\bigl(P[[\\Lambda,\\Gamma],Z]\\bigr)\n   =D[i[\\Lambda,\\Gamma]].\n","context":1659,"kind":"use","source":32},{"line":52,"start":2510,"end":2622,"expression":"\n \\{D[\\Lambda],H[N]\\}\n =-i\\operatorname{Tr}(N[\\Lambda,\\mathcal E])\n =\\operatorname{Tr}(i[\\Lambda,N]\\mathcal E).\n","context":1660,"kind":"use","source":32},{"line":59,"start":2794,"end":2808,"expression":"\\Lambda_h[\\xi]","context":1661,"kind":"use","source":32},{"line":62,"start":3136,"end":3154,"expression":"(\\mathcal C_F,s_F)","context":1662,"kind":"introduction-passage","source":32},{"line":62,"start":3187,"end":3194,"expression":"s_F^2=0","context":1663,"kind":"introduction-passage","source":32},{"line":62,"start":3202,"end":3231,"expression":"P:\\mathcal C_F\\to\\mathcal C_C","context":1663,"kind":"introduction-passage","source":32},{"line":62,"start":3238,"end":3267,"expression":"I:\\mathcal C_C\\to\\mathcal C_F","context":1663,"kind":"introduction-passage","source":32},{"line":62,"start":3301,"end":3317,"expression":"PI=\\mathrm{id}_C","context":1663,"kind":"introduction-passage","source":32},{"line":63,"start":3344,"end":3388,"expression":"\n Q=\\mathrm{id}_F-IP,\\qquad \\bar s_C=Ps_FI.\n","context":1663,"kind":"introduction-passage","source":32},{"line":66,"start":3409,"end":3410,"expression":"I","context":1663,"kind":"introduction-passage","source":32},{"line":66,"start":3443,"end":3454,"expression":"\\mathrm{id}","context":1663,"kind":"introduction-passage","source":32},{"line":66,"start":3500,"end":3518,"expression":"IP=\\mathrm{id}_F-Q","context":1664,"kind":"introduction-passage","source":32},{"line":67,"start":3562,"end":3635,"expression":"\n \\bar s_C^2\n =Ps_F(IP)s_FI\n =-Ps_FQs_FI.\n \\label{c:constraints:leakage}\n","context":1665,"kind":"introduction-passage","source":32},{"line":76,"start":3961,"end":4049,"expression":"\n \\bar s_C=\\begin{pmatrix}\n -1/25&-1/25&-26/625\\\\\n 1&0&1/25\\\\\n 0&1/25&0\n \\end{pmatrix}.\n","context":1666,"kind":"use","source":32},{"line":83,"start":4095,"end":4100,"expression":"(2,2)","context":1667,"kind":"use","source":32},{"line":83,"start":4112,"end":4117,"expression":"-1/25","context":1667,"kind":"use","source":32},{"line":84,"start":4179,"end":4254,"expression":"\n s_C=\\begin{pmatrix}0&0&0\\\\1&0&1/25\\\\0&0&0\\end{pmatrix},\n \\qquad s_C^2=0.\n","context":1668,"kind":"use","source":32},{"line":91,"start":4571,"end":4585,"expression":"s_F=s_0+\\delta","context":1669,"kind":"use","source":32},{"line":91,"start":4595,"end":4602,"expression":"s_0^2=0","context":1669,"kind":"use","source":32},{"line":91,"start":4609,"end":4615,"expression":"\\delta","context":1669,"kind":"use","source":32},{"line":91,"start":4642,"end":4664,"expression":"(\\mathcal C_C,s_{0,C})","context":1669,"kind":"use","source":32},{"line":92,"start":4731,"end":4769,"expression":"\n s_0I=Is_{0,C},\\qquad Ps_0=s_{0,C}P.\n","context":1669,"kind":"use","source":32},{"line":95,"start":4820,"end":4821,"expression":"H","context":1669,"kind":"use","source":32},{"line":96,"start":4849,"end":5007,"expression":"\n \\begin{gathered}\n PI=\\mathrm{id}_C,\\qquad\n \\mathrm{id}_F-IP=s_0H+Hs_0,\\\\\n PH=0,\\qquad HI=0,\\qquad H^2=0.\n \\end{gathered}\n \\label{c:constraints:contraction}\n","context":1670,"kind":"use","source":32},{"line":104,"start":5040,"end":5056,"expression":"(s_0+\\delta)^2=0","context":1671,"kind":"use","source":32},{"line":104,"start":5077,"end":5098,"expression":"\\mathrm{id}_F+H\\delta","context":1671,"kind":"use","source":32},{"line":108,"start":5403,"end":5524,"expression":"\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n","context":1672,"kind":"use","source":32},{"line":114,"start":5545,"end":5568,"expression":"PI_\\delta=\\mathrm{id}_C","context":1673,"kind":"use","source":32},{"line":115,"start":5572,"end":5605,"expression":"(s_0+\\delta)I_\\delta=I_\\delta s_C","context":1674,"kind":"use","source":32},{"line":115,"start":5613,"end":5620,"expression":"s_C^2=0","context":1675,"kind":"use","source":32},{"line":118,"start":5676,"end":5703,"expression":"I_\\delta=I-H\\delta I_\\delta","context":1676,"kind":"use","source":32},{"line":119,"start":5722,"end":5723,"expression":"P","context":1677,"kind":"use","source":32},{"line":119,"start":5733,"end":5759,"expression":"PI_\\delta=PI=\\mathrm{id}_C","context":1678,"kind":"use","source":32},{"line":121,"start":5782,"end":5821,"expression":"\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n","context":1679,"kind":"use","source":32},{"line":125,"start":5870,"end":5896,"expression":"\\mathrm{id}_F-IP=s_0H+Hs_0","context":1680,"kind":"use","source":32},{"line":126,"start":5939,"end":6036,"expression":"\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n","context":1681,"kind":"use","source":32},{"line":131,"start":6094,"end":6125,"expression":"s_0\\delta+\\delta s_0+\\delta^2=0","context":1682,"kind":"use","source":32},{"line":131,"start":6162,"end":6171,"expression":"-\\delta K","context":1683,"kind":"use","source":32},{"line":132,"start":6181,"end":6207,"expression":"(\\mathrm{id}_F+H\\delta)K=0","context":1684,"kind":"use","source":32},{"line":132,"start":6235,"end":6238,"expression":"K=0","context":1685,"kind":"use","source":32},{"line":134,"start":6309,"end":6325,"expression":"0=I_\\delta s_C^2","context":1686,"kind":"use","source":32},{"line":134,"start":6346,"end":6347,"expression":"P","context":1687,"kind":"use","source":32},{"line":134,"start":6361,"end":6368,"expression":"s_C^2=0","context":1688,"kind":"use","source":32},{"line":138,"start":6455,"end":6577,"expression":"\n \\begin{aligned}\n s_C={}&s_{0,C}+P\\delta I-P\\delta H\\delta I\\\\\n       &+P\\delta H\\delta H\\delta I-\\cdots.\n \\end{aligned}\n","context":1689,"kind":"use","source":32},{"line":144,"start":6648,"end":6649,"expression":"H","context":1690,"kind":"use","source":32},{"line":146,"start":6919,"end":6922,"expression":"c^A","context":1691,"kind":"use","source":32},{"line":146,"start":6977,"end":6985,"expression":"\\varpi_A","context":1691,"kind":"use","source":32},{"line":147,"start":7072,"end":7190,"expression":"\n \\Omega_F=c^AC_A-\\tfrac12c^Ac^BU_{AB}{}^C\\varpi_C\n             +\\Omega_F^{(\\ge3)}.\n \\label{c:constraints:bfv-charge}\n","context":1691,"kind":"use","source":32},{"line":154,"start":7523,"end":7546,"expression":"\\{\\Omega_F,\\Omega_F\\}=0","context":1692,"kind":"use","source":32},{"line":155,"start":7555,"end":7577,"expression":"s_F=\\{\\Omega_F,\\cdot\\}","context":1692,"kind":"use","source":32},{"line":155,"start":7868,"end":7876,"expression":"\\Omega_C","context":1692,"kind":"use","source":32},{"line":155,"start":7912,"end":7941,"expression":"\\tfrac12\\{\\Omega_C,\\Omega_C\\}","context":1692,"kind":"use","source":32},{"line":159,"start":8267,"end":8322,"expression":"\n \\|H\\delta\\|\\le\\rho<1,\\qquad\n \\|Q\\delta I\\|\\le C h^r.\n","context":1693,"kind":"use","source":32},{"line":163,"start":8365,"end":8369,"expression":"H=HQ","context":1693,"kind":"use","source":32},{"line":164,"start":8389,"end":8579,"expression":"\n s_C-(s_{0,C}+P\\delta I)\n &=-P\\delta(\\mathrm{id}_F+H\\delta)^{-1}H\\delta I,\\\\\n \\|s_C-(s_{0,C}+P\\delta I)\\|\n &\\le\\frac{\\|P\\delta\\|\\,\\|H\\|}{1-\\rho}\\,C h^r.\n \\label{c:constraints:return-bound}\n","context":1694,"kind":"use","source":32},{"line":171,"start":8702,"end":8703,"expression":"h","context":1695,"kind":"use","source":32},{"line":171,"start":8778,"end":8787,"expression":"Q\\delta I","context":1695,"kind":"use","source":32},{"line":177,"start":9102,"end":9141,"expression":"\\mathbb T^3=(\\mathbb R/2\\pi\\mathbb Z)^3","context":1696,"kind":"use","source":32},{"line":178,"start":9164,"end":9261,"expression":"\n \\mathcal T_K=\\left\\{\\sum_{\\|k\\|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n","context":1696,"kind":"use","source":32},{"line":183,"start":9320,"end":9421,"expression":"\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n","context":1696,"kind":"use","source":32},{"line":187,"start":9442,"end":9454,"expression":"\\mathcal T_K","context":1696,"kind":"use","source":32},{"line":188,"start":9484,"end":9505,"expression":"\\bigcup_K\\mathcal T_K","context":1697,"kind":"use","source":32},{"line":189,"start":9556,"end":9559,"expression":"S_N","context":1698,"kind":"use","source":32},{"line":189,"start":9575,"end":9578,"expression":"N^3","context":1699,"kind":"use","source":32},{"line":189,"start":9596,"end":9604,"expression":"\\delta_a","context":1700,"kind":"use","source":32},{"line":190,"start":9638,"end":9657,"expression":"\\tau_N=N^{-3}\\sum_x","context":1701,"kind":"use","source":32},{"line":192,"start":9686,"end":9708,"expression":"f_j\\in\\mathcal T_{K_j}","context":1702,"kind":"use","source":32},{"line":192,"start":9715,"end":9728,"expression":"\\sum_jK_j<N/2","context":1702,"kind":"use","source":32},{"line":193,"start":9853,"end":10046,"expression":"\n \\delta_a\\prod_jS_Nf_j\n &=S_N\\partial_a\\left(\\prod_jf_j\\right),\\\\\n \\tau_N\\left(\\prod_jS_Nf_j\\right)\n &=(2\\pi)^{-3}\\int_{\\mathbb T^3}\\prod_jf_j\\,\\mathrm d^3x.\n \\label{c:constraints:alias-free}\n","context":1703,"kind":"use","source":32},{"line":203,"start":10478,"end":10483,"expression":"A_a^i","context":1704,"kind":"use","source":32},{"line":203,"start":10505,"end":10510,"expression":"E^a_i","context":1704,"kind":"use","source":32},{"line":203,"start":10534,"end":10543,"expression":"\\lambda^i","context":1704,"kind":"use","source":32},{"line":203,"start":10554,"end":10559,"expression":"\\xi^a","context":1704,"kind":"use","source":32},{"line":203,"start":10584,"end":10585,"expression":"n","context":1704,"kind":"use","source":32},{"line":203,"start":10591,"end":10603,"expression":"\\mathcal T_K","context":1704,"kind":"use","source":32},{"line":204,"start":10669,"end":11012,"expression":"\n F_{ab}^i&=\\delta_aA_b^i-\\delta_bA_a^i\n                    +\\epsilon^i{}_{jk}A_a^jA_b^k,\\\\\n \\mathcal G_i&=\\delta_aE^a_i+\\epsilon_{ij}{}^kA_a^jE^a_k,\\\\\n \\mathcal V_a&=F_{ab}^iE^b_i,\\qquad\n \\mathcal D_a=\\mathcal V_a-A_a^i\\mathcal G_i,\\\\\n \\mathcal H&=\\tfrac12\\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k\n                                  +\\Lambda\\det E.\n","context":1705,"kind":"use","source":32},{"line":213,"start":11030,"end":11088,"expression":"\\det E=\\tfrac16\\epsilon_{abc}\\epsilon^{ijk}E^a_iE^b_jE^c_k","context":1706,"kind":"use","source":32},{"line":215,"start":11131,"end":11276,"expression":"\nG[\\lambda]&=\\tau_N(\\lambda^i\\mathcal G_i),&\nD[\\xi]&=\\tau_N(\\xi^a\\mathcal D_a),\\\\\nV[\\xi]&=\\tau_N(\\xi^a\\mathcal V_a),&\nH[n]&=\\tau_N(n\\mathcal H).\n","context":1707,"kind":"use","source":32},{"line":223,"start":11359,"end":11571,"expression":"\n \\{F,G\\}_N=N^3\\sum_{x,a,i}\n \\left(\\frac{\\partial F}{\\partial A_a^i(x)}\n       \\frac{\\partial G}{\\partial E^a_i(x)}\n      -\\frac{\\partial F}{\\partial E^a_i(x)}\n       \\frac{\\partial G}{\\partial A_a^i(x)}\\right).\n","context":1708,"kind":"use","source":32},{"line":230,"start":11598,"end":11601,"expression":"N^3","context":1708,"kind":"use","source":32},{"line":230,"start":11813,"end":11825,"expression":"\\mathcal T_K","context":1709,"kind":"use","source":32},{"line":233,"start":11945,"end":12167,"expression":"\n (\\lambda\\times\\mu)^i&=\\epsilon^i{}_{jk}\\lambda^j\\mu^k,\\\\\n [\\xi,\\eta]^a&=\\xi^b\\delta_b\\eta^a-\\eta^b\\delta_b\\xi^a,\\\\\n \\mathcal L_\\xi^{(-1)}n&=\\xi^a\\delta_an-n\\delta_a\\xi^a,\\\\\n S^a(n,m)&=E^a_iE^{bi}(m\\delta_bn-n\\delta_bm).\n","context":1710,"kind":"use","source":32},{"line":242,"start":12497,"end":12502,"expression":"16K<N","context":1711,"kind":"use","source":32},{"line":243,"start":12559,"end":12893,"expression":"\n \\{G[\\lambda],G[\\mu]\\}_N&=G[\\lambda\\times\\mu],\\\\\n \\{D[\\xi],G[\\lambda]\\}_N&=G[\\xi^a\\delta_a\\lambda],\\\\\n \\{D[\\xi],D[\\eta]\\}_N&=D[[\\xi,\\eta]],\\\\\n \\{G[\\lambda],H[n]\\}_N&=0,\\\\\n \\{D[\\xi],H[n]\\}_N&=H[\\mathcal L_\\xi^{(-1)}n],\\\\\n \\{H[n],H[m]\\}_N&=V[S(n,m)]\n                  =D[S(n,m)]+G[S^a(n,m)A_a].\n \\label{c:constraints:resolved-algebra}\n","context":1712,"kind":"use","source":32},{"line":255,"start":13063,"end":13064,"expression":"K","context":1713,"kind":"use","source":32},{"line":255,"start":13213,"end":13219,"expression":"8K<N/2","context":1714,"kind":"use","source":32},{"line":255,"start":13591,"end":13610,"expression":"V[S]=D[S]+G[S^aA_a]","context":1715,"kind":"use","source":32},{"line":255,"start":13653,"end":13665,"expression":"\\mathcal D_a","context":1716,"kind":"use","source":32},{"line":261,"start":14146,"end":14158,"expression":"\\mathcal T_K","context":1717,"kind":"use","source":32},{"line":262,"start":14353,"end":14388,"expression":"\n I_{L\\to M}I_{K\\to L}=I_{K\\to M}.\n","context":1717,"kind":"use","source":32},{"line":267,"start":14767,"end":14771,"expression":"N=18","context":1718,"kind":"use","source":32},{"line":267,"start":14775,"end":14778,"expression":"K=1","context":1718,"kind":"use","source":32},{"line":267,"start":14788,"end":14793,"expression":"16K<N","context":1718,"kind":"use","source":32},{"line":267,"start":14832,"end":14840,"expression":"10^{-17}","context":1718,"kind":"use","source":32},{"line":267,"start":14947,"end":14963,"expression":"3.3\\times10^{-8}","context":1718,"kind":"use","source":32},{"line":269,"start":15263,"end":15264,"expression":"E","context":1719,"kind":"use","source":32},{"line":270,"start":15271,"end":15295,"expression":"A+\\overline A=2\\Gamma(E)","context":1719,"kind":"use","source":32},{"line":273,"start":15637,"end":15641,"expression":"\\tau","context":1720,"kind":"use","source":32},{"line":273,"start":15658,"end":15671,"expression":"t=e^{-\\tau}>0","context":1720,"kind":"use","source":32},{"line":274,"start":15718,"end":15826,"expression":"\n \\mathcal H_t=\\frac{\\Pi_P^2+e^{-2P}\\Pi_Q^2}{2t}\n                   +\\frac t2(P_\\theta^2+e^{2P}Q_\\theta^2).\n","context":1720,"kind":"use","source":32},{"line":278,"start":15884,"end":15885,"expression":"t","context":1720,"kind":"use","source":32},{"line":279,"start":15915,"end":15953,"expression":"\\mathcal D=\\Pi_PP_\\theta+\\Pi_QQ_\\theta","context":1720,"kind":"use","source":32},{"line":279,"start":16084,"end":16096,"expression":"\\mathcal H_t","context":1721,"kind":"use","source":32},{"line":287,"start":16966,"end":17005,"expression":"\\eta_{IJ}=\\operatorname{diag}(-1,1,1,1)","context":1722,"kind":"use","source":32},{"line":287,"start":17012,"end":17029,"expression":"\\epsilon_{0123}=1","context":1722,"kind":"use","source":32},{"line":287,"start":17075,"end":17078,"expression":"e^I","context":1722,"kind":"use","source":32},{"line":287,"start":17122,"end":17133,"expression":"\\omega^{IJ}","context":1722,"kind":"use","source":32},{"line":287,"start":17159,"end":17160,"expression":"a","context":1722,"kind":"use","source":32},{"line":287,"start":17198,"end":17210,"expression":"\\varphi_{IJ}","context":1722,"kind":"use","source":32},{"line":287,"start":17231,"end":17244,"expression":"\\psi,\\bar\\psi","context":1722,"kind":"use","source":32},{"line":288,"start":17298,"end":17552,"expression":"\n R^{IJ}&=\\mathrm d\\omega^{IJ}+\\omega^I{}_K\\wedge\\omega^{KJ},\n &F&=\\mathrm da,\\\\\n \\operatorname{vol}_e&=\\tfrac1{4!}\\epsilon_{IJKL}\n                         e^I\\wedge e^J\\wedge e^K\\wedge e^L,\n &\\Sigma_I&=\\tfrac1{3!}\\epsilon_{IJKL}e^J\\wedge e^K\\wedge e^L.\n","context":1723,"kind":"use","source":32},{"line":295,"start":17569,"end":17601,"expression":"\\{\\gamma^I,\\gamma^J\\}=2\\eta^{IJ}","context":1724,"kind":"use","source":32},{"line":296,"start":17608,"end":17647,"expression":"\\gamma_{IJ}=\\tfrac12[\\gamma_I,\\gamma_J]","context":1724,"kind":"use","source":32},{"line":297,"start":17693,"end":17864,"expression":"\n \\mathcal D\\psi&=\\mathrm d\\psi+\\tfrac14\\omega^{IJ}\\gamma_{IJ}\\psi-iq a\\psi,\\\\\n \\mathcal D\\bar\\psi&=\\mathrm d\\bar\\psi-\\tfrac14\\bar\\psi\\omega^{IJ}\\gamma_{IJ}+iq a\\bar\\psi.\n","context":1725,"kind":"use","source":32},{"line":301,"start":17891,"end":17892,"expression":"q","context":1726,"kind":"use","source":32},{"line":301,"start":17978,"end":17990,"expression":"\\det(q_{ab})","context":1726,"kind":"use","source":32},{"line":303,"start":18048,"end":18099,"expression":"S_0=S_{\\mathrm G}+S_{\\mathrm M}^{(1)}+S_{\\mathrm D}","context":1727,"kind":"use","source":32},{"line":304,"start":18120,"end":18654,"expression":"\n S_{\\mathrm G}\n &=\\frac1{4\\kappa}\\int\\epsilon_{IJKL}e^I\\wedge e^J\\wedge R^{KL}\n                  -\\frac\\Lambda\\kappa\\int\\operatorname{vol}_e,\\\\\n S_{\\mathrm M}^{(1)}\n &=\\int\\left(\\tfrac12\\varphi_{IJ}F\\wedge e^I\\wedge e^J\n                    -\\tfrac14\\varphi_{IJ}\\varphi^{IJ}\\operatorname{vol}_e\\right),\\\\\n S_{\\mathrm D}\n &=\\frac i2\\int\\Sigma_I\\wedge\n       \\left(\\bar\\psi\\gamma^I\\mathcal D\\psi\n               -(\\mathcal D\\bar\\psi)\\gamma^I\\psi\\right)\n       -m\\int\\operatorname{vol}_e\\bar\\psi\\psi.\n \\label{c:constraints:parent-action}\n","context":1728,"kind":"use","source":32},{"line":320,"start":19110,"end":19111,"expression":"i","context":1729,"kind":"use","source":32},{"line":320,"start":19140,"end":19164,"expression":"i\\gamma^I\\partial_I\\pm m","context":1730,"kind":"use","source":32},{"line":321,"start":19195,"end":19307,"expression":"\n (i\\gamma^I\\partial_I+m)(i\\gamma^J\\partial_J-m)\n       =-\\Box-m^2,\n \\qquad \\Box=\\eta^{IJ}\\partial_I\\partial_J.\n","context":1730,"kind":"use","source":32},{"line":326,"start":19358,"end":19368,"expression":"\\Box+m^2=0","context":1731,"kind":"use","source":32},{"line":326,"start":19453,"end":19463,"expression":"\\Box-m^2=0","context":1732,"kind":"use","source":32},{"line":328,"start":19859,"end":19866,"expression":"\\varphi","context":1733,"kind":"use","source":32},{"line":329,"start":19921,"end":20020,"expression":"\n F\\wedge e^I\\wedge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n","context":1734,"kind":"use","source":32},{"line":334,"start":20083,"end":20152,"expression":"\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e","context":1734,"kind":"use","source":32},{"line":335,"start":20181,"end":20188,"expression":"\\varphi","context":1734,"kind":"use","source":32},{"line":335,"start":20229,"end":20236,"expression":"\\star F","context":1734,"kind":"use","source":32},{"line":335,"start":20253,"end":20270,"expression":"F=E e^0\\wedge e^1","context":1735,"kind":"use","source":32},{"line":336,"start":20277,"end":20301,"expression":"\\star F=-E e^2\\wedge e^3","context":1736,"kind":"use","source":32},{"line":336,"start":20338,"end":20352,"expression":"\\varphi^{23}=E","context":1737,"kind":"use","source":32},{"line":336,"start":20380,"end":20395,"expression":"\\varphi=\\star F","context":1738,"kind":"use","source":32},{"line":337,"start":20523,"end":20615,"expression":"\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n","context":1739,"kind":"use","source":32},{"line":344,"start":20800,"end":20815,"expression":"(\\xi,\\lambda,u)","context":1740,"kind":"use","source":32},{"line":345,"start":20904,"end":20940,"expression":"\\mathcal L_\\xi\\omega-D_\\omega\\lambda","context":1740,"kind":"use","source":32},{"line":345,"start":20973,"end":21000,"expression":"\\mathcal L_\\xi a+\\mathrm du","context":1740,"kind":"use","source":32},{"line":346,"start":21029,"end":21091,"expression":"\\mathcal L_\\xi\\psi+\\tfrac14\\lambda^{IJ}\\gamma_{IJ}\\psi+iqu\\psi","context":1740,"kind":"use","source":32},{"line":347,"start":21119,"end":21127,"expression":"\\bar\\psi","context":1740,"kind":"use","source":32},{"line":350,"start":21541,"end":21629,"expression":"\n [\\delta_U,\\delta_V]_{\\mathrm{fld}}\n       =-\\delta_{\\left[U,V\\right]_{\\mathrm{par}}},\n","context":1741,"kind":"use","source":32},{"line":355,"start":21666,"end":21861,"expression":"\n \\left[(\\xi,\\lambda,u),(\\eta,\\mu,v)\\right]_{\\mathrm{par}}\n =\\left([\\xi,\\eta],\\mathcal L_\\xi\\mu-\\mathcal L_\\eta\\lambda+[\\lambda,\\mu],\n                  \\mathcal L_\\xi v-\\mathcal L_\\eta u\\right).\n","context":1741,"kind":"use","source":32},{"line":362,"start":22003,"end":22018,"expression":"c,\\chi,\\upsilon","context":1742,"kind":"use","source":32},{"line":362,"start":22043,"end":22044,"expression":"s","context":1742,"kind":"use","source":32},{"line":363,"start":22156,"end":22309,"expression":"\n sc^\\mu=c^\\nu\\partial_\\nu c^\\mu,\\qquad\n s\\chi=\\mathcal L_c\\chi+\\chi^2,\\qquad\n s\\upsilon=\\mathcal L_c\\upsilon.\n \\label{c:constraints:ghost-differential}\n","context":1742,"kind":"use","source":32},{"line":370,"start":22484,"end":22709,"expression":"\n s\\psi&=\\mathcal L_c\\psi+\\tfrac14\\chi^{IJ}\\gamma_{IJ}\\psi+iq\\upsilon\\psi,\\\\\n s\\bar\\psi&=\\mathcal L_c\\bar\\psi\n                -\\tfrac14\\chi^{IJ}\\bar\\psi\\gamma_{IJ}-iq\\upsilon\\bar\\psi.\n \\label{c:constraints:dual-ghost-action}\n","context":1743,"kind":"use","source":32},{"line":376,"start":22768,"end":22772,"expression":"\\chi","context":1744,"kind":"use","source":32},{"line":376,"start":22779,"end":22787,"expression":"\\bar\\psi","context":1744,"kind":"use","source":32},{"line":376,"start":22814,"end":22827,"expression":"-\\bar\\psi\\chi","context":1744,"kind":"use","source":32},{"line":378,"start":23215,"end":23245,"expression":"g=\\tfrac14\\chi^{IJ}\\gamma_{IJ}","context":1745,"kind":"use","source":32},{"line":378,"start":23306,"end":23312,"expression":"sg=g^2","context":1745,"kind":"use","source":32},{"line":378,"start":23316,"end":23327,"expression":"s\\psi=g\\psi","context":1745,"kind":"use","source":32},{"line":378,"start":23352,"end":23372,"expression":"s\\bar\\psi=\\bar\\psi g","context":1745,"kind":"use","source":32},{"line":379,"start":23443,"end":23522,"expression":"\n s^2\\psi=g^2\\psi-g(g\\psi)=0,\\qquad\n s^2\\bar\\psi=(\\bar\\psi g)g-\\bar\\psi g^2=0.\n","context":1746,"kind":"use","source":32},{"line":383,"start":23544,"end":23565,"expression":"s\\bar\\psi=-\\bar\\psi g","context":1747,"kind":"use","source":32},{"line":383,"start":23623,"end":23636,"expression":"2\\bar\\psi g^2","context":1748,"kind":"use","source":32},{"line":386,"start":23785,"end":23788,"expression":"\\Xi","context":1749,"kind":"use","source":32},{"line":386,"start":23813,"end":23825,"expression":"\\Xi^\\ddagger","context":1749,"kind":"use","source":32},{"line":386,"start":23866,"end":23891,"expression":"-1-\\operatorname{gh}(\\Xi)","context":1749,"kind":"use","source":32},{"line":387,"start":24044,"end":24150,"expression":"\n S_{\\mathrm{BV}}=S_0+\n     \\int\\sum_\\Xi\\langle\\Xi^\\ddagger,s\\Xi\\rangle.\n \\label{c:constraints:bv-action}\n","context":1749,"kind":"use","source":32},{"line":392,"start":24276,"end":24279,"expression":"1/2","context":1750,"kind":"use","source":32},{"line":395,"start":24456,"end":24462,"expression":"sS_0=0","context":1751,"kind":"use","source":32},{"line":395,"start":24469,"end":24474,"expression":"s^2=0","context":1751,"kind":"use","source":32},{"line":396,"start":24498,"end":24537,"expression":"\n (S_{\\mathrm{BV}},S_{\\mathrm{BV}})=0.\n","context":1751,"kind":"use","source":32},{"line":401,"start":24600,"end":24603,"expression":"S_0","context":1751,"kind":"use","source":32},{"line":401,"start":24725,"end":24731,"expression":"s^2\\Xi","context":1751,"kind":"use","source":32},{"line":401,"start":24850,"end":24853,"expression":"S_0","context":1752,"kind":"use","source":32},{"line":409,"start":25634,"end":25652,"expression":"\\mathcal T_K^{(4)}","context":1753,"kind":"use","source":32},{"line":409,"start":25866,"end":25867,"expression":"K","context":1753,"kind":"use","source":32},{"line":409,"start":25899,"end":25904,"expression":"16K<N","context":1753,"kind":"use","source":32},{"line":411,"start":26143,"end":26144,"expression":"e","context":1754,"kind":"use","source":32},{"line":411,"start":26151,"end":26157,"expression":"\\omega","context":1754,"kind":"use","source":32},{"line":411,"start":26213,"end":26233,"expression":"\\star^2=-\\mathrm{id}","context":1754,"kind":"use","source":32},{"line":412,"start":26270,"end":26310,"expression":"\n P_\\pm=\\tfrac12(\\mathrm{id}\\mp i\\star)\n","context":1754,"kind":"use","source":32},{"line":415,"start":26334,"end":26354,"expression":"\\overline{P_+X}=P_-X","context":1754,"kind":"use","source":32},{"line":421,"start":26813,"end":26828,"expression":"M=I\\times\\Sigma","context":1755,"kind":"use","source":32},{"line":421,"start":26848,"end":26889,"expression":"\\mathrm dt\\wedge\\operatorname{or}(\\Sigma)","context":1755,"kind":"use","source":32},{"line":422,"start":26914,"end":27035,"expression":"\n e^I=e_\\perp^I\\mathrm dt+E^I,\\qquad\n \\omega^{IJ}=\\omega_\\perp^{IJ}\\mathrm dt+\\Omega^{IJ},\\qquad\n a=a_\\perp\\mathrm dt+A.\n","context":1755,"kind":"use","source":32},{"line":427,"start":27055,"end":27107,"expression":"\\mathcal R=\\mathrm d_\\Sigma\\Omega+\\Omega\\wedge\\Omega","context":1755,"kind":"use","source":32},{"line":427,"start":27114,"end":27143,"expression":"\\mathcal F=\\mathrm d_\\Sigma A","context":1755,"kind":"use","source":32},{"line":428,"start":27204,"end":27343,"expression":"\n R=\\mathrm dt\\wedge(\\dot\\Omega-D_\\Omega\\omega_\\perp)+\\mathcal R,\n \\qquad\n F=\\mathrm dt\\wedge(\\dot A-\\mathrm d_\\Sigma a_\\perp)+\\mathcal F.\n","context":1756,"kind":"use","source":32},{"line":434,"start":27378,"end":27544,"expression":"\n B_{IJ}&=\\tfrac1{4\\kappa}\\epsilon_{KLIJ}E^K\\wedge E^L,\\\\\n \\Pi&=\\tfrac12\\varphi_{IJ}E^I\\wedge E^J,\\qquad\n \\sigma_I=\\tfrac1{3!}\\epsilon_{IJKL}E^J\\wedge E^K\\wedge E^L.\n","context":1757,"kind":"use","source":32},{"line":440,"start":27642,"end":27909,"expression":"\n \\begin{aligned}\n \\Theta_\\Sigma=\\int_\\Sigma\\bigl[&B_{IJ}\\wedge\\delta\\Omega^{IJ}+\\Pi\\wedge\\delta A\\\\\n &-\\tfrac i2\\sigma_I(\\bar\\psi\\gamma^I\\delta\\psi\n                         -\\delta\\bar\\psi\\gamma^I\\psi)\\bigr].\n \\end{aligned}\n \\label{c:constraints:boundary-potential}\n","context":1758,"kind":"use","source":32},{"line":448,"start":28013,"end":28023,"expression":"\\mathrm dt","context":1759,"kind":"use","source":32},{"line":451,"start":28202,"end":28289,"expression":"\\mathcal C_I=\\mathcal C_I^{\\mathrm G}+\\mathcal C_I^{\\mathrm M}+\\mathcal C_I^{\\mathrm D}","context":1760,"kind":"use","source":32},{"line":452,"start":28311,"end":28788,"expression":"\n \\mathcal C_I^{\\mathrm G}\n &=\\tfrac1{2\\kappa}\\epsilon_{IJKL}E^J\\wedge\\mathcal R^{KL}\n                  -\\tfrac\\Lambda\\kappa\\sigma_I,\\\\\n \\mathcal C_I^{\\mathrm M}\n &=\\varphi_{IJ}\\mathcal F\\wedge E^J-\\tfrac14\\varphi_{KL}\\varphi^{KL}\\sigma_I,\\\\\n \\mathcal C_I^{\\mathrm D}\n &=\\tfrac i4\\epsilon_{JIKL}E^K\\wedge E^L\\wedge\n       \\bigl(\\bar\\psi\\gamma^J\\mathcal D_\\Sigma\\psi\n                  -(\\mathcal D_\\Sigma\\bar\\psi)\\gamma^J\\psi\\bigr)\n                      -m\\sigma_I\\bar\\psi\\psi.\n","context":1761,"kind":"use","source":32},{"line":465,"start":28890,"end":29070,"expression":"\n \\mathcal G_{IJ}\n &=D_\\Omega B_{IJ}\n       -\\tfrac i8\\sigma_K\\bar\\psi\\{\\gamma^K,\\gamma_{IJ}\\}\\psi,\\\\\n \\mathcal G_{\\mathrm U}\n &=\\mathrm d_\\Sigma\\Pi-q\\sigma_I\\bar\\psi\\gamma^I\\psi.\n","context":1762,"kind":"use","source":32},{"line":473,"start":29125,"end":29178,"expression":"\\mathcal C_I=\\mathcal G_{IJ}=\\mathcal G_{\\mathrm U}=0","context":1763,"kind":"use","source":32},{"line":478,"start":29590,"end":29750,"expression":"\n \\{G_{\\mathrm L}[\\lambda],G_{\\mathrm L}[\\mu]\\}\n       =G_{\\mathrm L}[[\\lambda,\\mu]],\\qquad\n \\{G_{\\mathrm L},G_{\\mathrm U}\\}=\\{G_{\\mathrm U},G_{\\mathrm U}\\}=0.\n","context":1764,"kind":"use","source":32},{"line":484,"start":29819,"end":29951,"expression":"\n \\Omega_{\\mathrm{int}}=G_{\\mathrm L}[\\chi]+G_{\\mathrm U}[\\upsilon]\n                   -\\tfrac12\\langle\\pi_\\chi,[\\chi,\\chi]\\rangle.\n","context":1765,"kind":"use","source":32},{"line":489,"start":30092,"end":30141,"expression":"\\{\\Omega_{\\mathrm{int}},\\Omega_{\\mathrm{int}}\\}=0","context":1766,"kind":"use","source":32},{"line":492,"start":30386,"end":30390,"expression":"\\eta","context":1767,"kind":"use","source":32},{"line":493,"start":30436,"end":30589,"expression":"\n \\mathcal L_\\eta\\Omega=\\iota_\\eta\\mathcal R+D_\\Omega(\\iota_\\eta\\Omega),\n \\qquad\n \\mathcal L_\\eta A=\\iota_\\eta\\mathcal F+\\mathrm d_\\Sigma(\\iota_\\eta A).\n","context":1768,"kind":"use","source":32},{"line":499,"start":30790,"end":30954,"expression":"\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n","context":1769,"kind":"use","source":32},{"line":505,"start":30985,"end":30986,"expression":"c","context":1770,"kind":"use","source":32},{"line":505,"start":30990,"end":31007,"expression":"\\zeta(c)=\\iota_cE","context":1771,"kind":"use","source":32},{"line":508,"start":31618,"end":31626,"expression":"J_\\Sigma","context":1772,"kind":"use","source":32},{"line":508,"start":31822,"end":31835,"expression":"\\Omega_\\Sigma","context":1773,"kind":"use","source":32},{"line":509,"start":31859,"end":31896,"expression":"\n \\{\\Omega_\\Sigma,\\Omega_\\Sigma\\}=0.\n","context":1774,"kind":"use","source":32},{"line":521,"start":32779,"end":32805,"expression":"q_{ab}=\\eta_{IJ}E_a^IE_b^J","context":1775,"kind":"use","source":32},{"line":522,"start":32856,"end":32930,"expression":"\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n","context":1776,"kind":"use","source":32},{"line":525,"start":32959,"end":32980,"expression":"\\widetilde n_IE_a^I=0","context":1776,"kind":"use","source":32},{"line":526,"start":32987,"end":33038,"expression":"-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})","context":1776,"kind":"use","source":32},{"line":527,"start":33054,"end":33091,"expression":"\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I","context":1776,"kind":"use","source":32},{"line":527,"start":33120,"end":33123,"expression":"\\nu","context":1777,"kind":"use","source":32},{"line":527,"start":33136,"end":33140,"expression":"\\eta","context":1778,"kind":"use","source":32},{"line":528,"start":33245,"end":33342,"expression":"\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n","context":1779,"kind":"use","source":32},{"line":534,"start":33658,"end":33668,"expression":"E,\\Omega,A","context":1780,"kind":"use","source":32},{"line":536,"start":34062,"end":34063,"expression":"K","context":1781,"kind":"use","source":32},{"line":536,"start":34168,"end":34173,"expression":"32K<N","context":1781,"kind":"use","source":32},{"line":543,"start":34605,"end":34712,"expression":"\n E^0=0,\\qquad E^i=e_a{}^i\\,\\mathrm dx^a,\n \\qquad e:=\\det(e_a{}^i)>0.\n \\label{c:constraints:regular-triad}\n","context":1782,"kind":"use","source":32},{"line":549,"start":34824,"end":34946,"expression":"\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n","context":1783,"kind":"use","source":32},{"line":553,"start":34966,"end":34981,"expression":"K^i=\\Omega^{0i}","context":1784,"kind":"use","source":32},{"line":554,"start":34988,"end":35034,"expression":"\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}","context":1785,"kind":"use","source":32},{"line":556,"start":35083,"end":35317,"expression":"\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n","context":1786,"kind":"use","source":32},{"line":564,"start":35351,"end":35353,"expression":"IJ","context":1787,"kind":"use","source":32},{"line":564,"start":35374,"end":35376,"expression":"0i","context":1788,"kind":"use","source":32},{"line":564,"start":35383,"end":35385,"expression":"i0","context":1789,"kind":"use","source":32},{"line":564,"start":35449,"end":35461,"expression":"\\delta\\Gamma","context":1790,"kind":"use","source":32},{"line":566,"start":35648,"end":35691,"expression":"\\mathcal E^a=\\tfrac12\\epsilon^{abc}\\Pi_{bc}","context":1791,"kind":"use","source":32},{"line":567,"start":35710,"end":35799,"expression":"\n \\int_\\Sigma\\Pi\\wedge\\delta A\n        =\\int_\\Sigma\\mathcal E^a\\delta A_a\\,\\mathrm d^3x.\n","context":1791,"kind":"use","source":32},{"line":572,"start":35895,"end":36057,"expression":"\n \\mathcal H_{\\mathrm M}=\\frac1{2e}q_{ab}\n       (\\mathcal E^a\\mathcal E^b+\\mathcal B^a\\mathcal B^b),\n \\qquad \\mathcal B^a=\\tfrac12\\epsilon^{abc}\\mathcal F_{bc}.\n","context":1791,"kind":"use","source":32},{"line":577,"start":36085,"end":36088,"expression":"1/e","context":1792,"kind":"use","source":32},{"line":580,"start":36260,"end":36289,"expression":"\\bar\\psi=\\psi^\\dagger\\gamma^0","context":1793,"kind":"use","source":32},{"line":580,"start":36299,"end":36324,"expression":"(\\gamma^0)^2=-\\mathrm{id}","context":1794,"kind":"use","source":32},{"line":580,"start":36424,"end":36448,"expression":"\\sigma_0=e\\,\\mathrm d^3x","context":1795,"kind":"use","source":32},{"line":580,"start":36455,"end":36465,"expression":"\\sigma_i=0","context":1795,"kind":"use","source":32},{"line":581,"start":36542,"end":36656,"expression":"\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n","context":1795,"kind":"use","source":32},{"line":586,"start":36767,"end":36854,"expression":"\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n","context":1796,"kind":"use","source":32},{"line":590,"start":36901,"end":36906,"expression":"e\\ne0","context":1797,"kind":"use","source":32},{"line":591,"start":36914,"end":36930,"expression":"\\Psi=e^{1/2}\\psi","context":1798,"kind":"use","source":32},{"line":591,"start":36937,"end":36969,"expression":"\\Psi^\\dagger=e^{1/2}\\psi^\\dagger","context":1799,"kind":"use","source":32},{"line":593,"start":37007,"end":37188,"expression":"\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,\\\\\n \\delta\\Psi^\\dagger\\Psi\n   &=e\\delta\\psi^\\dagger\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e.\n","context":1800,"kind":"use","source":32},{"line":600,"start":37249,"end":37455,"expression":"\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma\n       (\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\,\\mathrm d^3x,\n \\qquad\n \\{\\Psi_A(x),\\Psi_B^\\dagger(y)\\}_{\\mathrm D}\n             =-i\\delta_{AB}\\delta^3(x-y).\n","context":1801,"kind":"use","source":32},{"line":607,"start":37658,"end":37659,"expression":"e","context":1802,"kind":"use","source":32},{"line":611,"start":37837,"end":37879,"expression":"\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3","context":1803,"kind":"use","source":32},{"line":612,"start":37886,"end":37920,"expression":"J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi","context":1804,"kind":"use","source":32},{"line":614,"start":38016,"end":38132,"expression":"\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n","context":1805,"kind":"use","source":32},{"line":619,"start":38153,"end":38180,"expression":"\\omega=\\mathring\\omega(e)+C","context":1806,"kind":"use","source":32},{"line":621,"start":38268,"end":38357,"expression":"\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n","context":1807,"kind":"use","source":32},{"line":625,"start":38399,"end":38400,"expression":"K","context":1808,"kind":"use","source":32},{"line":626,"start":38591,"end":38647,"expression":"-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e","context":1809,"kind":"use","source":32},{"line":630,"start":38855,"end":38941,"expression":"\n (K_a{}^i,\\widetilde E^a{}_i),\\qquad\n (A_a,\\mathcal E^a),\\qquad (\\Psi,\\Psi^\\dagger),\n","context":1810,"kind":"use","source":32},{"line":635,"start":38985,"end":39230,"expression":"\n \\Theta_{\\mathrm{red}}=\\int_\\Sigma\\mathrm d^3x\\,\n \\left[\\frac1\\kappa\\widetilde E^a{}_i\\delta K_a{}^i\n       +\\mathcal E^a\\delta A_a\n       +\\frac i2(\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\right].\n \\label{c:constraints:reduced-one-form}\n","context":1811,"kind":"use","source":32},{"line":642,"start":39278,"end":39280,"expression":"24","context":1812,"kind":"use","source":32},{"line":645,"start":39750,"end":39912,"expression":"\n \\mathcal G_{\\mathrm U}^{\\mathrm{red}}=\\partial_a\\mathcal E^a+q\\Psi^\\dagger\\Psi,\n \\qquad Q[u]=-\\int_\\Sigma u\\mathcal G_{\\mathrm U}^{\\mathrm{red}}\\,\\mathrm d^3x.\n","context":1813,"kind":"use","source":32},{"line":649,"start":39941,"end":39964,"expression":"\\delta_uA_a=\\partial_au","context":1813,"kind":"use","source":32},{"line":649,"start":39971,"end":39991,"expression":"\\delta_u\\Psi=iqu\\Psi","context":1813,"kind":"use","source":32},{"line":650,"start":40070,"end":40161,"expression":"\n \\mathcal L_\\eta^{(1/2)}\\Psi\n       =\\eta^a\\partial_a\\Psi+\\tfrac12(\\partial_a\\eta^a)\\Psi.\n","context":1813,"kind":"use","source":32},{"line":654,"start":40229,"end":40233,"expression":"\\Psi","context":1814,"kind":"use","source":32},{"line":657,"start":40339,"end":40344,"expression":"\\iota","context":1815,"kind":"introduction-passage","source":32},{"line":657,"start":40416,"end":40443,"expression":"\\{\\cdot,\\cdot\\}_{\\mathrm D}","context":1816,"kind":"introduction-passage","source":32},{"line":658,"start":40644,"end":40762,"expression":"\n \\Omega_{\\mathrm{red}}=\\iota^*\\Omega_\\Sigma,\n \\qquad\n \\{\\Omega_{\\mathrm{red}},\\Omega_{\\mathrm{red}}\\}_{\\mathrm D}=0.\n","context":1816,"kind":"introduction-passage","source":32},{"line":665,"start":41198,"end":41201,"expression":"e=0","context":1817,"kind":"use","source":32},{"line":665,"start":41262,"end":41273,"expression":"e^{1/2}\\psi","context":1817,"kind":"use","source":32},{"line":667,"start":41537,"end":41544,"expression":"e_a{}^i","context":1818,"kind":"use","source":32},{"line":667,"start":41578,"end":41584,"expression":"e^{-1}","context":1818,"kind":"use","source":32},{"line":667,"start":41588,"end":41595,"expression":"e^{1/2}","context":1818,"kind":"use","source":32},{"line":667,"start":41704,"end":41709,"expression":"16K<N","context":1818,"kind":"use","source":32},{"line":667,"start":41716,"end":41721,"expression":"32K<N","context":1818,"kind":"use","source":32},{"line":675,"start":42366,"end":42384,"expression":"z\\in\\mathbb R^{2n}","context":1819,"kind":"use","source":32},{"line":675,"start":42413,"end":42414,"expression":"J","context":1820,"kind":"use","source":32},{"line":675,"start":42422,"end":42452,"expression":"\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})","context":1821,"kind":"use","source":32},{"line":675,"start":42703,"end":42704,"expression":"F","context":1822,"kind":"use","source":32},{"line":676,"start":42760,"end":42875,"expression":"\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n","context":1823,"kind":"use","source":32},{"line":681,"start":42957,"end":43055,"expression":"\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n","context":1824,"kind":"use","source":32},{"line":685,"start":43075,"end":43076,"expression":"G","context":1825,"kind":"use","source":32},{"line":685,"start":43089,"end":43112,"expression":"\\overline\\nabla\\Phi_j^T","context":1826,"kind":"use","source":32},{"line":685,"start":43150,"end":43169,"expression":"\\Delta_\\Delta=GJG^T","context":1827,"kind":"use","source":32},{"line":686,"start":43236,"end":43336,"expression":"\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n","context":1828,"kind":"use","source":32},{"line":691,"start":43381,"end":43406,"expression":"GJ_{\\mathrm D}^{\\Delta}=0","context":1829,"kind":"use","source":32},{"line":691,"start":43416,"end":43429,"expression":"\\Delta_\\Delta","context":1830,"kind":"use","source":32},{"line":692,"start":43479,"end":43529,"expression":"(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}","context":1831,"kind":"use","source":32},{"line":697,"start":43888,"end":43889,"expression":"H","context":1832,"kind":"use","source":32},{"line":698,"start":43967,"end":44104,"expression":"\n z_1-z_0=\\Delta t\\,J_{\\mathrm D}^{\\Delta}(z_0,z_1)\n                     \\overline\\nabla H(z_0,z_1)\n \\label{c:constraints:implicit-step}\n","context":1832,"kind":"use","source":32},{"line":703,"start":44136,"end":44149,"expression":"\\Delta_\\Delta","context":1832,"kind":"use","source":32},{"line":704,"start":44177,"end":44221,"expression":"\n \\Phi(z_1)=\\Phi(z_0),\\qquad H(z_1)=H(z_0).\n","context":1832,"kind":"use","source":32},{"line":710,"start":44300,"end":44330,"expression":"\\Phi(z_1)-\\Phi(z_0)=G(z_1-z_0)","context":1833,"kind":"use","source":32},{"line":710,"start":44357,"end":44382,"expression":"GJ_{\\mathrm D}^{\\Delta}=0","context":1834,"kind":"use","source":32},{"line":712,"start":44422,"end":44512,"expression":"\n H(z_1)-H(z_0)\n =\\Delta t\\,\\overline\\nabla H^T J_{\\mathrm D}^{\\Delta}\\overline\\nabla H=0\n","context":1835,"kind":"use","source":32},{"line":719,"start":44701,"end":44707,"expression":"H(z,t)","context":1836,"kind":"use","source":32},{"line":722,"start":44962,"end":44973,"expression":"\\mathbb R^6","context":1837,"kind":"use","source":32},{"line":722,"start":44981,"end":44988,"expression":"z=(q,p)","context":1837,"kind":"use","source":32},{"line":722,"start":45005,"end":45006,"expression":"J","context":1837,"kind":"use","source":32},{"line":723,"start":45029,"end":45073,"expression":"\n \\Phi_1=q\\cdot q-1,\\qquad \\Phi_2=q\\cdot p.\n","context":1837,"kind":"use","source":32},{"line":727,"start":45146,"end":45211,"expression":"\n \\Delta(z)=\\begin{pmatrix}0&2\\|q\\|^2\\\\-2\\|q\\|^2&0\\end{pmatrix}.\n","context":1838,"kind":"use","source":32},{"line":730,"start":45253,"end":45258,"expression":"q\\ne0","context":1839,"kind":"use","source":32},{"line":730,"start":45298,"end":45299,"expression":"q","context":1839,"kind":"use","source":32},{"line":730,"start":45330,"end":45331,"expression":"p","context":1839,"kind":"use","source":32},{"line":731,"start":45369,"end":45387,"expression":"\\bar q=(q_0+q_1)/2","context":1839,"kind":"use","source":32},{"line":731,"start":45394,"end":45412,"expression":"\\bar p=(p_0+p_1)/2","context":1839,"kind":"use","source":32},{"line":732,"start":45464,"end":45547,"expression":"\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n","context":1839,"kind":"use","source":32},{"line":736,"start":45576,"end":45589,"expression":"\\Delta_\\Delta","context":1839,"kind":"use","source":32},{"line":736,"start":45625,"end":45626,"expression":"q","context":1840,"kind":"use","source":32},{"line":736,"start":45641,"end":45647,"expression":"\\bar q","context":1841,"kind":"use","source":32},{"line":736,"start":45689,"end":45697,"expression":"\\bar q=0","context":1842,"kind":"use","source":32},{"line":738,"start":45917,"end":45944,"expression":"H(q,p)=\\tfrac12\\|p\\|^2+V(q)","context":1843,"kind":"use","source":32},{"line":738,"start":46030,"end":46031,"expression":"V","context":1843,"kind":"use","source":32},{"line":738,"start":46162,"end":46163,"expression":"H","context":1843,"kind":"use","source":32},{"line":744,"start":46701,"end":46766,"expression":"\n r=(z_1-z_0)-\\Delta t\\,J_{\\mathrm D}^{\\Delta}\\overline\\nabla H.\n","context":1844,"kind":"use","source":32},{"line":748,"start":46878,"end":46985,"expression":"\n \\Phi(z_1)-\\Phi(z_0)=Gr,\\qquad\n H(z_1)-H(z_0)=\\overline\\nabla H^Tr.\n \\label{c:constraints:residual-bound}\n","context":1844,"kind":"use","source":32},{"line":753,"start":47006,"end":47044,"expression":"\\|\\Phi(z_1)-\\Phi(z_0)\\|\\le\\|G\\|\\,\\|r\\|","context":1845,"kind":"use","source":32},{"line":754,"start":47051,"end":47097,"expression":"|H(z_1)-H(z_0)|\\le\\|\\overline\\nabla H\\|\\,\\|r\\|","context":1846,"kind":"use","source":32},{"line":757,"start":47456,"end":47460,"expression":"1000","context":1847,"kind":"use","source":32},{"line":757,"start":47472,"end":47484,"expression":"\\Delta t=0.1","context":1847,"kind":"use","source":32},{"line":757,"start":47512,"end":47529,"expression":"7.8\\times10^{-16}","context":1847,"kind":"use","source":32},{"line":757,"start":47556,"end":47573,"expression":"3.9\\times10^{-15}","context":1847,"kind":"use","source":32},{"line":757,"start":47605,"end":47622,"expression":"1.1\\times10^{-16}","context":1847,"kind":"use","source":32},{"line":757,"start":47713,"end":47720,"expression":"10^{-6}","context":1847,"kind":"use","source":32},{"line":762,"start":48264,"end":48274,"expression":"\\Delta t=0","context":1848,"kind":"use","source":32},{"line":762,"start":48281,"end":48288,"expression":"z_1=z_0","context":1848,"kind":"use","source":32},{"line":762,"start":48426,"end":48429,"expression":"z_1","context":1848,"kind":"use","source":32},{"line":762,"start":48698,"end":48706,"expression":"\\Delta t","context":1849,"kind":"use","source":32},{"line":765,"start":48886,"end":48929,"expression":"\n H(z)\\ge a\\|z\\|^p-b,\\qquad a>0,\\quad p>0,\n","context":1850,"kind":"use","source":32},{"line":769,"start":48986,"end":49036,"expression":"\n \\|z_n\\|\\le\\left(\\frac{H(z_0)+b}{a}\\right)^{1/p}\n","context":1850,"kind":"use","source":32},{"line":774,"start":49394,"end":49404,"expression":"I_{K\\to L}","context":1851,"kind":"use","source":32},{"line":774,"start":49419,"end":49420,"expression":"L","context":1851,"kind":"use","source":32},{"line":774,"start":49428,"end":49431,"expression":"U_L","context":1851,"kind":"use","source":32},{"line":774,"start":49455,"end":49474,"expression":"U_L\\circ I_{K\\to L}","context":1851,"kind":"use","source":32},{"line":774,"start":49488,"end":49489,"expression":"L","context":1851,"kind":"use","source":32},{"line":782,"start":50553,"end":50558,"expression":"m,n>0","context":1852,"kind":"use","source":32},{"line":783,"start":50580,"end":50677,"expression":"\n S_F(x_1,x_2,y,z)=\\tfrac12(x_2-x_1y)^2\n                         +\\tfrac m2(y-z)^2+\\tfrac n2z^2.\n","context":1852,"kind":"use","source":32},{"line":787,"start":50715,"end":50749,"expression":"R_F=\\partial_{x_1}+y\\partial_{x_2}","context":1852,"kind":"use","source":32},{"line":787,"start":50759,"end":50762,"expression":"S_F","context":1853,"kind":"use","source":32},{"line":788,"start":50846,"end":50927,"expression":"\n M_D=\\begin{pmatrix}x_1^2+m&-m\\\\-m&m+n\\end{pmatrix},\n \\qquad D=(m+n)x_1^2+mn>0.\n","context":1854,"kind":"use","source":32},{"line":792,"start":50951,"end":50963,"expression":"\\mu=mn/(m+n)","context":1855,"kind":"use","source":32},{"line":792,"start":50970,"end":50981,"expression":"d=x_1^2+\\mu","context":1855,"kind":"use","source":32},{"line":793,"start":51018,"end":51030,"expression":"y_*=x_1x_2/d","context":1855,"kind":"use","source":32},{"line":793,"start":51037,"end":51052,"expression":"z_*=my_* /(m+n)","context":1855,"kind":"use","source":32},{"line":794,"start":51090,"end":51176,"expression":"\n S_C=\\frac{\\mu x_2^2}{2d},\\qquad\n R_C=\\partial_{x_1}+\\frac{x_1x_2}{d}\\partial_{x_2}.\n","context":1855,"kind":"use","source":32},{"line":798,"start":51215,"end":51223,"expression":"R_CS_C=0","context":1855,"kind":"use","source":32},{"line":798,"start":51315,"end":51337,"expression":"\\rho_C\\propto D^{-1/2}","context":1856,"kind":"use","source":32},{"line":799,"start":51387,"end":51463,"expression":"\n \\operatorname{div}R_C=\\frac{x_1}{d},\\qquad\n R_C\\log\\rho_C=-\\frac{x_1}{d}.\n","context":1857,"kind":"use","source":32},{"line":805,"start":51623,"end":51624,"expression":"z","context":1858,"kind":"use","source":32},{"line":805,"start":51665,"end":51668,"expression":"\\mu","context":1858,"kind":"use","source":32},{"line":805,"start":51675,"end":51678,"expression":"y^2","context":1858,"kind":"use","source":32},{"line":805,"start":51696,"end":51708,"expression":"(m+n)^{-1/2}","context":1858,"kind":"use","source":32},{"line":805,"start":51724,"end":51725,"expression":"y","context":1858,"kind":"use","source":32},{"line":805,"start":51739,"end":51747,"expression":"d^{-1/2}","context":1858,"kind":"use","source":32},{"line":805,"start":51768,"end":51776,"expression":"D^{-1/2}","context":1858,"kind":"use","source":32},{"line":805,"start":51839,"end":51851,"expression":"e^{-S/\\hbar}","context":1858,"kind":"use","source":32},{"line":806,"start":51919,"end":51953,"expression":"S_{C,\\hbar}=S_C+\\tfrac\\hbar2\\log D","context":1858,"kind":"use","source":32},{"line":810,"start":52273,"end":52341,"expression":"\n \\tfrac12(S_F,S_F)-i\\hbar\\Delta_FS_F=0.\n \\label{c:constraints:qme}\n","context":1859,"kind":"use","source":32},{"line":815,"start":52423,"end":52449,"expression":"\\Delta_F=\\Delta_C+\\Delta_D","context":1860,"kind":"use","source":32},{"line":815,"start":52496,"end":52508,"expression":"\\mathcal L_D","context":1861,"kind":"use","source":32},{"line":816,"start":52555,"end":52622,"expression":"\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n","context":1862,"kind":"use","source":32},{"line":820,"start":52793,"end":52889,"expression":"\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n","context":1863,"kind":"use","source":32},{"line":827,"start":53420,"end":53471,"expression":"\n L_{\\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},\n","context":1864,"kind":"use","source":32},{"line":13,"start":592,"end":598,"expression":"N\\geq3","context":1865,"kind":"use","source":33},{"line":13,"start":616,"end":619,"expression":"h>0","context":1866,"kind":"use","source":33},{"line":14,"start":642,"end":643,"expression":"N","context":1867,"kind":"use","source":33},{"line":15,"start":700,"end":768,"expression":"\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n","context":1868,"kind":"use","source":33},{"line":19,"start":792,"end":793,"expression":"N","context":1869,"kind":"use","source":33},{"line":20,"start":845,"end":846,"expression":"N","context":1870,"kind":"use","source":33},{"line":21,"start":927,"end":928,"expression":"w","context":1871,"kind":"use","source":33},{"line":21,"start":932,"end":933,"expression":"c","context":1872,"kind":"use","source":33},{"line":21,"start":941,"end":942,"expression":"v","context":1873,"kind":"use","source":33},{"line":22,"start":958,"end":1058,"expression":"\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n","context":1874,"kind":"use","source":33},{"line":27,"start":1067,"end":1070,"expression":"N/a","context":1875,"kind":"use","source":33},{"line":29,"start":1245,"end":1257,"expression":"\\varphi_j(t)","context":1876,"kind":"use","source":33},{"line":30,"start":1264,"end":1296,"expression":"\\Delta_j=\\varphi_{j+1}-\\varphi_j","context":1877,"kind":"use","source":33},{"line":31,"start":1325,"end":1464,"expression":"\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n","context":1878,"kind":"use","source":33},{"line":38,"start":1537,"end":1682,"expression":"\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n","context":1879,"kind":"use","source":33},{"line":46,"start":1832,"end":1848,"expression":"w_j\\dot\\varphi_j","context":1880,"kind":"use","source":33},{"line":46,"start":1866,"end":1887,"expression":"\\dot w_j\\dot\\varphi_j","context":1881,"kind":"use","source":33},{"line":53,"start":2107,"end":2530,"expression":"\n e_j&=\\frac h2w_j\\dot\\varphi_j^2+\\frac h2v_j\\varphi_j^2\n +\\frac{c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{c_{j-1/2}}{4h}\\Delta_{j-1}^2,\\nonumber\\\\\n F_{j+1/2}&=-\\frac{c_{j+1/2}}{2h}\\Delta_j\n             (\\dot\\varphi_j+\\dot\\varphi_{j+1}),\\nonumber\\\\\n s_j&=-\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2\n +\\frac{\\dot c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{\\dot c_{j-1/2}}{4h}\\Delta_{j-1}^2.\n \\label{c:core:scalar-densities}\n","context":1882,"kind":"use","source":33},{"line":66,"start":2563,"end":2630,"expression":"\n \\dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.\n \\label{c:core:scalar-balance}\n","context":1883,"kind":"use","source":33},{"line":70,"start":2661,"end":2674,"expression":"E_h=\\sum_je_j","context":1883,"kind":"use","source":33},{"line":71,"start":2687,"end":2720,"expression":"\\dot E_h=\\sum_js_j=-\\partial_tL_h","context":1883,"kind":"use","source":33},{"line":72,"start":2759,"end":2766,"expression":"\\varphi","context":1884,"kind":"use","source":33},{"line":72,"start":2773,"end":2784,"expression":"\\dot\\varphi","context":1885,"kind":"use","source":33},{"line":76,"start":2831,"end":2852,"expression":"B_j=c_{j+1/2}\\Delta_j","context":1886,"kind":"use","source":33},{"line":77,"start":2939,"end":2940,"expression":"j","context":1887,"kind":"use","source":33},{"line":78,"start":2947,"end":3051,"expression":"\n \\frac{\\dot\\varphi_j}{h}(B_j-B_{j-1})\n -\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2.\n","context":1887,"kind":"use","source":33},{"line":84,"start":3102,"end":3191,"expression":"\n \\frac{\\dot c_{j+1/2}\\Delta_j^2}{4h}\n +\\frac{B_j}{2h}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n","context":1887,"kind":"use","source":33},{"line":90,"start":3269,"end":3310,"expression":"B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)","context":1888,"kind":"use","source":33},{"line":91,"start":3373,"end":3376,"expression":"s_j","context":1889,"kind":"use","source":33},{"line":94,"start":3555,"end":3561,"expression":"1/(4h)","context":1890,"kind":"use","source":33},{"line":94,"start":3572,"end":3586,"expression":"-\\partial_tL_h","context":1891,"kind":"use","source":33},{"line":97,"start":3609,"end":3612,"expression":"e_j","context":1892,"kind":"use","source":33},{"line":98,"start":3693,"end":3703,"expression":"r,\\ldots,s","context":1892,"kind":"use","source":33},{"line":99,"start":3738,"end":3807,"expression":"\n \\frac{d}{dt}\\sum_{j=r}^se_j\n +F_{s+1/2}-F_{r-1/2}=\\sum_{j=r}^ss_j.\n","context":1892,"kind":"use","source":33},{"line":109,"start":4167,"end":4173,"expression":"a(t)>0","context":1893,"kind":"use","source":33},{"line":109,"start":4206,"end":4209,"expression":"m=0","context":1893,"kind":"use","source":33},{"line":110,"start":4218,"end":4221,"expression":"w=a","context":1893,"kind":"use","source":33},{"line":110,"start":4225,"end":4230,"expression":"c=1/a","context":1893,"kind":"use","source":33},{"line":111,"start":4239,"end":4327,"expression":"\n \\dot E_h=-\\frac h2\\dot a\\sum_j\\dot\\varphi_j^2\n -\\frac{\\dot a}{2ha^2}\\sum_j\\Delta_j^2.\n","context":1893,"kind":"use","source":33},{"line":115,"start":4345,"end":4346,"expression":"a","context":1893,"kind":"use","source":33},{"line":123,"start":4736,"end":4766,"expression":"\\partial_x(c\\partial_x\\varphi)","context":1894,"kind":"use","source":33},{"line":123,"start":4791,"end":4794,"expression":"x_j","context":1894,"kind":"use","source":33},{"line":124,"start":4804,"end":4921,"expression":"\n c(x_j+h/2)[\\varphi(x_j+h)-\\varphi(x_j)]\n -c(x_j-h/2)[\\varphi(x_j)-\\varphi(x_j-h)]\n =h^2(c\\varphi_x)_x(x_j)+O(h^4).\n","context":1894,"kind":"use","source":33},{"line":143,"start":5509,"end":5513,"expression":"\\phi","context":1895,"kind":"use","source":33},{"line":144,"start":5531,"end":5536,"expression":"(t,x)","context":1895,"kind":"use","source":33},{"line":145,"start":5567,"end":5641,"expression":"\n e^0=e^\\phi dt,\\qquad e^1=e^\\phi dx,\n \\qquad ds^2=e^{2\\phi}(-dt^2+dx^2).\n","context":1895,"kind":"use","source":33},{"line":150,"start":5652,"end":5730,"expression":"\n K=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n \\omega=(\\phi_xdt+\\phi_tdx)K.\n","context":1896,"kind":"use","source":33},{"line":156,"start":5875,"end":5895,"expression":"\\omega\\wedge\\omega=0","context":1897,"kind":"use","source":33},{"line":157,"start":5924,"end":6058,"expression":"\n \\Omega=d\\omega=(\\phi_{tt}-\\phi_{xx})K\\,dt\\wedge dx,\n \\qquad R=2e^{-2\\phi}(\\phi_{tt}-\\phi_{xx}).\n \\label{c:core:conformal-curvature}\n","context":1897,"kind":"use","source":33},{"line":165,"start":6234,"end":6257,"expression":"U_e=\\exp(+\\int_e\\omega)","context":1898,"kind":"use","source":33},{"line":168,"start":6390,"end":6560,"expression":"\n W_\\square\n =\\exp\\left(K\\oint_{\\partial\\square}(\\phi_xdt+\\phi_tdx)\\right)\n =\\exp\\left(\\frac K2\\int_\\square e^{2\\phi}R\\,dt\\,dx\\right).\n \\label{c:core:conformal-holonomy}\n","context":1898,"kind":"use","source":33},{"line":174,"start":6623,"end":6632,"expression":"\\phi_{tt}","context":1899,"kind":"use","source":33},{"line":175,"start":6674,"end":6683,"expression":"\\phi_{xx}","context":1900,"kind":"use","source":33},{"line":176,"start":6755,"end":6766,"expression":"dt\\wedge dx","context":1901,"kind":"use","source":33},{"line":180,"start":6931,"end":6935,"expression":"\\chi","context":1902,"kind":"use","source":33},{"line":181,"start":6945,"end":6985,"expression":"\n e^{\\chi K}=\\cosh\\chi\\,I+\\sinh\\chi\\,K.\n","context":1902,"kind":"use","source":33},{"line":184,"start":7021,"end":7046,"expression":"\\operatorname{diag}(-1,1)","context":1902,"kind":"use","source":33},{"line":185,"start":7078,"end":7086,"expression":"e^{\\chi}","context":1902,"kind":"use","source":33},{"line":185,"start":7093,"end":7102,"expression":"e^{-\\chi}","context":1902,"kind":"use","source":33},{"line":186,"start":7145,"end":7151,"expression":"\\chi K","context":1902,"kind":"use","source":33},{"line":191,"start":7338,"end":7360,"expression":"\\phi(t,x)=b(t^2-x^2)/2","context":1903,"kind":"use","source":33},{"line":192,"start":7373,"end":7482,"expression":"\n \\phi_{tt}-\\phi_{xx}=2b,\n \\qquad R=4b\\,e^{-2\\phi},\n \\qquad W_\\square=e^{2b\\,\\operatorname{area}(\\square)K}.\n","context":1903,"kind":"use","source":33},{"line":197,"start":7508,"end":7520,"expression":"e^{2\\phi}R/2","context":1903,"kind":"use","source":33},{"line":197,"start":7544,"end":7545,"expression":"R","context":1903,"kind":"use","source":33},{"line":203,"start":7724,"end":7793,"expression":"\n G(t,x)=e^{\\chi(t,x)K},\\qquad\n U_{a\\to b}'=G(b)U_{a\\to b}G(a)^{-1}.\n","context":1904,"kind":"use","source":33},{"line":208,"start":7842,"end":7869,"expression":"\\omega_+'=\\omega_++d\\chi\\,K","context":1904,"kind":"use","source":33},{"line":210,"start":7993,"end":8003,"expression":"D=d+\\omega","context":1904,"kind":"use","source":33},{"line":210,"start":8032,"end":8043,"expression":"e\\mapsto Ge","context":1904,"kind":"use","source":33},{"line":211,"start":8060,"end":8083,"expression":"\\omega'=\\omega-d\\chi\\,K","context":1904,"kind":"use","source":33},{"line":212,"start":8109,"end":8128,"expression":"\\exp(-\\int_e\\omega)","context":1905,"kind":"use","source":33},{"line":213,"start":8192,"end":8201,"expression":"W_\\square","context":1906,"kind":"use","source":33},{"line":229,"start":8786,"end":8789,"expression":"G>0","context":1907,"kind":"use","source":33},{"line":229,"start":8797,"end":8804,"expression":"\\rho(r)","context":1907,"kind":"use","source":33},{"line":230,"start":8864,"end":8896,"expression":"\\lim_{r\\downarrow0}r^2\\Phi'(r)=0","context":1907,"kind":"use","source":33},{"line":232,"start":8935,"end":9062,"expression":"\n (r^2\\Phi')'=4\\pi G r^2\\rho,\n \\qquad M_{\\rm enc}(r)=4\\pi\\int_0^rs^2\\rho(s)\\,ds,\n \\qquad \\Phi'(r)=\\frac{GM_{\\rm enc}(r)}{r^2}.\n","context":1907,"kind":"use","source":33},{"line":237,"start":9086,"end":9096,"expression":"a_r=-\\Phi'","context":1907,"kind":"use","source":33},{"line":240,"start":9175,"end":9188,"expression":"u(r)=r\\Phi(r)","context":1908,"kind":"use","source":33},{"line":241,"start":9229,"end":9296,"expression":"\n r^2\\Phi'=ru'-u,\\qquad\n (r^2\\Phi')'=ru'',\\qquad u''=4\\pi G r\\rho.\n","context":1908,"kind":"use","source":33},{"line":246,"start":9373,"end":9439,"expression":"\n S[u]=\\int_0^{R_0}\\left[\\frac{(u')^2}{2G}+4\\pi r\\rho u\\right]dr.\n","context":1908,"kind":"use","source":33},{"line":249,"start":9446,"end":9452,"expression":"r_i=ih","context":1908,"kind":"use","source":33},{"line":249,"start":9456,"end":9468,"expression":"i=0,\\ldots,M","context":1908,"kind":"use","source":33},{"line":249,"start":9477,"end":9483,"expression":"Mh=R_0","context":1908,"kind":"use","source":33},{"line":250,"start":9528,"end":9651,"expression":"\n S_h[u]=\\sum_{i=0}^{M-1}\\frac{(u_{i+1}-u_i)^2}{2Gh}\n +4\\pi h\\sum_{i=1}^{M-1}r_i\\rho_i u_i.\n \\label{c:core:poisson-action}\n","context":1908,"kind":"use","source":33},{"line":258,"start":9755,"end":9762,"expression":"u_0,u_M","context":1909,"kind":"use","source":33},{"line":258,"start":9772,"end":9775,"expression":"S_h","context":1909,"kind":"use","source":33},{"line":260,"start":9831,"end":9884,"expression":"\n \\frac{u_{i+1}-2u_i+u_{i-1}}{h^2}=4\\pi G r_i\\rho_i.\n","context":1909,"kind":"use","source":33},{"line":266,"start":9953,"end":9956,"expression":"u_i","context":1910,"kind":"use","source":33},{"line":267,"start":9965,"end":10009,"expression":"(2u_i-u_{i-1}-u_{i+1})/(Gh)+4\\pi h r_i\\rho_i","context":1910,"kind":"use","source":33},{"line":268,"start":10068,"end":10069,"expression":"v","context":1910,"kind":"use","source":33},{"line":268,"start":10077,"end":10086,"expression":"v_0=v_M=0","context":1910,"kind":"use","source":33},{"line":270,"start":10121,"end":10166,"expression":"\n \\frac1{Gh}\\sum_{i=0}^{M-1}(v_{i+1}-v_i)^2.\n","context":1910,"kind":"use","source":33},{"line":273,"start":10191,"end":10192,"expression":"v","context":1910,"kind":"use","source":33},{"line":278,"start":10388,"end":10393,"expression":"u_0=0","context":1911,"kind":"use","source":33},{"line":279,"start":10458,"end":10461,"expression":"u/r","context":1911,"kind":"use","source":33},{"line":279,"start":10481,"end":10484,"expression":"r>0","context":1911,"kind":"use","source":33},{"line":280,"start":10539,"end":10542,"expression":"r=0","context":1911,"kind":"use","source":33},{"line":283,"start":10584,"end":10666,"expression":"\n \\rho(r)=\\frac{M_0}{(2\\pi)^{3/2}\\sigma^3}e^{-r^2/(2\\sigma^2)},\n \\qquad \\sigma>0,\n","context":1912,"kind":"use","source":33},{"line":288,"start":10715,"end":10941,"expression":"\n M_{\\rm enc}(r)&=M_0\\left[\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right)\n -\\sqrt{\\frac2\\pi}\\frac r\\sigma e^{-r^2/(2\\sigma^2)}\\right],\\\\\n \\Phi(r)&=-\\frac{GM_0}{r}\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right).\n","context":1913,"kind":"use","source":33},{"line":296,"start":11028,"end":11054,"expression":"-\\sqrt{2/\\pi}\\,GM_0/\\sigma","context":1914,"kind":"use","source":33},{"line":302,"start":11255,"end":11408,"expression":"\n ds^2=-\\left(1+\\frac{2\\Phi}{c_0^2}\\right)c_0^2dt^2\n       +\\left(1-\\frac{2\\Phi}{c_0^2}\\right)d\\mathbf x^2,\n \\qquad \\left|\\frac{\\Phi}{c_0^2}\\right|\\ll1,\n","context":1915,"kind":"use","source":33},{"line":307,"start":11437,"end":11468,"expression":"\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}","context":1915,"kind":"use","source":33},{"line":309,"start":11525,"end":11704,"expression":"\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n","context":1915,"kind":"use","source":33},{"line":315,"start":11712,"end":11742,"expression":"\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}","context":1916,"kind":"use","source":33},{"line":315,"start":11746,"end":11749,"expression":"b>0","context":1917,"kind":"use","source":33},{"line":316,"start":11808,"end":11814,"expression":"[-L,L]","context":1918,"kind":"use","source":33},{"line":317,"start":11836,"end":12085,"expression":"\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),\\\\\n \\alpha(L,b)\n &=\\frac2{c_0^2}\\left|\\int_{-L}^L\\partial_b\\Phi(x,b)\\,dx\\right|\n =\\frac{4GM_0L}{c_0^2b\\sqrt{L^2+b^2}}.\n","context":1919,"kind":"use","source":33},{"line":325,"start":12104,"end":12110,"expression":"\\alpha","context":1920,"kind":"use","source":33},{"line":326,"start":12177,"end":12187,"expression":"L\\to\\infty","context":1920,"kind":"use","source":33},{"line":326,"start":12196,"end":12210,"expression":"4GM_0/(c_0^2b)","context":1920,"kind":"use","source":33},{"line":341,"start":12787,"end":12790,"expression":"P,Q","context":1921,"kind":"use","source":33},{"line":341,"start":12818,"end":12827,"expression":"I\\times\\T","context":1921,"kind":"use","source":33},{"line":342,"start":12836,"end":12856,"expression":"\\theta\\in\\R/(2\\pi\\Z)","context":1921,"kind":"use","source":33},{"line":342,"start":12863,"end":12886,"expression":"\\gamma(\\tau)=e^{-2\\tau}","context":1921,"kind":"use","source":33},{"line":343,"start":12912,"end":13067,"expression":"\n S_G=\\frac12\\int_I\\int_0^{2\\pi}\n \\left[P_\\tau^2-\\gamma P_\\theta^2\n +e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2)\\right]d\\theta\\,d\\tau.\n \\label{c:core:gowdy-action}\n","context":1921,"kind":"use","source":33},{"line":349,"start":13126,"end":13127,"expression":"I","context":1921,"kind":"use","source":33},{"line":352,"start":13242,"end":13469,"expression":"\n \\mathcal E_P&=P_{\\tau\\tau}-\\gamma P_{\\theta\\theta}\n -e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2),\\nonumber\\\\\n \\mathcal E_Q&=Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta}\n +2(P_\\tau Q_\\tau-\\gamma P_\\theta Q_\\theta).\n \\label{c:core:gowdy-euler}\n","context":1922,"kind":"use","source":33},{"line":359,"start":13501,"end":13502,"expression":"Q","context":1923,"kind":"use","source":33},{"line":360,"start":13538,"end":13606,"expression":"\\partial_\\tau(e^{2P}Q_\\tau)-\\partial_\\theta(e^{2P}\\gamma Q_\\theta)=0","context":1923,"kind":"use","source":33},{"line":361,"start":13639,"end":13666,"expression":"\\mathcal E_P=\\mathcal E_Q=0","context":1923,"kind":"use","source":33},{"line":364,"start":13704,"end":13867,"expression":"\n H_G&=P_\\tau^2+\\gamma P_\\theta^2\n +e^{2P}(Q_\\tau^2+\\gamma Q_\\theta^2),\\nonumber\\\\\n M_G&=2(P_\\tau P_\\theta+e^{2P}Q_\\tau Q_\\theta).\n \\label{c:core:gowdy-densities}\n","context":1924,"kind":"use","source":33},{"line":371,"start":13923,"end":13939,"expression":"\\lambda_\\tau=H_G","context":1925,"kind":"use","source":33},{"line":371,"start":13946,"end":13964,"expression":"\\lambda_\\theta=M_G","context":1925,"kind":"use","source":33},{"line":376,"start":14134,"end":14137,"expression":"P,Q","context":1926,"kind":"use","source":33},{"line":377,"start":14156,"end":14280,"expression":"\n \\partial_\\tau M_G-\\partial_\\theta H_G\n =2P_\\theta\\mathcal E_P+2e^{2P}Q_\\theta\\mathcal E_Q.\n \\label{c:core:gowdy-identity}\n","context":1926,"kind":"use","source":33},{"line":382,"start":14314,"end":14323,"expression":"I\\times\\T","context":1926,"kind":"use","source":33},{"line":382,"start":14352,"end":14359,"expression":"\\lambda","context":1926,"kind":"use","source":33},{"line":384,"start":14411,"end":14438,"expression":"\\int_0^{2\\pi}M_G\\,d\\theta=0","context":1927,"kind":"use","source":33},{"line":389,"start":14594,"end":14615,"expression":"P_\\tau\nP_{\\tau\\theta}","context":1928,"kind":"use","source":33},{"line":390,"start":14622,"end":14643,"expression":"Q_\\tau Q_{\\tau\\theta}","context":1928,"kind":"use","source":33},{"line":391,"start":14663,"end":14931,"expression":"\n\\begin{aligned}\n \\partial_\\tau M_G-\\partial_\\theta H_G\n ={}&2P_\\theta(P_{\\tau\\tau}-\\gamma P_{\\theta\\theta})\\\\\n &+2e^{2P}\\big[\n Q_\\theta(Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta})\n +2P_\\tau Q_\\tau Q_\\theta\n -P_\\theta Q_\\tau^2-\\gamma P_\\theta Q_\\theta^2\\big].\n\\end{aligned}\n","context":1929,"kind":"use","source":33},{"line":402,"start":15008,"end":15034,"expression":"\\gamma P_\\theta Q_\\theta^2","context":1930,"kind":"use","source":33},{"line":404,"start":15138,"end":15164,"expression":"d\\int M_G\\,d\\theta/d\\tau=0","context":1930,"kind":"use","source":33},{"line":406,"start":15183,"end":15190,"expression":"\\lambda","context":1931,"kind":"use","source":33},{"line":406,"start":15213,"end":15231,"expression":"\\lambda_\\theta=M_G","context":1931,"kind":"use","source":33},{"line":407,"start":15284,"end":15290,"expression":"\\tau_0","context":1931,"kind":"use","source":33},{"line":409,"start":15325,"end":15433,"expression":"\n \\lambda(\\tau,\\theta)=\\lambda_0\n +\\int_{\\tau_0}^{\\tau}H_G(s,0)\\,ds\n +\\int_0^{\\theta}M_G(\\tau,\\eta)\\,d\\eta.\n","context":1931,"kind":"use","source":33},{"line":414,"start":15441,"end":15447,"expression":"\\theta","context":1931,"kind":"use","source":33},{"line":414,"start":15464,"end":15467,"expression":"M_G","context":1931,"kind":"use","source":33},{"line":414,"start":15475,"end":15479,"expression":"\\tau","context":1931,"kind":"use","source":33},{"line":414,"start":15496,"end":15499,"expression":"H_G","context":1931,"kind":"use","source":33},{"line":427,"start":15988,"end":15991,"expression":"Q=0","context":1932,"kind":"use","source":33},{"line":427,"start":16014,"end":16020,"expression":"n\\geq1","context":1932,"kind":"use","source":33},{"line":428,"start":16052,"end":16112,"expression":"\n J_0(x)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m(x^2/4)^m}{(m!)^2}.\n","context":1932,"kind":"use","source":33},{"line":433,"start":16222,"end":16245,"expression":"x^2J_0''+xJ_0'+x^2J_0=0","context":1932,"kind":"use","source":33},{"line":434,"start":16269,"end":16344,"expression":"\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n","context":1932,"kind":"use","source":33},{"line":438,"start":16404,"end":16416,"expression":"x=ne^{-\\tau}","context":1933,"kind":"use","source":33},{"line":439,"start":16420,"end":16473,"expression":"\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)","context":1934,"kind":"use","source":33},{"line":440,"start":16485,"end":16507,"expression":"P_{\\theta\\theta}=-n^2P","context":1935,"kind":"use","source":33},{"line":441,"start":16516,"end":16536,"expression":"M_G=2P_\\tau P_\\theta","context":1936,"kind":"use","source":33},{"line":442,"start":16578,"end":16585,"expression":"\\lambda","context":1937,"kind":"use","source":33},{"line":445,"start":16688,"end":16695,"expression":"y=x^2/4","context":1938,"kind":"use","source":33},{"line":445,"start":16699,"end":16726,"expression":"\\mathcal B(y)=J_0(2\\sqrt y)","context":1938,"kind":"use","source":33},{"line":446,"start":16749,"end":16755,"expression":"r\\geq2","context":1938,"kind":"use","source":33},{"line":447,"start":16766,"end":16874,"expression":"\n \\mathcal B_{r,j}(y)=\\sum_{q=0}^{\\infty}\n \\frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},\n \\qquad j=0,\\ldots,r-1.\n","context":1938,"kind":"use","source":33},{"line":452,"start":16883,"end":16916,"expression":"\\mathcal B=\\sum_j\\mathcal B_{r,j}","context":1938,"kind":"use","source":33},{"line":453,"start":16925,"end":16942,"expression":"\\vartheta=y\\,d/dy","context":1938,"kind":"use","source":33},{"line":454,"start":16972,"end":17043,"expression":"\n \\vartheta^2\\mathcal B_{r,j}+y\\mathcal B_{r,j-1}=0,\n \\qquad j\\pmod r,\n","context":1938,"kind":"use","source":33},{"line":458,"start":17060,"end":17083,"expression":"m^2/(m!)^2=1/((m-1)!)^2","context":1939,"kind":"use","source":33},{"line":458,"start":17090,"end":17096,"expression":"m\\geq1","context":1940,"kind":"use","source":33},{"line":469,"start":17412,"end":17420,"expression":"h=2\\pi/N","context":1941,"kind":"use","source":33},{"line":470,"start":17465,"end":17559,"expression":"\n (D_hf)_j=\\frac{f_{j+1}-f_{j-1}}{2h},\\qquad\n (\\Delta_hf)_j=\\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.\n","context":1941,"kind":"use","source":33},{"line":474,"start":17568,"end":17588,"expression":"D_h^{\\mathsf T}=-D_h","context":1941,"kind":"use","source":33},{"line":474,"start":17597,"end":17605,"expression":"U=\\dot P","context":1941,"kind":"use","source":33},{"line":474,"start":17609,"end":17617,"expression":"V=\\dot Q","context":1941,"kind":"use","source":33},{"line":475,"start":17655,"end":17666,"expression":"f_h(\\tau,z)","context":1941,"kind":"use","source":33},{"line":475,"start":17673,"end":17684,"expression":"z=(P,Q,U,V)","context":1941,"kind":"use","source":33},{"line":476,"start":17702,"end":17878,"expression":"\n \\dot P&=U,&\\dot Q&=V,\\nonumber\\\\\n \\dot U&=\\gamma\\Delta_hP+e^{2P}[V^2-\\gamma(D_hQ)^2],\\nonumber\\\\\n \\dot V&=\\gamma\\Delta_hQ-2[UV-\\gamma(D_hP)(D_hQ)].\n \\label{c:core:gowdy-raw}\n","context":1942,"kind":"use","source":33},{"line":484,"start":17999,"end":18082,"expression":"\n C_h(z)=2h\\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].\n \\label{c:core:gowdy-momentum}\n","context":1943,"kind":"use","source":33},{"line":488,"start":18160,"end":18163,"expression":"P,Q","context":1943,"kind":"use","source":33},{"line":491,"start":18235,"end":18238,"expression":"D_h","context":1944,"kind":"use","source":33},{"line":492,"start":18260,"end":18398,"expression":"\n \\nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\\\\n \\nabla_QC_h&=-2hD_h(e^{2P}V),\\\\\n g_U:=\\nabla_UC_h&=2hD_hP,\n &g_V:=\\nabla_VC_h&=2h e^{2P}D_hQ.\n","context":1945,"kind":"use","source":33},{"line":498,"start":18416,"end":18433,"expression":"G_h=(0,0,g_U,g_V)","context":1946,"kind":"use","source":33},{"line":498,"start":18440,"end":18473,"expression":"d_h=\\norm{g_U}_2^2+\\norm{g_V}_2^2","context":1946,"kind":"use","source":33},{"line":499,"start":18496,"end":18501,"expression":"d_h>0","context":1946,"kind":"use","source":33},{"line":500,"start":18513,"end":18596,"expression":"\n \\Lambda_h=\\frac{\\nabla C_h\\cdot f_h}{d_h},\\qquad\n \\widehat f_h=f_h-\\Lambda_hG_h.\n","context":1946,"kind":"use","source":33},{"line":505,"start":18651,"end":18719,"expression":"\n \\nabla C_h\\cdot\\widehat f_h\n =\\nabla C_h\\cdot f_h-\\Lambda_hd_h=0.\n","context":1946,"kind":"use","source":33},{"line":510,"start":18797,"end":18800,"expression":"C_h","context":1947,"kind":"use","source":33},{"line":514,"start":18900,"end":18903,"expression":"z_n","context":1948,"kind":"use","source":33},{"line":515,"start":18941,"end":19007,"expression":"\\widetilde z=(\\widetilde P,\\widetilde Q,\\widetilde U,\\widetilde V)","context":1948,"kind":"use","source":33},{"line":516,"start":19016,"end":19045,"expression":"\\widetilde g_U,\\widetilde g_V","context":1948,"kind":"use","source":33},{"line":517,"start":19080,"end":19096,"expression":"\\widetilde d_h>0","context":1948,"kind":"use","source":33},{"line":518,"start":19116,"end":19282,"expression":"\n \\mu=\\frac{C_h(\\widetilde z)-C_h(z_n)}{\\widetilde d_h},\\qquad\n z_{n+1}=(\\widetilde P,\\widetilde Q,\n \\widetilde U-\\mu\\widetilde g_U,\n \\widetilde V-\\mu\\widetilde g_V)\n","context":1948,"kind":"use","source":33},{"line":524,"start":19296,"end":19317,"expression":"C_h(z_{n+1})=C_h(z_n)","context":1949,"kind":"use","source":33},{"line":530,"start":19481,"end":19529,"expression":"C_h(z_{n+1})=C_h(\\widetilde z)-\\mu\\widetilde d_h","context":1950,"kind":"use","source":33},{"line":531,"start":19555,"end":19588,"expression":"g=(\\widetilde g_U,\\widetilde g_V)","context":1950,"kind":"use","source":33},{"line":532,"start":19595,"end":19623,"expression":"b=C_h(z_n)-C_h(\\widetilde z)","context":1950,"kind":"use","source":33},{"line":533,"start":19666,"end":19667,"expression":"v","context":1950,"kind":"use","source":33},{"line":533,"start":19680,"end":19690,"expression":"g\\cdot v=b","context":1950,"kind":"use","source":33},{"line":534,"start":19718,"end":19740,"expression":"\\norm v\\geq|b|/\\norm g","context":1950,"kind":"use","source":33},{"line":535,"start":19766,"end":19780,"expression":"v=bg/\\norm g^2","context":1950,"kind":"use","source":33},{"line":541,"start":19954,"end":19955,"expression":"h","context":1951,"kind":"use","source":33},{"line":542,"start":19994,"end":20021,"expression":"\\dot z=\\widehat f_h(\\tau,z)","context":1951,"kind":"use","source":33},{"line":543,"start":20095,"end":20108,"expression":"d_h\\geq d_0>0","context":1951,"kind":"use","source":33},{"line":544,"start":20159,"end":20171,"expression":"\\widehat f_h","context":1951,"kind":"use","source":33},{"line":545,"start":20223,"end":20226,"expression":"C_h","context":1951,"kind":"use","source":33},{"line":547,"start":20389,"end":20404,"expression":"O(\\Delta\\tau^3)","context":1952,"kind":"use","source":33},{"line":548,"start":20456,"end":20471,"expression":"O(\\Delta\\tau^2)","context":1953,"kind":"use","source":33},{"line":554,"start":20670,"end":20685,"expression":"O(\\Delta\\tau^3)","context":1954,"kind":"use","source":33},{"line":555,"start":20716,"end":20719,"expression":"C_h","context":1954,"kind":"use","source":33},{"line":555,"start":20735,"end":20738,"expression":"C_h","context":1954,"kind":"use","source":33},{"line":556,"start":20814,"end":20829,"expression":"O(\\Delta\\tau^3)","context":1954,"kind":"use","source":33},{"line":557,"start":20891,"end":20892,"expression":"g","context":1954,"kind":"use","source":33},{"line":558,"start":20915,"end":20930,"expression":"O(\\Delta\\tau^3)","context":1955,"kind":"use","source":33},{"line":560,"start":21074,"end":21087,"expression":"1+L\\Delta\\tau","context":1956,"kind":"use","source":33},{"line":561,"start":21110,"end":21113,"expression":"e_n","context":1957,"kind":"use","source":33},{"line":562,"start":21163,"end":21210,"expression":"\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n","context":1958,"kind":"use","source":33},{"line":565,"start":21244,"end":21261,"expression":"n\\Delta\\tau\\leq T","context":1959,"kind":"use","source":33},{"line":566,"start":21284,"end":21305,"expression":"MT e^{LT}\\Delta\\tau^2","context":1960,"kind":"use","source":33},{"line":566,"start":21313,"end":21318,"expression":"e_0=0","context":1961,"kind":"use","source":33},{"line":570,"start":21411,"end":21420,"expression":"g_U=g_V=0","context":1962,"kind":"use","source":33},{"line":571,"start":21478,"end":21483,"expression":"C_h=0","context":1962,"kind":"use","source":33},{"line":572,"start":21535,"end":21536,"expression":"h","context":1962,"kind":"use","source":33},{"line":573,"start":21636,"end":21637,"expression":"h","context":1962,"kind":"use","source":33},{"line":576,"start":21708,"end":21723,"expression":"H_{G,h},M_{G,h}","context":1963,"kind":"use","source":33},{"line":577,"start":21798,"end":21801,"expression":"D_h","context":1963,"kind":"use","source":33},{"line":578,"start":21808,"end":21848,"expression":"\\mathcal E_{P,h}=\\dot U-\\dot U_{\\rm raw}","context":1963,"kind":"use","source":33},{"line":579,"start":21855,"end":21895,"expression":"\\mathcal E_{Q,h}=\\dot V-\\dot V_{\\rm raw}","context":1963,"kind":"use","source":33},{"line":580,"start":21905,"end":22011,"expression":"\n \\mathcal I_h=\\partial_\\tau M_{G,h}-D_hH_{G,h}\n -2(D_hP)\\mathcal E_{P,h}\n -2e^{2P}(D_hQ)\\mathcal E_{Q,h}\n","context":1963,"kind":"use","source":33},{"line":586,"start":22119,"end":22131,"expression":"\\mathcal I_h","context":1964,"kind":"use","source":33},{"line":604,"start":22800,"end":22803,"expression":"V_h","context":1965,"kind":"use","source":33},{"line":605,"start":22848,"end":22874,"expression":"\\delta_1,\\delta_2,\\delta_3","context":1965,"kind":"use","source":33},{"line":606,"start":22921,"end":23047,"expression":"\n \\delta_i\\delta_j=\\delta_j\\delta_i,\n \\qquad\\inner{\\delta_iX}{Y}_h=-\\inner{X}{\\delta_iY}_h,\n \\qquad\\delta^2=\\sum_i\\delta_i^2.\n","context":1965,"kind":"use","source":33},{"line":615,"start":23253,"end":23266,"expression":"h_{ij}=h_{ji}","context":1966,"kind":"use","source":33},{"line":615,"start":23273,"end":23290,"expression":"\\pi_{ij}=\\pi_{ji}","context":1966,"kind":"use","source":33},{"line":615,"start":23308,"end":23311,"expression":"V_h","context":1966,"kind":"use","source":33},{"line":616,"start":23356,"end":23373,"expression":"h_{\\rm tr}=h_{kk}","context":1966,"kind":"use","source":33},{"line":617,"start":23377,"end":23398,"expression":"\\pi_{\\rm tr}=\\pi_{kk}","context":1966,"kind":"use","source":33},{"line":618,"start":23424,"end":23686,"expression":"\n C&=\\delta_i\\delta_jh_{ij}-\\delta^2h_{\\rm tr},\n &C_i&=-2\\delta_j\\pi_{ij},\\nonumber\\\\\n \\mathcal G_{ij}(h)&=\\frac12\\big[\n \\delta_k\\delta_ih_{jk}+\\delta_k\\delta_jh_{ik}\n -\\delta^2h_{ij}-\\delta_i\\delta_jh_{\\rm tr}\n -\\delta_{ij}C\\big].\n \\label{c:core:adm-operators}\n","context":1967,"kind":"use","source":33},{"line":630,"start":23818,"end":23846,"expression":"\\delta_j\\mathcal G_{ij}(h)=0","context":1968,"kind":"use","source":33},{"line":636,"start":23963,"end":24169,"expression":"\n 2\\delta_j\\mathcal G_{ij}\n ={}&\\delta_i\\delta_j\\delta_kh_{jk}\n +\\delta^2\\delta_kh_{ik}-\\delta^2\\delta_jh_{ij}\n -\\delta_i\\delta^2h_{\\rm tr}\\\\\n &-\\delta_i\\delta_k\\delta_lh_{kl}\n +\\delta_i\\delta^2h_{\\rm tr}.\n","context":1969,"kind":"use","source":33},{"line":645,"start":24244,"end":24265,"expression":"B_i(a)=\\delta_ja_{ij}","context":1970,"kind":"use","source":33},{"line":645,"start":24272,"end":24299,"expression":"A(a)=\\delta_i\\delta_ja_{ij}","context":1970,"kind":"use","source":33},{"line":647,"start":24362,"end":24578,"expression":"\n 2\\inner{a}{\\mathcal G(b)}_h\n ={}&-2\\inner{B(a)}{B(b)}_h\n +\\sum_k\\inner{\\delta_ka}{\\delta_kb}_h\\\\\n &-\\inner{A(a)}{b_{\\rm tr}}_h\n -\\inner{a_{\\rm tr}}{A(b)}_h\n -\\sum_k\\inner{\\delta_ka_{\\rm tr}}{\\delta_kb_{\\rm tr}}_h.\n","context":1971,"kind":"use","source":33},{"line":655,"start":24624,"end":24627,"expression":"a,b","context":1972,"kind":"use","source":33},{"line":659,"start":24649,"end":24768,"expression":"\n T_h=\\inner{\\pi}{\\pi}_h-\\tfrac12\\inner{\\pi_{\\rm tr}}{\\pi_{\\rm tr}}_h,\n \\qquad U_h=\\tfrac12\\inner{h}{\\mathcal G(h)}_h.\n","context":1973,"kind":"use","source":33},{"line":663,"start":24796,"end":24797,"expression":"n","context":1973,"kind":"use","source":33},{"line":663,"start":24823,"end":24826,"expression":"n_i","context":1973,"kind":"use","source":33},{"line":664,"start":24859,"end":24986,"expression":"\n S_h^{(2)}=\\int\\left[\n \\inner{\\pi}{\\dot h}_h-T_h-U_h\n +\\inner{n}{C}_h-\\inner{n_i}{C_i}_h\\right]dt.\n \\label{c:core:adm-action}\n","context":1973,"kind":"use","source":33},{"line":670,"start":25015,"end":25020,"expression":"n,n_i","context":1973,"kind":"use","source":33},{"line":670,"start":25031,"end":25038,"expression":"C=C_i=0","context":1973,"kind":"use","source":33},{"line":671,"start":25082,"end":25301,"expression":"\n \\dot h_{ij}&=2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}\n             +\\delta_in_j+\\delta_jn_i,\\nonumber\\\\\n \\dot\\pi_{ij}&=-\\mathcal G_{ij}(h)\n              +\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n.\n \\label{c:core:adm-evolution}\n","context":1974,"kind":"use","source":33},{"line":678,"start":25338,"end":25371,"expression":"2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}","context":1975,"kind":"use","source":33},{"line":680,"start":25435,"end":25448,"expression":"\\mathcal G(h)","context":1975,"kind":"use","source":33},{"line":682,"start":25555,"end":25604,"expression":"-\\langle\\delta_in_j+\\delta_jn_i,\\pi_{ij}\\rangle_h","context":1975,"kind":"use","source":33},{"line":692,"start":25915,"end":25955,"expression":"\n \\dot C=-\\delta_iC_i,\\qquad\\dot C_i=0.\n","context":1976,"kind":"use","source":33},{"line":697,"start":26060,"end":26085,"expression":"C[n]=\\langle n,C\\rangle_h","context":1976,"kind":"use","source":33},{"line":697,"start":26089,"end":26121,"expression":"D[\\xi]=\\langle\\xi_i,C_i\\rangle_h","context":1976,"kind":"use","source":33},{"line":698,"start":26144,"end":26157,"expression":"\\{C[n],C[m]\\}","context":1976,"kind":"use","source":33},{"line":698,"start":26161,"end":26179,"expression":"\\{D[\\xi],D[\\eta]\\}","context":1977,"kind":"use","source":33},{"line":699,"start":26187,"end":26202,"expression":"\\{C[n],D[\\xi]\\}","context":1978,"kind":"use","source":33},{"line":703,"start":26288,"end":26294,"expression":"\\dot h","context":1979,"kind":"use","source":33},{"line":703,"start":26300,"end":26313,"expression":"-\\pi_{\\rm tr}","context":1979,"kind":"use","source":33},{"line":704,"start":26337,"end":26343,"expression":"\\dot C","context":1979,"kind":"use","source":33},{"line":705,"start":26359,"end":26397,"expression":"2\\delta_i\\delta_j\\pi_{ij}=-\\delta_iC_i","context":1979,"kind":"use","source":33},{"line":706,"start":26428,"end":26500,"expression":"\n \\delta_i\\delta_j(\\delta_in_j+\\delta_jn_i)\n -\\delta^2(2\\delta_kn_k)=0.\n","context":1979,"kind":"use","source":33},{"line":710,"start":26508,"end":26516,"expression":"\\dot C_i","context":1979,"kind":"use","source":33},{"line":710,"start":26538,"end":26548,"expression":"\\mathcal G","context":1979,"kind":"use","source":33},{"line":712,"start":26599,"end":26649,"expression":"\\delta_j(\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n)=0","context":1979,"kind":"use","source":33},{"line":716,"start":26743,"end":26829,"expression":"\\{F,G\\}=\\langle\\nabla_hF,\\nabla_\\pi G\\rangle_h\n-\\langle\\nabla_\\pi F,\\nabla_hG\\rangle_h","context":1980,"kind":"use","source":33},{"line":718,"start":26871,"end":26872,"expression":"h","context":1980,"kind":"use","source":33},{"line":719,"start":26919,"end":26922,"expression":"\\pi","context":1980,"kind":"use","source":33},{"line":721,"start":26993,"end":27116,"expression":"\n (\\nabla_hC[n])_{ij}=\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n,\n \\qquad(\\nabla_\\pi D[\\xi])_{ij}=\\delta_i\\xi_j+\\delta_j\\xi_i.\n","context":1980,"kind":"use","source":33},{"line":730,"start":27294,"end":27342,"expression":"h_{ij}\\mapsto h_{ij}+\\delta_i\\xi_j+\\delta_j\\xi_i","context":1981,"kind":"use","source":33},{"line":732,"start":27399,"end":27446,"expression":"2\\delta^2\\delta_k\\xi_k-2\\delta^2\\delta_k\\xi_k=0","context":1981,"kind":"use","source":33},{"line":735,"start":27557,"end":27558,"expression":"A","context":1982,"kind":"use","source":33},{"line":737,"start":27590,"end":27663,"expression":"\n E_A^{\\rm con}=\\sum_{x\\in A}h^3\n \\left(|C_x|^2+\\sum_i|C_{i,x}|^2\\right)\n","context":1982,"kind":"use","source":33},{"line":749,"start":27929,"end":27936,"expression":"n=n_i=0","context":1983,"kind":"use","source":33},{"line":750,"start":27991,"end":28077,"expression":"\n \\delta_jh_{ij}=0,\\quad h_{\\rm tr}=0,\n \\qquad\\delta_j\\pi_{ij}=0,\\quad\\pi_{\\rm tr}=0.\n","context":1983,"kind":"use","source":33},{"line":755,"start":28126,"end":28250,"expression":"\n \\mathcal G_{ij}(h)=-\\tfrac12\\delta^2h_{ij},\\qquad\n \\dot h=2\\pi,\\qquad\\dot\\pi=\\tfrac12\\delta^2h,\n \\qquad\\ddot h=\\delta^2h.\n","context":1983,"kind":"use","source":33},{"line":761,"start":28292,"end":28362,"expression":"\n H_{\\rm TT}=\\norm{\\pi}_h^2\n +\\frac14\\sum_k\\norm{\\delta_kh}_h^2\\geq0.\n","context":1983,"kind":"use","source":33},{"line":767,"start":28447,"end":28523,"expression":"\\langle\\pi,\\delta^2h\\rangle_h\n+\\sum_k\\langle\\delta_kh,\\delta_k\\pi\\rangle_h=0","context":1984,"kind":"use","source":33},{"line":771,"start":28585,"end":28612,"expression":"\\widetilde k_i=\\sin(k_ih)/h","context":1985,"kind":"use","source":33},{"line":771,"start":28619,"end":28645,"expression":"\\widetilde{\\mathbf k}\\neq0","context":1985,"kind":"use","source":33},{"line":773,"start":28657,"end":28844,"expression":"\n \\Pi_{ij}=\\delta_{ij}\n -\\frac{\\widetilde k_i\\widetilde k_j}{|\\widetilde{\\mathbf k}|^2},\n \\qquad\n \\Lambda^{\\rm TT}_{ij,kl}\n =\\tfrac12(\\Pi_{ik}\\Pi_{jl}+\\Pi_{il}\\Pi_{jk}-\\Pi_{ij}\\Pi_{kl}).\n","context":1985,"kind":"use","source":33},{"line":780,"start":28871,"end":28872,"expression":"X","context":1985,"kind":"use","source":33},{"line":781,"start":28889,"end":28927,"expression":"\\Pi X\\Pi-\\Pi\\operatorname{tr}(\\Pi X)/2","context":1985,"kind":"use","source":33},{"line":782,"start":28970,"end":28979,"expression":"\\Pi^2=\\Pi","context":1986,"kind":"use","source":33},{"line":782,"start":28986,"end":29008,"expression":"\\operatorname{tr}\\Pi=2","context":1987,"kind":"use","source":33},{"line":788,"start":29250,"end":29326,"expression":"\n \\omega_h^2=\\sum_i\\frac{\\sin^2(k_ih)}{h^2}.\n \\label{c:core:adm-dispersion}\n","context":1988,"kind":"use","source":33},{"line":792,"start":29352,"end":29366,"expression":"\\mathbf k\\neq0","context":1988,"kind":"use","source":33},{"line":792,"start":29372,"end":29377,"expression":"h\\to0","context":1988,"kind":"use","source":33},{"line":793,"start":29382,"end":29520,"expression":"\n \\omega_h^2=|\\mathbf k|^2-\\frac{h^2}{3}\\sum_i k_i^4+O(h^4),\n \\qquad\n \\omega_h=|\\mathbf k|\n -\\frac{h^2}{6|\\mathbf k|}\\sum_i k_i^4+O(h^4).\n","context":1988,"kind":"use","source":33},{"line":800,"start":29611,"end":29620,"expression":"\\mathbf k","context":1989,"kind":"use","source":33},{"line":830,"start":30971,"end":30998,"expression":"\\{q_a,p_b\\}_h=\\delta_{ab}/h","context":1990,"kind":"use","source":33},{"line":830,"start":31019,"end":31020,"expression":"D","context":1990,"kind":"use","source":33},{"line":831,"start":31032,"end":31122,"expression":"\n \\mathcal D_h[\\xi]=h\\sum_a\\xi_ap_a(Dq)_a,\n \\qquad\n \\mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).\n","context":1990,"kind":"use","source":33},{"line":840,"start":31284,"end":31296,"expression":"q,p,\\xi,\\eta","context":1991,"kind":"use","source":33},{"line":841,"start":31312,"end":31537,"expression":"\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n ={}&\\mathcal D_h[\\xi D\\eta-\\eta D\\xi]\\nonumber\\\\\n &+h\\sum_ap_a\\big[\n \\xi_a\\mathfrak L_D(\\eta,Dq)_a\n -\\eta_a\\mathfrak L_D(\\xi,Dq)_a\\big].\n \\label{c:core:nonlinear-defect-identity}\n","context":1992,"kind":"use","source":33},{"line":853,"start":31616,"end":31651,"expression":"\\partial_p\\mathcal D_h[\\xi]=h\\xi Dq","context":1993,"kind":"use","source":33},{"line":854,"start":31658,"end":31696,"expression":"\\partial_q\\mathcal D_h[\\xi]=-hD(\\xi p)","context":1993,"kind":"use","source":33},{"line":855,"start":31746,"end":31762,"expression":"D^{\\mathsf T}=-D","context":1993,"kind":"use","source":33},{"line":856,"start":31767,"end":31893,"expression":"\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n =h\\sum_ap_a\\left[\\xi_aD(\\eta Dq)_a\n                  -\\eta_aD(\\xi Dq)_a\\right].\n","context":1993,"kind":"use","source":33},{"line":861,"start":31947,"end":31960,"expression":"\\mathfrak L_D","context":1993,"kind":"use","source":33},{"line":862,"start":31979,"end":31991,"expression":"\\xi\\eta D^2q","context":1994,"kind":"use","source":33},{"line":870,"start":32296,"end":32300,"expression":"\\C^N","context":1995,"kind":"use","source":33},{"line":876,"start":32453,"end":32456,"expression":"e_a","context":1996,"kind":"use","source":33},{"line":876,"start":32493,"end":32494,"expression":"D","context":1996,"kind":"use","source":33},{"line":877,"start":32519,"end":32545,"expression":"D(e_a)=D(e_a^2)=2e_aD(e_a)","context":1996,"kind":"use","source":33},{"line":877,"start":32563,"end":32564,"expression":"a","context":1996,"kind":"use","source":33},{"line":879,"start":32685,"end":32693,"expression":"D(e_a)=0","context":1996,"kind":"use","source":33},{"line":879,"start":32706,"end":32707,"expression":"a","context":1996,"kind":"use","source":33},{"line":880,"start":32752,"end":32755,"expression":"D=0","context":1996,"kind":"use","source":33},{"line":891,"start":33154,"end":33157,"expression":"N=5","context":1997,"kind":"use","source":33},{"line":891,"start":33161,"end":33164,"expression":"h=1","context":1997,"kind":"use","source":33},{"line":891,"start":33172,"end":33198,"expression":"(Df)_a=(f_{a+1}-f_{a-1})/2","context":1997,"kind":"use","source":33},{"line":892,"start":33207,"end":33212,"expression":"q=e_0","context":1997,"kind":"use","source":33},{"line":892,"start":33216,"end":33225,"expression":"p=\\xi=e_1","context":1997,"kind":"use","source":33},{"line":892,"start":33233,"end":33241,"expression":"\\eta=e_2","context":1997,"kind":"use","source":33},{"line":893,"start":33250,"end":33261,"expression":"(Dq)_1=-1/2","context":1997,"kind":"use","source":33},{"line":893,"start":33265,"end":33278,"expression":"(D\\eta)_1=1/2","context":1997,"kind":"use","source":33},{"line":893,"start":33286,"end":33295,"expression":"\\eta Dq=0","context":1997,"kind":"use","source":33},{"line":895,"start":33357,"end":33404,"expression":"\n \\mathcal D_h[\\xi D\\eta-\\eta D\\xi]=-\\tfrac14.\n","context":1997,"kind":"use","source":33},{"line":898,"start":33440,"end":33444,"expression":"+1/4","context":1997,"kind":"use","source":33},{"line":907,"start":33746,"end":33782,"expression":"\n D_hf=f'+\\frac{h^2}{6}f'''+O(h^4).\n","context":1998,"kind":"use","source":33},{"line":910,"start":33789,"end":33792,"expression":"f,g","context":1998,"kind":"use","source":33},{"line":912,"start":33907,"end":33999,"expression":"\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n","context":1998,"kind":"use","source":33},{"line":917,"start":34032,"end":34044,"expression":"p,\\xi,\\eta,q","context":1998,"kind":"use","source":33},{"line":918,"start":34105,"end":34106,"expression":"q","context":1999,"kind":"use","source":33},{"line":918,"start":34120,"end":34124,"expression":"D_hq","context":2000,"kind":"use","source":33},{"line":919,"start":34191,"end":34197,"expression":"O(h^2)","context":2001,"kind":"use","source":33},{"line":920,"start":34256,"end":34262,"expression":"O(h^2)","context":2002,"kind":"use","source":33},{"line":920,"start":34269,"end":34271,"expression":"hN","context":2003,"kind":"use","source":33},{"line":930,"start":34766,"end":34767,"expression":"K","context":2004,"kind":"use","source":33},{"line":931,"start":34778,"end":34794,"expression":"\\delta_KX=i[K,X]","context":2004,"kind":"use","source":33},{"line":932,"start":34841,"end":34916,"expression":"\n \\delta_K(XY)=(\\delta_KX)Y+X(\\delta_KY).\n \\label{c:core:inner-derivation}\n","context":2004,"kind":"use","source":33},{"line":936,"start":34946,"end":34973,"expression":"\\delta_K(X^*)=(\\delta_KX)^*","context":2004,"kind":"use","source":33},{"line":937,"start":34977,"end":35005,"expression":"\\operatorname{Tr}\\delta_KX=0","context":2004,"kind":"use","source":33},{"line":939,"start":35105,"end":35106,"expression":"K","context":2004,"kind":"use","source":33},{"line":24,"start":3085,"end":3130,"expression":"\\{F,G\\}=\\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})","context":2005,"kind":"use","source":34},{"line":24,"start":3137,"end":3151,"expression":"\\dot F=\\{F,H\\}","context":2005,"kind":"use","source":34},{"line":27,"start":3622,"end":3623,"expression":"K","context":2006,"kind":"introduction-passage","source":34},{"line":27,"start":3654,"end":3655,"expression":"N","context":2006,"kind":"introduction-passage","source":34},{"line":27,"start":3732,"end":3737,"expression":"16K<N","context":2006,"kind":"introduction-passage","source":34},{"line":6,"start":841,"end":861,"expression":"G=c=4\\pi\\epsilon_0=1","context":2007,"kind":"introduction-passage","source":36},{"line":6,"start":965,"end":968,"expression":"R>0","context":2008,"kind":"introduction-passage","source":36},{"line":6,"start":975,"end":984,"expression":"\\Lambda>0","context":2009,"kind":"introduction-passage","source":36},{"line":7,"start":1009,"end":1098,"expression":"\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n","context":2010,"kind":"introduction-passage","source":36},{"line":11,"start":1119,"end":1120,"expression":"R","context":2011,"kind":"introduction-passage","source":36},{"line":11,"start":1144,"end":1151,"expression":"\\Lambda","context":2012,"kind":"introduction-passage","source":36},{"line":11,"start":1189,"end":1190,"expression":"N","context":2013,"kind":"introduction-passage","source":36},{"line":11,"start":1211,"end":1214,"expression":"N^r","context":2014,"kind":"introduction-passage","source":36},{"line":11,"start":1272,"end":1278,"expression":"\\Phi_e","context":2015,"kind":"introduction-passage","source":36},{"line":12,"start":1380,"end":1518,"expression":"\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n","context":2016,"kind":"use","source":36},{"line":20,"start":1834,"end":1894,"expression":"\n \\mathcal D=P_RR'-\\Lambda P_\\Lambda',\\qquad \\mathcal G=Q',\n","context":2017,"kind":"use","source":36},{"line":24,"start":1959,"end":2213,"expression":"\n\\begin{aligned}\n \\mathcal H={}&-\\frac{P_RP_\\Lambda}{R}\n +\\frac{\\Lambda P_\\Lambda^2}{2R^2}\n +\\frac{RR''}{\\Lambda}-\\frac{RR'\\Lambda'}{\\Lambda^2}\n +\\frac{R'^2}{2\\Lambda}-\\frac\\Lambda2\n +\\frac{\\Lambda Q^2}{2R^2}.\n\\end{aligned}\n\\label{c:strong:electrovac-H}\n","context":2017,"kind":"use","source":36},{"line":35,"start":2275,"end":2423,"expression":"\n P_\\Lambda&=-\\frac RN(\\dot R-N^rR'),\\\\\n P_R&=-\\frac1N\\left[\\Lambda(\\dot R-N^rR')\n                  +R\\bigl(\\dot\\Lambda-(N^r\\Lambda)'\\bigr)\\right].\n","context":2018,"kind":"use","source":36},{"line":44,"start":2653,"end":2838,"expression":"\n \\mathcal M=\\frac R2\\left[1-\\left(\\frac{R'}\\Lambda\\right)^2\n                    +\\left(\\frac{P_\\Lambda}R\\right)^2\\right]\n             +\\frac{Q^2}{2R}.\n\\label{c:strong:mass-coordinate}\n","context":2019,"kind":"definition-passage","definition_title":"Charged mass coordinate","source":36},{"line":50,"start":2893,"end":2903,"expression":"\\mathcal M","context":2019,"kind":"definition-passage","definition_title":"Charged mass coordinate","source":36},{"line":55,"start":3127,"end":3285,"expression":"\n \\mathcal M'=-\\frac{R'}\\Lambda\\mathcal H\n             -\\frac{P_\\Lambda}{R\\Lambda}\\mathcal D\n             +\\frac QR\\mathcal G.\n\\label{c:strong:mass-identity}\n","context":2020,"kind":"use","source":36},{"line":61,"start":3332,"end":3346,"expression":"R'/\\Lambda\\ne0","context":2020,"kind":"use","source":36},{"line":62,"start":3362,"end":3398,"expression":"(\\mathcal H,\\mathcal D,\\mathcal G)=0","context":2020,"kind":"use","source":36},{"line":63,"start":3405,"end":3442,"expression":"(\\mathcal M',\\mathcal D,\\mathcal G)=0","context":2021,"kind":"use","source":36},{"line":67,"start":3571,"end":3807,"expression":"\n\\begin{aligned}\n \\mathcal M'={}&\\frac{R'}2-\\frac{R'^3}{2\\Lambda^2}\n -\\frac{RR'R''}{\\Lambda^2}+\\frac{RR'^2\\Lambda'}{\\Lambda^3}\n +\\frac{P_\\Lambda P_\\Lambda'}R-\\frac{P_\\Lambda^2R'}{2R^2}\\\\\n &+\\frac{QQ'}R-\\frac{Q^2R'}{2R^2}.\n\\end{aligned}\n","context":2022,"kind":"use","source":36},{"line":75,"start":3898,"end":3901,"expression":"P_R","context":2023,"kind":"use","source":36},{"line":75,"start":4014,"end":4031,"expression":"(-R'/\\Lambda,1,1)","context":2023,"kind":"use","source":36},{"line":83,"start":4600,"end":4603,"expression":"M>0","context":2024,"kind":"use","source":36},{"line":83,"start":4621,"end":4622,"expression":"Q","context":2024,"kind":"use","source":36},{"line":83,"start":4639,"end":4662,"expression":"\\varepsilon\\in\\{+1,-1\\}","context":2024,"kind":"use","source":36},{"line":83,"start":4669,"end":4678,"expression":"2MR-Q^2>0","context":2024,"kind":"use","source":36},{"line":84,"start":4704,"end":4763,"expression":"\n p(R)=\\varepsilon\\sqrt{2MR-Q^2},\\qquad v(R)=\\frac{p(R)}R.\n","context":2024,"kind":"use","source":36},{"line":88,"start":4843,"end":4911,"expression":"\n ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2d\\Omega^2.\n\\label{c:strong:RN-PG}\n","context":2024,"kind":"use","source":36},{"line":92,"start":4948,"end":4962,"expression":"1-2M/R+Q^2/R^2","context":2024,"kind":"use","source":36},{"line":94,"start":5099,"end":5107,"expression":"R=F(t,r)","context":2025,"kind":"use","source":36},{"line":94,"start":5115,"end":5120,"expression":"F_r>0","context":2025,"kind":"use","source":36},{"line":94,"start":5136,"end":5141,"expression":"t_P=t","context":2025,"kind":"use","source":36},{"line":94,"start":5155,"end":5169,"expression":"dR=F_tdt+F_rdr","context":2025,"kind":"use","source":36},{"line":95,"start":5198,"end":5256,"expression":"\n N=1,\\qquad \\Lambda=F_r,\\qquad N^r=\\frac{F_t+v(F)}{F_r}.\n","context":2025,"kind":"use","source":36},{"line":99,"start":5324,"end":5445,"expression":"\n P_\\Lambda=p(F),\\qquad P_R=\\frac{MF_r}{p(F)},\n \\qquad A_r=0,\\quad \\Phi_e=-\\frac QF,\\quad Q'=0.\n\\label{c:strong:RN-data}\n","context":2025,"kind":"use","source":36},{"line":104,"start":5473,"end":5476,"expression":"P_R","context":2025,"kind":"use","source":36},{"line":104,"start":5495,"end":5499,"expression":"p(F)","context":2026,"kind":"use","source":36},{"line":107,"start":5679,"end":5684,"expression":"F_r>0","context":2027,"kind":"use","source":36},{"line":107,"start":5688,"end":5691,"expression":"F>0","context":2027,"kind":"use","source":36},{"line":107,"start":5699,"end":5708,"expression":"2MF-Q^2>0","context":2027,"kind":"use","source":36},{"line":110,"start":5751,"end":5763,"expression":"R'/\\Lambda=1","context":2027,"kind":"use","source":36},{"line":110,"start":5770,"end":5784,"expression":"p(F)^2=2MF-Q^2","context":2027,"kind":"use","source":36},{"line":110,"start":5793,"end":5805,"expression":"\\mathcal M=M","context":2027,"kind":"use","source":36},{"line":111,"start":5830,"end":5866,"expression":"\n P_\\Lambda'=\\frac{MF_r}{p(F)}=P_R,\n","context":2027,"kind":"use","source":36},{"line":114,"start":5938,"end":5942,"expression":"Q'=0","context":2028,"kind":"use","source":36},{"line":114,"start":5955,"end":5967,"expression":"\\mathcal H=0","context":2029,"kind":"use","source":36},{"line":117,"start":6028,"end":6091,"expression":"\n \\dot R=-\\frac{P_\\Lambda}R+N^rR'\n        =-v(F)+F_t+v(F)=F_t.\n","context":2030,"kind":"use","source":36},{"line":122,"start":6152,"end":6237,"expression":"\n \\dot\\Lambda=\\frac{\\Lambda P_\\Lambda}{R^2}\n             -\\frac{P_R}R+(N^r\\Lambda)'.\n","context":2030,"kind":"use","source":36},{"line":126,"start":6257,"end":6264,"expression":"p_R=M/p","context":2030,"kind":"use","source":36},{"line":126,"start":6271,"end":6287,"expression":"v_R=M/(pR)-p/R^2","context":2030,"kind":"use","source":36},{"line":126,"start":6299,"end":6305,"expression":"F_{tr}","context":2030,"kind":"use","source":36},{"line":126,"start":6330,"end":6348,"expression":"\\dot\\Lambda=F_{tr}","context":2030,"kind":"use","source":36},{"line":126,"start":6382,"end":6385,"expression":"N=1","context":2031,"kind":"use","source":36},{"line":127,"start":6415,"end":6688,"expression":"\n \\dot P_\\Lambda&=N^rP_\\Lambda'\n +\\frac12\\left[1-\\frac{P_\\Lambda^2+Q^2}{R^2}\n                 -\\left(\\frac{R'}\\Lambda\\right)^2\\right],\\\\\n \\dot P_R&=\\frac{\\Lambda(P_\\Lambda^2+Q^2)}{R^3}\n -\\frac{P_\\Lambda P_R}{R^2}+(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n","context":2032,"kind":"use","source":36},{"line":135,"start":6743,"end":6757,"expression":"p(F)^2=2MF-Q^2","context":2033,"kind":"use","source":36},{"line":136,"start":6804,"end":6898,"expression":"\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n","context":2033,"kind":"use","source":36},{"line":142,"start":6981,"end":6997,"expression":"\\Phi_e'=QF_r/F^2","context":2034,"kind":"use","source":36},{"line":142,"start":7008,"end":7022,"expression":"N\\Lambda Q/R^2","context":2035,"kind":"use","source":36},{"line":143,"start":7028,"end":7059,"expression":"\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'","context":2036,"kind":"use","source":36},{"line":146,"start":7100,"end":7101,"expression":"F","context":2037,"kind":"use","source":36},{"line":150,"start":7416,"end":7453,"expression":"\n f(R)=1-\\frac{2M}R+\\frac{Q^2}{R^2}.\n","context":2038,"kind":"use","source":36},{"line":153,"start":7473,"end":7480,"expression":"0<|Q|<M","context":2038,"kind":"use","source":36},{"line":154,"start":7526,"end":7554,"expression":"\n r_\\pm=M\\pm\\sqrt{M^2-Q^2}.\n","context":2038,"kind":"use","source":36},{"line":157,"start":7585,"end":7595,"expression":"p(R)^2=R^2","context":2038,"kind":"use","source":36},{"line":157,"start":7672,"end":7675,"expression":"F_r","context":2038,"kind":"use","source":36},{"line":157,"start":7688,"end":7693,"expression":"|Q|=M","context":2038,"kind":"use","source":36},{"line":157,"start":7723,"end":7726,"expression":"R=M","context":2038,"kind":"use","source":36},{"line":157,"start":7734,"end":7739,"expression":"|Q|>M","context":2038,"kind":"use","source":36},{"line":157,"start":7768,"end":7771,"expression":"Q=0","context":2039,"kind":"use","source":36},{"line":157,"start":7793,"end":7795,"expression":"2M","context":2040,"kind":"use","source":36},{"line":157,"start":7824,"end":7825,"expression":"0","context":2041,"kind":"use","source":36},{"line":160,"start":7957,"end":7988,"expression":"\n R_{\\rm turn}=\\frac{Q^2}{2M}.\n","context":2042,"kind":"use","source":36},{"line":163,"start":8010,"end":8013,"expression":"p=0","context":2042,"kind":"use","source":36},{"line":163,"start":8021,"end":8031,"expression":"P_R=MF_r/p","context":2042,"kind":"use","source":36},{"line":163,"start":8064,"end":8067,"expression":"F_r","context":2042,"kind":"use","source":36},{"line":163,"start":8304,"end":8307,"expression":"R=0","context":2043,"kind":"use","source":36},{"line":166,"start":8365,"end":8463,"expression":"\n \\mathcal K=\\frac{48M^2}{R^6}-\\frac{96MQ^2}{R^7}\n                            +\\frac{56Q^4}{R^8}.\n","context":2044,"kind":"use","source":36},{"line":170,"start":8483,"end":8486,"expression":"M=1","context":2044,"kind":"use","source":36},{"line":170,"start":8490,"end":8495,"expression":"Q=0.8","context":2044,"kind":"use","source":36},{"line":170,"start":8513,"end":8520,"expression":"r_+=1.6","context":2044,"kind":"use","source":36},{"line":170,"start":8524,"end":8531,"expression":"r_-=0.4","context":2044,"kind":"use","source":36},{"line":170,"start":8539,"end":8556,"expression":"R_{\\rm turn}=0.32","context":2044,"kind":"use","source":36},{"line":170,"start":8583,"end":8594,"expression":"R\\in[0.6,6]","context":2044,"kind":"use","source":36},{"line":170,"start":8646,"end":8673,"expression":"\\mathcal K\\simeq199.6646853","context":2044,"kind":"use","source":36},{"line":170,"start":8683,"end":8715,"expression":"\\mathcal K(r_+)\\simeq1.106262207","context":2044,"kind":"use","source":36},{"line":175,"start":9212,"end":9230,"expression":"d=\\sqrt{M^2-Q^2}>0","context":2045,"kind":"use","source":36},{"line":176,"start":9254,"end":9282,"expression":"\n f'(r_+)=\\frac{2d}{r_+^2}.\n","context":2045,"kind":"use","source":36},{"line":180,"start":9430,"end":9566,"expression":"\n |\\delta r_+|\\le\\frac{r_+^2}{2d}\\,\\norm{\\delta f}_\\infty\n                    +O(\\norm{\\delta f}_\\infty^2).\n\\label{c:strong:root-bound}\n","context":2045,"kind":"use","source":36},{"line":185,"start":9811,"end":9828,"expression":"2.133333333\\ldots","context":2046,"kind":"use","source":36},{"line":185,"start":9864,"end":9876,"expression":"d\\downarrow0","context":2047,"kind":"use","source":36},{"line":187,"start":9896,"end":9910,"expression":"f(R)=(1-M/R)^2","context":2048,"kind":"use","source":36},{"line":187,"start":9955,"end":9963,"expression":"\\epsilon","context":2048,"kind":"use","source":36},{"line":187,"start":9993,"end":10006,"expression":"\\sqrt\\epsilon","context":2048,"kind":"use","source":36},{"line":193,"start":10526,"end":10585,"expression":"\n (D_4u)_j=\\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.\n","context":2049,"kind":"use","source":36},{"line":196,"start":10634,"end":10635,"expression":"M","context":2049,"kind":"use","source":36},{"line":196,"start":10639,"end":10640,"expression":"Q","context":2049,"kind":"use","source":36},{"line":196,"start":10663,"end":10674,"expression":"\\varepsilon","context":2049,"kind":"use","source":36},{"line":196,"start":10704,"end":10712,"expression":"D_4R_j>0","context":2049,"kind":"use","source":36},{"line":196,"start":10719,"end":10730,"expression":"2MR_j-Q^2>0","context":2049,"kind":"use","source":36},{"line":197,"start":10759,"end":10937,"expression":"\n \\Pi_h[R]_j=\\left(R_j,D_4R_j,\n                 \\varepsilon\\sqrt{2MR_j-Q^2},\n                 \\frac{M D_4R_j}{\\varepsilon\\sqrt{2MR_j-Q^2}},Q\\right).\n\\label{c:strong:reconstruct}\n","context":2049,"kind":"use","source":36},{"line":203,"start":10987,"end":11014,"expression":"(R,\\Lambda,P_\\Lambda,P_R,Q)","context":2050,"kind":"use","source":36},{"line":203,"start":11151,"end":11152,"expression":"R","context":2051,"kind":"use","source":36},{"line":208,"start":11371,"end":11536,"expression":"\n\\begin{aligned}\n C_{\\Lambda,j}&=\\Lambda_j-D_4R_j,&\n C_{M,j}&=P_{\\Lambda,j}^2-2MR_j+Q^2,\\\\\n C_{P,j}&=P_{R,j}P_{\\Lambda,j}-M\\Lambda_j,&\n C_{Q,j}&=Q_j-Q\n\\end{aligned}\n","context":2052,"kind":"use","source":36},{"line":216,"start":11631,"end":11633,"expression":"R'","context":2053,"kind":"use","source":36},{"line":216,"start":11641,"end":11645,"expression":"D_4R","context":2053,"kind":"use","source":36},{"line":216,"start":11691,"end":11692,"expression":"M","context":2053,"kind":"use","source":36},{"line":219,"start":11837,"end":11851,"expression":"D_4R/\\Lambda=1","context":2053,"kind":"use","source":36},{"line":219,"start":11858,"end":11877,"expression":"P_\\Lambda^2=2MR-Q^2","context":2053,"kind":"use","source":36},{"line":220,"start":11889,"end":11926,"expression":"\\tfrac{R}{2}(2M/R-Q^2/R^2)+Q^2/(2R)=M","context":2054,"kind":"use","source":36},{"line":223,"start":12167,"end":12184,"expression":"D_4\\sqrt{2MR-Q^2}","context":2055,"kind":"use","source":36},{"line":223,"start":12202,"end":12222,"expression":"MD_4R/\\sqrt{2MR-Q^2}","context":2055,"kind":"use","source":36},{"line":227,"start":12493,"end":12567,"expression":"\n R_j\\ge R_*>0,\\quad 2MR_j-Q^2\\ge p_*^2>0,\n \\quad 0<d_*\\le D_4R_j\\le d^*.\n","context":2056,"kind":"use","source":36},{"line":231,"start":12625,"end":12642,"expression":"(M,Q,\\varepsilon)","context":2056,"kind":"use","source":36},{"line":232,"start":12711,"end":13018,"expression":"\n \\norm{\\delta\\Lambda}_\\infty&\\le\\norm{D_4\\delta R}_\\infty,\\\\\n \\norm{\\delta P_\\Lambda}_\\infty&\\le\\frac M{p_*}\\norm{\\delta R}_\\infty,\\\\\n \\norm{\\delta P_R}_\\infty&\\le\\frac M{p_*}\\norm{D_4\\delta R}_\\infty\n                          +\\frac{M^2d^*}{p_*^3}\\norm{\\delta R}_\\infty.\n\\label{c:strong:projection-bound}\n","context":2057,"kind":"use","source":36},{"line":239,"start":13044,"end":13059,"expression":"\\partial_Rp=M/p","context":2058,"kind":"use","source":36},{"line":239,"start":13097,"end":13107,"expression":"M\\Lambda/p","context":2058,"kind":"use","source":36},{"line":239,"start":13134,"end":13139,"expression":"M/p_*","context":2058,"kind":"use","source":36},{"line":239,"start":13146,"end":13158,"expression":"M^2d^*/p_*^3","context":2058,"kind":"use","source":36},{"line":239,"start":13279,"end":13280,"expression":"h","context":2059,"kind":"use","source":36},{"line":241,"start":13505,"end":13517,"expression":"W^{1,\\infty}","context":2060,"kind":"use","source":36},{"line":241,"start":13612,"end":13615,"expression":"L^2","context":2060,"kind":"use","source":36},{"line":245,"start":13787,"end":13887,"expression":"\n \\dot R_j=g(R_j)+b_j(D_4R)_j,\n \\qquad g(R)=-\\frac{\\sqrt{2MR-Q^2}}R,\n\\label{c:strong:direct-radius}\n","context":2061,"kind":"use","source":36},{"line":250,"start":13909,"end":13914,"expression":"b=N^r","context":2061,"kind":"use","source":36},{"line":251,"start":13998,"end":14108,"expression":"\n \\dot R=g(R)+A_h[b]R,\\qquad\n A_h[b]u=\\frac12\\bigl(bD_4u+D_4(bu)-(D_4b)u\\bigr).\n\\label{c:strong:split-radius}\n","context":2061,"kind":"use","source":36},{"line":258,"start":14370,"end":14375,"expression":"\\Pi_h","context":2062,"kind":"use","source":36},{"line":261,"start":15169,"end":15170,"expression":"D","context":2063,"kind":"use","source":36},{"line":261,"start":15216,"end":15230,"expression":"\\inner{u}{w}_h","context":2064,"kind":"use","source":36},{"line":261,"start":15275,"end":15276,"expression":"b","context":2065,"kind":"use","source":36},{"line":262,"start":15316,"end":15405,"expression":"\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n","context":2066,"kind":"use","source":36},{"line":267,"start":15486,"end":15487,"expression":"b","context":2067,"kind":"use","source":36},{"line":267,"start":15539,"end":15550,"expression":"|g'|\\le L_g","context":2068,"kind":"use","source":36},{"line":268,"start":15581,"end":15734,"expression":"\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n","context":2069,"kind":"use","source":36},{"line":276,"start":15807,"end":15842,"expression":"\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h","context":2070,"kind":"use","source":36},{"line":276,"start":15907,"end":15923,"expression":"e=R-\\widetilde R","context":2071,"kind":"use","source":36},{"line":277,"start":15962,"end":16000,"expression":"|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|","context":2072,"kind":"use","source":36},{"line":278,"start":16025,"end":16067,"expression":"\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n","context":2073,"kind":"use","source":36},{"line":281,"start":16088,"end":16142,"expression":"\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2","context":2074,"kind":"use","source":36},{"line":284,"start":16273,"end":16276,"expression":"D_4","context":2075,"kind":"use","source":36},{"line":284,"start":16350,"end":16358,"expression":"D_4R_j>0","context":2075,"kind":"use","source":36},{"line":286,"start":16690,"end":16691,"expression":"H","context":2076,"kind":"use","source":36},{"line":286,"start":16698,"end":16707,"expression":"HD+D^TH=B","context":2076,"kind":"use","source":36},{"line":286,"start":16729,"end":16730,"expression":"b","context":2076,"kind":"use","source":36},{"line":286,"start":16747,"end":16748,"expression":"H","context":2076,"kind":"use","source":36},{"line":287,"start":16799,"end":16891,"expression":"\n \\inner{u}{A_h[b]u}_H\n =\\frac12u^TB\\,\\operatorname{diag}(b)u\n  -\\frac12\\inner{u}{(Db)u}_H.\n","context":2076,"kind":"use","source":36},{"line":295,"start":17174,"end":17243,"expression":"\n \\widehat D_4(\\theta)=\\frac{i}{h}\\frac{8\\sin\\theta-\\sin2\\theta}{6}.\n","context":2077,"kind":"use","source":36},{"line":298,"start":17298,"end":17316,"expression":"1.3722219798\\ldots","context":2077,"kind":"use","source":36},{"line":299,"start":17421,"end":17467,"expression":"\n \\frac{|b|\\Delta t}{h}\\le2.0612023174\\ldots.\n","context":2077,"kind":"use","source":36},{"line":302,"start":17740,"end":17743,"expression":"L^2","context":2078,"kind":"use","source":36},{"line":302,"start":17812,"end":17824,"expression":"W^{1,\\infty}","context":2079,"kind":"use","source":36},{"line":308,"start":17965,"end":18071,"expression":"\n F(t,r)=r+0.08\\sin(0.7t)\n                \\sin^2\\!\\left(\\pi\\frac{r-0.6}{5.4}\\right),\n \\quad 0\\le t\\le0.4,\n","context":2080,"kind":"use","source":36},{"line":313,"start":18092,"end":18095,"expression":"M=1","context":2080,"kind":"use","source":36},{"line":313,"start":18099,"end":18104,"expression":"Q=0.8","context":2080,"kind":"use","source":36},{"line":313,"start":18112,"end":18122,"expression":"\\Delta t/h","context":2080,"kind":"use","source":36},{"line":313,"start":18134,"end":18137,"expression":"0.1","context":2080,"kind":"use","source":36},{"line":313,"start":18180,"end":18192,"expression":"0.9871380683","context":2080,"kind":"use","source":36},{"line":313,"start":18232,"end":18236,"expression":"0.56","context":2080,"kind":"use","source":36},{"line":319,"start":18517,"end":18518,"expression":"N","context":2081,"kind":"use","source":36},{"line":319,"start":18521,"end":18543,"expression":"\\norm{\\delta R}_\\infty","context":2081,"kind":"use","source":36},{"line":319,"start":18546,"end":18573,"expression":"\\norm{\\delta\\Lambda}_\\infty","context":2081,"kind":"use","source":36},{"line":319,"start":18576,"end":18606,"expression":"\\norm{\\delta P_\\Lambda}_\\infty","context":2081,"kind":"use","source":36},{"line":319,"start":18609,"end":18633,"expression":"\\norm{\\delta P_R}_\\infty","context":2081,"kind":"use","source":36},{"line":321,"start":18650,"end":18665,"expression":"8.3247\\,10^{-9}","context":2082,"kind":"use","source":36},{"line":321,"start":18668,"end":18683,"expression":"4.1362\\,10^{-8}","context":2082,"kind":"use","source":36},{"line":321,"start":18686,"end":18701,"expression":"5.7367\\,10^{-9}","context":2082,"kind":"use","source":36},{"line":321,"start":18704,"end":18719,"expression":"2.3732\\,10^{-8}","context":2082,"kind":"use","source":36},{"line":322,"start":18728,"end":18744,"expression":"5.0495\\,10^{-10}","context":2083,"kind":"use","source":36},{"line":322,"start":18747,"end":18762,"expression":"2.5079\\,10^{-9}","context":2083,"kind":"use","source":36},{"line":322,"start":18765,"end":18781,"expression":"3.3079\\,10^{-10}","context":2083,"kind":"use","source":36},{"line":322,"start":18784,"end":18799,"expression":"1.3795\\,10^{-9}","context":2083,"kind":"use","source":36},{"line":323,"start":18808,"end":18824,"expression":"3.1077\\,10^{-11}","context":2084,"kind":"use","source":36},{"line":323,"start":18827,"end":18843,"expression":"1.5438\\,10^{-10}","context":2084,"kind":"use","source":36},{"line":323,"start":18846,"end":18862,"expression":"2.0248\\,10^{-11}","context":2084,"kind":"use","source":36},{"line":323,"start":18865,"end":18881,"expression":"8.4854\\,10^{-11}","context":2084,"kind":"use","source":36},{"line":324,"start":18890,"end":18906,"expression":"1.9296\\,10^{-12}","context":2085,"kind":"use","source":36},{"line":324,"start":18909,"end":18925,"expression":"1.0485\\,10^{-11}","context":2085,"kind":"use","source":36},{"line":324,"start":18928,"end":18944,"expression":"1.2559\\,10^{-12}","context":2085,"kind":"use","source":36},{"line":324,"start":18947,"end":18963,"expression":"5.3844\\,10^{-12}","context":2085,"kind":"use","source":36},{"line":328,"start":19049,"end":19056,"expression":"3.99856","context":2086,"kind":"use","source":36},{"line":328,"start":19060,"end":19067,"expression":"3.96010","context":2086,"kind":"use","source":36},{"line":328,"start":19071,"end":19078,"expression":"4.02393","context":2086,"kind":"use","source":36},{"line":328,"start":19086,"end":19093,"expression":"4.00788","context":2086,"kind":"use","source":36},{"line":328,"start":19167,"end":19174,"expression":"3.70410","context":2086,"kind":"use","source":36},{"line":330,"start":19412,"end":19417,"expression":"N=512","context":2087,"kind":"use","source":36},{"line":330,"start":19470,"end":19484,"expression":"4.44\\,10^{-16}","context":2087,"kind":"use","source":36},{"line":330,"start":19554,"end":19568,"expression":"2.22\\,10^{-15}","context":2087,"kind":"use","source":36},{"line":330,"start":19609,"end":19623,"expression":"1.11\\,10^{-16}","context":2087,"kind":"use","source":36},{"line":332,"start":19905,"end":19910,"expression":"N=128","context":2088,"kind":"use","source":36},{"line":332,"start":19936,"end":19952,"expression":"2.25125\\,10^{-4}","context":2088,"kind":"use","source":36},{"line":332,"start":19981,"end":19997,"expression":"1.27688\\,10^{-3}","context":2088,"kind":"use","source":36},{"line":332,"start":20026,"end":20042,"expression":"1.25735\\,10^{-2}","context":2088,"kind":"use","source":36},{"line":334,"start":20251,"end":20258,"expression":"10^{-8}","context":2089,"kind":"use","source":36},{"line":334,"start":20302,"end":20305,"expression":"L^2","context":2089,"kind":"use","source":36},{"line":334,"start":20330,"end":20340,"expression":"1.03345462","context":2089,"kind":"use","source":36},{"line":334,"start":20347,"end":20357,"expression":"1.03348106","context":2089,"kind":"use","source":36},{"line":334,"start":20363,"end":20381,"expression":"N=128,256,512,1024","context":2089,"kind":"use","source":36},{"line":334,"start":20399,"end":20415,"expression":"2.64438\\,10^{-5}","context":2089,"kind":"use","source":36},{"line":336,"start":20618,"end":20621,"expression":"1.6","context":2090,"kind":"use","source":36},{"line":336,"start":20668,"end":20669,"expression":"F","context":2090,"kind":"use","source":36},{"line":336,"start":20725,"end":20730,"expression":"N=256","context":2090,"kind":"use","source":36},{"line":336,"start":20739,"end":20782,"expression":"|r_{+,h}-r_+^{\\rm exact}|=1.89262\\,10^{-11}","context":2090,"kind":"use","source":36},{"line":343,"start":21447,"end":21452,"expression":"G=c=1","context":2091,"kind":"use","source":36},{"line":343,"start":21478,"end":21514,"expression":"\\varphi=\\sqrt{4\\pi}\\,\\phi_{\\rm phys}","context":2091,"kind":"use","source":36},{"line":343,"start":21547,"end":21566,"expression":"(\\varphi,P_\\varphi)","context":2091,"kind":"use","source":36},{"line":344,"start":21658,"end":21822,"expression":"\n \\mathcal H_{\\rm matt}=\\frac{P_\\varphi^2}{2\\Lambda R^2}\n                         +\\frac{R^2\\varphi'^2}{2\\Lambda},\n \\qquad \\mathcal D_{\\rm matt}=P_\\varphi\\varphi'.\n","context":2091,"kind":"use","source":36},{"line":349,"start":21899,"end":21902,"expression":"Q=0","context":2092,"kind":"use","source":36},{"line":351,"start":21914,"end":21917,"expression":"R=r","context":2093,"kind":"use","source":36},{"line":351,"start":21924,"end":21933,"expression":"\\Lambda=1","context":2093,"kind":"use","source":36},{"line":352,"start":21960,"end":22056,"expression":"\n N=\\sigma>0,\\quad N^r=\\sigma v,\\quad\n P_\\Lambda=rv,\\quad P_\\varphi=r^2\\Pi,\\quad \\Phi=\\varphi'.\n","context":2093,"kind":"use","source":36},{"line":356,"start":22077,"end":22080,"expression":"\\Pi","context":2093,"kind":"use","source":36},{"line":356,"start":22149,"end":22154,"expression":"\\Pi_h","context":2093,"kind":"use","source":36},{"line":357,"start":22180,"end":22226,"expression":"\n m=\\frac r2v^2,\\qquad v=\\sqrt{\\frac{2m}{r}}.\n","context":2093,"kind":"use","source":36},{"line":361,"start":22281,"end":22365,"expression":"\n ds^2=-\\sigma^2dt^2+(dr+\\sigma vdt)^2+r^2d\\Omega^2.\n\\label{c:strong:scalar-metric}\n","context":2093,"kind":"use","source":36},{"line":365,"start":22437,"end":22440,"expression":"r>0","context":2094,"kind":"use","source":36},{"line":365,"start":22444,"end":22452,"expression":"\\sigma>0","context":2095,"kind":"use","source":36},{"line":365,"start":22460,"end":22463,"expression":"v>0","context":2096,"kind":"use","source":36},{"line":367,"start":22549,"end":22552,"expression":"P_R","context":2097,"kind":"use","source":36},{"line":368,"start":22585,"end":22617,"expression":"\n P_R=P_\\Lambda'-P_\\varphi\\Phi.\n","context":2097,"kind":"use","source":36},{"line":371,"start":22639,"end":22683,"expression":"m'=P_\\Lambda P_\\Lambda'/r-P_\\Lambda^2/(2r^2)","context":2097,"kind":"use","source":36},{"line":372,"start":22764,"end":22863,"expression":"\n m'=r^2\\mathcal E,\\qquad\n \\mathcal E=\\frac12(\\Pi^2+\\Phi^2)+v\\Pi\\Phi.\n\\label{c:strong:scalar-mass}\n","context":2097,"kind":"use","source":36},{"line":377,"start":22895,"end":22904,"expression":"\\Lambda=1","context":2097,"kind":"use","source":36},{"line":378,"start":22928,"end":23014,"expression":"\n 0=\\sigma\\left(\\frac vr-\\frac{P_R}r\\right)+(\\sigma v)'\n   =\\sigma r\\Pi\\Phi+\\sigma'v,\n","context":2098,"kind":"use","source":36},{"line":383,"start":23051,"end":23118,"expression":"\n (\\log\\sigma)'=-\\frac{r\\Pi\\Phi}{v}.\n\\label{c:strong:scalar-lapse}\n","context":2099,"kind":"use","source":36},{"line":388,"start":23181,"end":23345,"expression":"\n \\dot\\varphi&=\\sigma(\\Pi+v\\Phi),\\\\\n \\dot\\Phi&=\\partial_r[\\sigma(\\Pi+v\\Phi)],\\\\\n \\dot\\Pi&=r^{-2}\\partial_r[r^2\\sigma(\\Phi+v\\Pi)].\n\\label{c:strong:scalar-evolution}\n","context":2100,"kind":"use","source":36},{"line":394,"start":23361,"end":23374,"expression":"\\Phi=\\varphi'","context":2101,"kind":"use","source":36},{"line":396,"start":23498,"end":23501,"expression":"v=0","context":2102,"kind":"use","source":36},{"line":396,"start":23581,"end":23584,"expression":"\\Pi","context":2102,"kind":"use","source":36},{"line":396,"start":23591,"end":23595,"expression":"\\Phi","context":2102,"kind":"use","source":36},{"line":396,"start":23628,"end":23638,"expression":"r\\Pi\\Phi/v","context":2102,"kind":"use","source":36},{"line":396,"start":23893,"end":23894,"expression":"v","context":2102,"kind":"use","source":36},{"line":399,"start":24027,"end":24038,"expression":"A=\\Pi+v\\Phi","context":2103,"kind":"use","source":36},{"line":399,"start":24045,"end":24056,"expression":"B=\\Phi+v\\Pi","context":2104,"kind":"use","source":36},{"line":400,"start":24123,"end":24204,"expression":"\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n","context":2105,"kind":"use","source":36},{"line":404,"start":24264,"end":24273,"expression":"P_\\Lambda","context":2105,"kind":"use","source":36},{"line":404,"start":24340,"end":24380,"expression":"\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'","context":2105,"kind":"use","source":36},{"line":404,"start":24455,"end":24481,"expression":"\\sigma r^2(\\Pi^2+\\Phi^2)/2","context":2106,"kind":"use","source":36},{"line":404,"start":24488,"end":24510,"expression":"\\dot m=v\\dot P_\\Lambda","context":2107,"kind":"use","source":36},{"line":405,"start":24547,"end":24619,"expression":"\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n","context":2108,"kind":"use","source":36},{"line":409,"start":24711,"end":24759,"expression":"\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]","context":2109,"kind":"use","source":36},{"line":413,"start":24969,"end":25062,"expression":"\n \\partial_t(r^2\\mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n","context":2110,"kind":"use","source":36},{"line":417,"start":25083,"end":25113,"expression":"\\partial_t(m'-r^2\\mathcal E)=0","context":2111,"kind":"use","source":36},{"line":422,"start":25310,"end":25330,"expression":"\\dot\\Phi=(\\sigma A)'","context":2112,"kind":"use","source":36},{"line":422,"start":25337,"end":25368,"expression":"\\dot\\Pi=(\\sigma B)'+2\\sigma B/r","context":2113,"kind":"use","source":36},{"line":423,"start":25386,"end":25396,"expression":"\\mathcal E","context":2114,"kind":"use","source":36},{"line":423,"start":25463,"end":25467,"expression":"\\Pi'","context":2115,"kind":"use","source":36},{"line":423,"start":25471,"end":25476,"expression":"\\Phi'","context":2116,"kind":"use","source":36},{"line":423,"start":25484,"end":25486,"expression":"v'","context":2117,"kind":"use","source":36},{"line":424,"start":25520,"end":25616,"expression":"\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n","context":2118,"kind":"use","source":36},{"line":428,"start":25638,"end":25646,"expression":"m=rv^2/2","context":2119,"kind":"use","source":36},{"line":428,"start":25676,"end":25695,"expression":"\\dot v=r\\sigma AB/v","context":2120,"kind":"use","source":36},{"line":429,"start":25742,"end":25798,"expression":"\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n","context":2121,"kind":"use","source":36},{"line":435,"start":25988,"end":25997,"expression":"[r_i,r_o]","context":2122,"kind":"use","source":36},{"line":436,"start":26037,"end":26130,"expression":"\n \\frac{d}{dt}\\bigl(m(t,r_o)-m(t,r_i)\\bigr)\n =r_o^2\\mathcal J(t,r_o)-r_i^2\\mathcal J(t,r_i).\n","context":2122,"kind":"use","source":36},{"line":443,"start":26481,"end":26493,"expression":"w_+=\\Pi+\\Phi","context":2123,"kind":"use","source":36},{"line":443,"start":26500,"end":26512,"expression":"w_-=\\Pi-\\Phi","context":2123,"kind":"use","source":36},{"line":444,"start":26548,"end":26741,"expression":"\n \\dot w_+={}&\\sigma(v+1)w_+'\n +[\\sigma v'+\\sigma'(v+1)]w_+\n +\\frac\\sigma r[(v+1)w_++(v-1)w_-],\\\\\n \\dot w_-={}&\\sigma(v-1)w_-'\n +[\\sigma v'+\\sigma'(v-1)]w_-\n +\\frac\\sigma r[(v-1)w_-+(v+1)w_+].\n","context":2124,"kind":"use","source":36},{"line":452,"start":26785,"end":26811,"expression":"\\dot w+c\\,w'=\\text{source}","context":2125,"kind":"use","source":36},{"line":453,"start":26858,"end":26934,"expression":"\n c_+=-\\sigma(v+1),\\qquad c_-=\\sigma(1-v).\n\\label{c:strong:characteristics}\n","context":2125,"kind":"use","source":36},{"line":457,"start":27035,"end":27038,"expression":"v<1","context":2125,"kind":"use","source":36},{"line":457,"start":27056,"end":27059,"expression":"v=1","context":2126,"kind":"use","source":36},{"line":457,"start":27079,"end":27082,"expression":"v>1","context":2127,"kind":"use","source":36},{"line":460,"start":27153,"end":27284,"expression":"\n \\mathcal E&=\\frac14[(1+v)w_+^2+(1-v)w_-^2],\\\\\n \\mathcal J&=\\frac\\sigma4[(1+v)^2w_+^2-(1-v)^2w_-^2].\n\\label{c:strong:null-energy}\n","context":2128,"kind":"use","source":36},{"line":465,"start":27301,"end":27309,"expression":"0\\le v<1","context":2129,"kind":"use","source":36},{"line":465,"start":27313,"end":27346,"expression":"\\mathcal E\\ge(1-v)(w_+^2+w_-^2)/4","context":2129,"kind":"use","source":36},{"line":465,"start":27397,"end":27400,"expression":"v>1","context":2129,"kind":"use","source":36},{"line":465,"start":27531,"end":27547,"expression":"(\\Pi^2+\\Phi^2)/2","context":2129,"kind":"use","source":36},{"line":465,"start":27597,"end":27607,"expression":"\\mathcal E","context":2129,"kind":"use","source":36},{"line":467,"start":27933,"end":27936,"expression":"v=1","context":2130,"kind":"use","source":36},{"line":473,"start":28205,"end":28258,"expression":"\n \\ell_\\pm=\\sigma^{-1}\\partial_t+(-v\\pm1)\\partial_r.\n","context":2131,"kind":"use","source":36},{"line":476,"start":28301,"end":28309,"expression":"4\\pi r^2","context":2131,"kind":"use","source":36},{"line":476,"start":28333,"end":28337,"expression":"\\ell","context":2131,"kind":"use","source":36},{"line":476,"start":28343,"end":28353,"expression":"2\\ell(r)/r","context":2131,"kind":"use","source":36},{"line":477,"start":28385,"end":28469,"expression":"\n \\theta_+=\\frac2r(1-v),\\qquad\n \\theta_-=-\\frac2r(1+v).\n\\label{c:strong:expansions}\n","context":2131,"kind":"use","source":36},{"line":482,"start":28509,"end":28512,"expression":"v=1","context":2131,"kind":"use","source":36},{"line":482,"start":28557,"end":28560,"expression":"v>1","context":2132,"kind":"use","source":36},{"line":482,"start":28625,"end":28629,"expression":"2m/r","context":2133,"kind":"use","source":36},{"line":485,"start":29045,"end":29061,"expression":"m(r_{\\rm exc})=0","context":2134,"kind":"use","source":36},{"line":485,"start":29129,"end":29147,"expression":"\\sigma(r_{\\max})=1","context":2134,"kind":"use","source":36},{"line":488,"start":29450,"end":29546,"expression":"\n (D_+u)_j=\\frac{-3u_j+4u_{j+1}-u_{j+2}}{2h},\\qquad\n (D_-u)_j=\\frac{3u_j-4u_{j-1}+u_{j-2}}{2h}.\n","context":2135,"kind":"use","source":36},{"line":492,"start":29586,"end":29589,"expression":"D_+","context":2135,"kind":"use","source":36},{"line":492,"start":29622,"end":29625,"expression":"D_-","context":2135,"kind":"use","source":36},{"line":492,"start":29660,"end":29663,"expression":"D_+","context":2135,"kind":"use","source":36},{"line":492,"start":29793,"end":29795,"expression":"v'","context":2136,"kind":"use","source":36},{"line":496,"start":30418,"end":30419,"expression":"v","context":2137,"kind":"use","source":36},{"line":496,"start":30426,"end":30445,"expression":"\\sqrt{\\max(2m/r,0)}","context":2137,"kind":"use","source":36},{"line":496,"start":30491,"end":30503,"expression":"v\\le10^{-13}","context":2137,"kind":"use","source":36},{"line":496,"start":30729,"end":30732,"expression":"v>0","context":2138,"kind":"use","source":36},{"line":496,"start":30810,"end":30822,"expression":"h\\downarrow0","context":2139,"kind":"use","source":36},{"line":499,"start":30896,"end":30928,"expression":"[r_{\\rm exc},r_{\\max}]=[0.05,30]","context":2140,"kind":"use","source":36},{"line":499,"start":31014,"end":31032,"expression":"w_-(r_{\\rm exc})=0","context":2140,"kind":"use","source":36},{"line":499,"start":31039,"end":31054,"expression":"w_+(r_{\\max})=0","context":2140,"kind":"use","source":36},{"line":502,"start":31316,"end":31437,"expression":"\n \\varphi(0,r)=A e^{-(r-10)^2},\\qquad\n \\Phi(0,r)=\\partial_r\\varphi(0,r),\\qquad\n \\Pi(0,r)=\\Phi(0,r)+\\frac{\\varphi(0,r)}r.\n","context":2141,"kind":"use","source":36},{"line":507,"start":31511,"end":31524,"expression":"\\chi=r\\varphi","context":2141,"kind":"use","source":36},{"line":509,"start":31751,"end":31757,"expression":"A=0.10","context":2142,"kind":"use","source":36},{"line":509,"start":31789,"end":31795,"expression":"A=0.05","context":2142,"kind":"use","source":36},{"line":509,"start":31828,"end":31833,"expression":"r=3.1","context":2142,"kind":"use","source":36},{"line":512,"start":32223,"end":32227,"expression":"2m/r","context":2143,"kind":"use","source":36},{"line":516,"start":32407,"end":32410,"expression":"N_r","context":2143,"kind":"use","source":36},{"line":516,"start":32413,"end":32414,"expression":"h","context":2143,"kind":"use","source":36},{"line":516,"start":32417,"end":32420,"expression":"t_H","context":2143,"kind":"use","source":36},{"line":516,"start":32423,"end":32426,"expression":"r_H","context":2143,"kind":"use","source":36},{"line":516,"start":32429,"end":32441,"expression":"\\Delta M/M_0","context":2143,"kind":"use","source":36},{"line":518,"start":32489,"end":32505,"expression":"-1.1205\\,10^{-2}","context":2144,"kind":"use","source":36},{"line":519,"start":32544,"end":32560,"expression":"-1.6427\\,10^{-2}","context":2145,"kind":"use","source":36},{"line":520,"start":32600,"end":32616,"expression":"-1.0696\\,10^{-2}","context":2146,"kind":"use","source":36},{"line":521,"start":32656,"end":32672,"expression":"-7.6261\\,10^{-3}","context":2147,"kind":"use","source":36},{"line":522,"start":32712,"end":32728,"expression":"-4.6604\\,10^{-3}","context":2148,"kind":"use","source":36},{"line":523,"start":32768,"end":32784,"expression":"-3.3322\\,10^{-3}","context":2149,"kind":"use","source":36},{"line":524,"start":32824,"end":32840,"expression":"-2.6241\\,10^{-3}","context":2150,"kind":"use","source":36},{"line":529,"start":33027,"end":33144,"expression":"\n t_H(h)&\\simeq4.6893522370+33.88577338\\,h^{2.173853282},\\\\\n r_H(h)&\\simeq3.1204965744-65.59310603\\,h^{2.084394965}.\n","context":2151,"kind":"use","source":36},{"line":535,"start":33367,"end":33375,"expression":"N_r=2400","context":2152,"kind":"use","source":36},{"line":536,"start":33421,"end":33498,"expression":"\n t_H=4.69182094580,\\quad r_H=3.11341154797,\n \\quad m_H=r_H/2=1.55670577399.\n","context":2152,"kind":"use","source":36},{"line":540,"start":33527,"end":33536,"expression":"m_H=r_H/2","context":2152,"kind":"use","source":36},{"line":542,"start":33714,"end":33729,"expression":"t=4.69206969694","context":2153,"kind":"use","source":36},{"line":542,"start":33770,"end":33773,"expression":"t_H","context":2153,"kind":"use","source":36},{"line":542,"start":33806,"end":33819,"expression":"1.72538868709","context":2153,"kind":"use","source":36},{"line":542,"start":33856,"end":33869,"expression":"1.72992821318","context":2153,"kind":"use","source":36},{"line":542,"start":33896,"end":33915,"expression":"-2.6241124\\,10^{-3}","context":2153,"kind":"use","source":36},{"line":542,"start":33961,"end":33975,"expression":"0.999829382946","context":2153,"kind":"use","source":36},{"line":542,"start":34005,"end":34018,"expression":"1.02582462096","context":2153,"kind":"use","source":36},{"line":542,"start":34066,"end":34079,"expression":"7.35797274672","context":2154,"kind":"use","source":36},{"line":545,"start":34285,"end":34315,"expression":"\n \\mathcal R=2(\\Phi^2-\\Pi^2).\n","context":2155,"kind":"use","source":36},{"line":550,"start":34718,"end":34721,"expression":"v=1","context":2156,"kind":"use","source":36},{"line":553,"start":34907,"end":34915,"expression":"N_r=1200","context":2157,"kind":"use","source":36},{"line":553,"start":34944,"end":34947,"expression":"t=5","context":2157,"kind":"use","source":36},{"line":553,"start":34976,"end":34990,"expression":"0.146722872367","context":2157,"kind":"use","source":36},{"line":553,"start":35026,"end":35045,"expression":"-4.5106042\\,10^{-3}","context":2157,"kind":"use","source":36},{"line":555,"start":35273,"end":35276,"expression":"t=3","context":2158,"kind":"use","source":36},{"line":555,"start":35284,"end":35292,"expression":"N_r=1200","context":2158,"kind":"use","source":36},{"line":561,"start":35434,"end":35453,"expression":"2.27319745\\,10^{-6}","context":2159,"kind":"use","source":36},{"line":562,"start":35475,"end":35494,"expression":"2.80774843\\,10^{-6}","context":2160,"kind":"use","source":36},{"line":563,"start":35541,"end":35560,"expression":"1.26971871\\,10^{-3}","context":2161,"kind":"use","source":36},{"line":567,"start":35833,"end":35846,"expression":"0.00230311127","context":2162,"kind":"use","source":36},{"line":578,"start":37869,"end":37872,"expression":"v=0","context":2163,"kind":"use","source":36},{"line":580,"start":38273,"end":38276,"expression":"3+1","context":2164,"kind":"use","source":36}],"entries":[{"id":"symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","occurrences":[2537,2542,1970,2216,2099,2502,2504,2896,3412,78,83,188,189,213,218,221,226,229,231,234,235,239,240,241,242,262,268,414,444,500,501,529,625,629,635,649,665,667,669,671,692,736,739,770,777,778,798,799,805,806,807,809,810,812,814,815,816,817,818,820,828,841,857,865,870,873,877,879,881,889,890,892,898,899,904,908,919,923,925,926,1017,1021,1062,1064,1070,1083,1166,1170,1172,1185,1433,1435,1436,1450,1467,1469,1548,1556,1588,1607,1646,1654,1659,1660,1726,1885,1907,1934,1996,1998,2005,2008,2014,2030,2071,2072,2073,2078,2079,2082,2083,2084,2085,2086,2087,2089,2101,2109,2155,2160,2211,2255,2258,2259,2264,2268,2269,2270,2271,2274,2278,2309,2310,2369,2429,2437,2438,2508,2511,2606,2624,2626,2657,2662,2718,2719,2725,2726,2730,2731,2734,2738,2739,2741,2742,2754,2755,2806,2807,2812,2813,2816,2870,3487,3488,3833,3952,3954,3957,3958]},{"id":"symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","occurrences":[2537,2538,2541,2542,2543,2544,2545,2549,2551,2555,2556,1418,1970,1972,1973,2216,2218,2221,2562,2564,3854,155,159,324,325,326,327,329,330,333,407,722,723,725,727,728,742,743,744,745,746,763,1047,1048,1049,1051,1052,1053,1225,1230,1231,1378,1379,1382,1805,1811,2099,2100,2286,2289,2293,2502,2503,2504,2824,2825,2829,2830,2831,2835,2892,2893,2896,2897,2899,2900,2904,2905,2906,3166,3169,3170,3173,3174,3412,3841,3844,5,6,7,16,20,21,31,45,47,49,51,53,54,72,76,77,78,81,82,83,84,85,86,87,95,100,107,110,111,112,113,114,117,123,129,133,134,136,140,141,143,150,152,153,162,165,170,173,174,175,178,184,186,187,188,189,190,192,195,196,197,201,202,206,207,209,211,212,213,214,215,218,221,226,227,229,231,235,238,239,240,241,242,245,249,257,260,262,264,267,268,270,274,275,276,283,286,290,294,295,296,297,299,300,301,307,309,310,311,312,314,315,321,322,323,336,339,341,346,349,362,368,377,378,379,382,384,385,388,391,392,393,394,395,397,403,414,422,427,429,431,438,443,444,447,448,453,457,459,460,461,463,464,467,468,469,471,472,474,476,479,488,494,497,499,501,503,519,526,528,529,530,531,536,542,547,552,556,558,560,563,566,567,568,571,574,575,577,593,594,597,603,605,606,607,609,611,615,616,617,622,623,625,627,629,635,639,644,646,649,651,652,655,656,663,664,665,667,668,669,670,671,672,674,675,677,679,684,685,687,690,691,692,703,706,715,716,720,732,736,737,738,739,748,751,753,755,770,775,777,778,779,780,783,791,794,797,798,799,805,806,809,810,812,814,815,816,817,818,819,820,821,822,824,825,827,828,829,830,841,843,844,845,847,848,850,854,855,858,859,861,863,865,870,873,876,877,878,879,880,881,882,883,884,887,889,890,892,897,898,902,904,908,911,913,919,922,923,925,926,928,934,949,952,954,957,961,962,966,969,970,971,972,976,987,989,997,999,1000,1001,1002,1003,1010,1012,1013,1014,1015,1016,1019,1021,1023,1024,1025,1026,1029,1033,1034,1037,1038,1039,1040,1059,1062,1063,1064,1070,1071,1072,1073,1074,1076,1080,1081,1082,1083,1084,1085,1088,1090,1095,1096,1097,1098,1099,1104,1105,1106,1107,1111,1114,1118,1120,1126,1127,1128,1130,1131,1133,1134,1135,1142,1143,1144,1145,1146,1147,1148,1152,1153,1154,1155,1158,1160,1164,1166,1167,1169,1170,1171,1172,1174,1183,1185,1190,1191,1192,1198,1205,1206,1208,1210,1216,1218,1219,1221,1222,1223,1224,1232,1234,1235,1236,1237,1238,1240,1241,1243,1245,1249,1251,1254,1255,1260,1261,1262,1263,1266,1267,1270,1271,1272,1274,1276,1280,1282,1283,1284,1285,1286,1293,1298,1300,1303,1307,1308,1312,1313,1316,1318,1322,1323,1324,1326,1327,1329,1330,1331,1332,1334,1336,1339,1340,1349,1350,1351,1355,1356,1357,1362,1366,1374,1377,1383,1391,1392,1393,1413,1414,1415,1417,1420,1421,1426,1432,1436,1437,1440,1442,1443,1445,1446,1455,1458,1460,1466,1467,1468,1469,1474,1475,1477,1479,1481,1483,1487,1489,1491,1493,1494,1496,1498,1500,1502,1508,1515,1516,1518,1520,1521,1522,1525,1526,1530,1531,1532,1534,1542,1543,1544,1545,1546,1548,1549,1554,1555,1556,1557,1558,1561,1562,1564,1565,1566,1570,1571,1572,1577,1580,1583,1584,1588,1591,1592,1601,1602,1603,1606,1607,1608,1609,1610,1611,1612,1613,1614,1615,1616,1617,1619,1620,1621,1622,1624,1625,1626,1627,1628,1629,1631,1633,1634,1638,1642,1643,1645,1646,1647,1651,1654,1655,1656,1657,1659,1660,1661,1665,1668,1670,1671,1676,1678,1680,1681,1684,1691,1694,1696,1701,1702,1705,1706,1710,1712,1714,1719,1722,1725,1726,1731,1742,1743,1749,1750,1751,1752,1757,1758,1761,1764,1766,1768,1772,1773,1775,1776,1777,1779,1786,1788,1793,1795,1798,1799,1800,1802,1807,1808,1809,1810,1812,1825,1826,1842,1843,1844,1845,1846,1847,1848,1849,1850,1854,1855,1856,1857,1858,1859,1860,1861,1862,1863,1865,1866,1867,1868,1874,1875,1880,1881,1882,1883,1884,1885,1886,1889,1903,1904,1907,1912,1913,1914,1915,1916,1920,1925,1929,1930,1934,1941,1947,1954,1955,1959,1960,1961,1962,1963,1974,1976,1977,1980,1982,1983,1984,1985,1986,1987,1988,1991,1996,1998,1999,2001,2003,2004,2005,2006,2007,2008,2014,2016,2017,2019,2021,2022,2024,2025,2028,2029,2030,2031,2032,2033,2035,2036,2037,2038,2039,2044,2045,2048,2049,2050,2051,2052,2053,2055,2056,2058,2059,2062,2063,2064,2071,2072,2073,2076,2077,2078,2079,2082,2083,2084,2086,2087,2089,2091,2092,2093,2095,2096,2097,2098,2102,2104,2105,2106,2107,2108,2109,2110,2112,2113,2114,2115,2116,2119,2120,2122,2124,2131,2132,2135,2136,2139,2142,2143,2144,2145,2155,2156,2157,2158,2160,2165,2167,2168,2169,2170,2171,2174,2176,2181,2184,2185,2186,2188,2189,2191,2192,2195,2196,2197,2198,2199,2200,2206,2208,2209,2211,2212,2222,2223,2229,2235,2236,2237,2238,2239,2240,2241,2243,2245,2246,2248,2250,2251,2255,2256,2257,2258,2259,2264,2265,2266,2268,2269,2270,2271,2274,2275,2276,2278,2279,2280,2281,2282,2300,2301,2302,2303,2304,2306,2310,2312,2316,2317,2322,2323,2325,2329,2331,2335,2336,2337,2338,2341,2343,2344,2345,2348,2349,2350,2353,2360,2361,2366,2368,2369,2375,2384,2388,2389,2392,2395,2398,2399,2400,2401,2406,2407,2421,2433,2437,2438,2442,2443,2448,2449,2450,2453,2457,2458,2459,2460,2461,2462,2472,2475,2484,2485,2487,2488,2490,2491,2492,2493,2496,2497,2499,2508,2511,2513,2514,2516,2517,2521,2522,2523,2524,2531,2532,2567,2568,2569,2573,2575,2576,2577,2581,2588,2590,2593,2594,2595,2596,2600,2606,2607,2608,2609,2612,2617,2618,2619,2622,2624,2625,2638,2644,2647,2648,2649,2650,2652,2653,2654,2657,2658,2659,2660,2661,2662,2665,2666,2669,2670,2671,2678,2682,2686,2688,2693,2694,2697,2698,2701,2706,2707,2708,2709,2718,2719,2723,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2738,2742,2745,2746,2750,2751,2752,2753,2754,2755,2756,2772,2778,2780,2783,2785,2790,2797,2798,2799,2800,2802,2806,2810,2812,2813,2815,2836,2837,2840,2845,2849,2851,2852,2858,2859,2860,2885,2888,2910,2911,2912,2913,2916,2918,2920,2921,2923,2924,2927,2929,2931,2935,2938,2940,2943,2951,2952,2956,2963,2964,2972,2974,2978,2979,2981,2982,2983,2984,2985,2993,2994,2995,2996,2997,2998,3003,3006,3011,3012,3014,3015,3019,3024,3025,3026,3029,3038,3039,3041,3043,3046,3051,3056,3057,3058,3063,3065,3067,3070,3074,3075,3076,3077,3079,3080,3095,3097,3100,3103,3108,3109,3110,3111,3112,3113,3115,3117,3118,3124,3126,3135,3136,3137,3151,3153,3154,3155,3156,3157,3159,3160,3161,3162,3163,3175,3178,3179,3180,3183,3185,3186,3188,3189,3190,3191,3192,3194,3195,3196,3197,3198,3200,3201,3202,3204,3205,3209,3210,3211,3215,3216,3219,3220,3227,3230,3237,3238,3239,3240,3243,3245,3248,3251,3252,3253,3258,3260,3261,3263,3265,3266,3272,3273,3274,3275,3278,3279,3282,3283,3284,3286,3287,3290,3291,3292,3293,3294,3300,3301,3304,3305,3309,3310,3311,3312,3313,3314,3319,3321,3322,3323,3332,3333,3334,3335,3337,3338,3339,3340,3341,3342,3344,3345,3346,3347,3348,3349,3350,3352,3353,3355,3358,3359,3360,3361,3362,3363,3366,3370,3371,3372,3373,3374,3379,3380,3381,3383,3384,3385,3386,3387,3388,3390,3391,3392,3393,3395,3396,3397,3398,3399,3403,3405,3406,3407,3408,3413,3422,3424,3425,3428,3429,3430,3432,3434,3436,3437,3438,3439,3442,3444,3445,3449,3450,3451,3455,3456,3459,3460,3464,3469,3470,3481,3482,3484,3485,3486,3487,3488,3489,3490,3492,3501,3502,3503,3505,3506,3507,3511,3517,3520,3521,3522,3525,3526,3527,3528,3531,3541,3543,3544,3545,3546,3550,3553,3554,3555,3556,3557,3558,3563,3568,3569,3572,3573,3574,3576,3581,3582,3583,3584,3585,3586,3587,3588,3589,3590,3593,3597,3598,3600,3603,3604,3605,3607,3608,3609,3610,3613,3620,3621,3623,3625,3626,3627,3628,3629,3631,3634,3637,3639,3641,3643,3648,3650,3651,3652,3653,3654,3655,3656,3658,3659,3661,3662,3663,3664,3665,3667,3668,3669,3670,3671,3673,3674,3675,3678,3679,3680,3682,3683,3686,3689,3690,3691,3692,3693,3695,3697,3699,3716,3717,3718,3723,3724,3725,3729,3730,3731,3733,3734,3735,3736,3737,3738,3739,3740,3742,3745,3747,3748,3752,3753,3754,3761,3762,3765,3766,3769,3771,3773,3774,3775,3776,3777,3778,3779,3781,3784,3785,3786,3789,3791,3793,3795,3796,3797,3798,3799,3805,3807,3809,3810,3811,3812,3813,3814,3815,3816,3817,3818,3819,3821,3823,3831,3832,3833,3834,3836,3837,3850,3851,3852,3853,3856,3858,3859,3860,3867,3868,3870,3872,3873,3874,3875,3881,3882,3883,3884,3885,3886,3887,3888,3889,3890,3892,3893,3894,3895,3897,3899,3901,3903,3904,3906,3907,3909,3912,3913,3914,3916,3917,3918,3919,3920,3921,3922,3926,3927,3931,3934,3940,3943,3947,3948,3949,3955,3962,3963,3964,3969,3974,3976,3981,3984,3985,3990,3991,3992,4023,4027,4035,4039,4040,4041,4042,4044,4045,4046,4047,4048,4051,4052,4057,4058,4059,4060,4061,4062,4063,4064,4065,4070,4071,4072,4076,4077,4078,4079,4080,4081,4082,4087,4088,4089,4090,4092,4093,4094,4095,4096,4097,4099,4101,4107,4108,4112,4113,4116,4117,4118,4128,4129,4130,4131,4132,4133,4134,4135,4150,4151,4152,4153,4161,4162,4163,4164,4167,4168,4173]},{"id":"symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","occurrences":[2537,2538,2539,2545,2546,2548,2549,1969,1970,1972,1973,2216,2218,2219,2220,2221,157,159,1379,2100,2283,2286,2289,2291,2292,12,47,51,63,65,69,80,81,82,83,84,86,87,88,89,91,93,95,99,101,102,104,105,106,173,184,187,188,192,197,206,207,312,314,377,383,384,385,388,389,390,391,392,393,397,398,401,403,468,635,636,968,971,974,975,1206,1210,1211,1251,1252,1254,1316,1317,1318,1319,1349,1350,1351,1352,1353,1356,1357,1359,1360,1362,1366,1374,1375,1377,1386,1388,1389,1390,1393,1395,1398,1399,1403,1405,1412,1415,1417,1424,1426,1431,1432,1437,1439,1440,1451,1452,1455,1457,1458,1459,1460,1461,1462,1465,1466,1470,1471,1472,1474,1476,1481,1483,1490,1491,1492,1493,1506,1507,1508,1529,1530,1531,1532,1567,1568,1670,1682,1684,1690,1691,1694,1695,1696,1706,1707,1763,1766,1780,1781,1812,1855,1900,1902,1905,1915,1925,1926,1929,1930,1932,1933,1934,1935,1946,1947,1948,1950,1952,1954,1955,1956,1959,1960,1962,1963,1976,1977,1979,1980,1991,1992,1994,1996,1997,2001,2003,2005,2011,2016,2019,2021,2022,2024,2025,2032,2045,2048,2051,2052,2073,2075,2076,2094,2105,2106,2107,2108,2110,2112,2113,2114,2115,2120,2121,2129,2131,2132,2135,2136,2143,2145,2151,2153,2154,2155,2156,2158,2159,2160,2161,2163,2174,2180,2181,2184,2185,2186,2188,2189,2194,2195,2196,2197,2198,2199,2200,2202,2205,2206,2207,2208,2213,2222,2223,2224,2225,2226,2227,2237,2238,2239,2241,2244,2245,2246,2250,2251,2253,2254,2255,2256,2257,2258,2263,2264,2265,2266,2268,2269,2275,2276,2277,2280,2331,2336,2337,2341,2342,2344,2345,2346,2347,2349,2350,2353,2354,2360,2361,2486,2487,2488,2489,2490,2491,2492,2496,2520,2521,2522,2524,2531,2532,2701,2706,2707,2709,2710,2711,2712,2713,2714,2749,2750,2792,2793,2812,2813,2908,2910,2911,2913,2916,2917,2918,2920,2923,2924,2926,2927,2928,2930,2931,2932,2933,2935,2936,2940,2943,3002,3003,3006,3008,3009,3032,3063,3064,3067,3082,3102,3117,3118,3133,3134,3135,3136,3137,3138,3221,3226,3230,3231,3232,3234,3237,3238,3239,3240,3243,3245,3248,3249,3269,3272,3275,3277,3297,3337,3340,3359,3360,3361,3362,3363,3366,3370,3371,3374,3379,3380,3381,3399,3402,3403,3405,3406,3408,3415,3473,3474,3511,3517,3518,3542,3544,3545,3546,3547,3572,3583,3613,3723,3724,3738,3739,3740,3741,3791,3793,3795,3799,3803,3805,3806,3807,3808,3812,3836,4040,4044]},{"id":"symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","occurrences":[2538,2540,2546,2547,2549,1970,2564,1226,1227,1228,2100,80,81,82,83,85,86,87,88,89,92,93,385,650,651,652,655,656,657,988,989,990,997,1357,1387,1388,1389,1390,1393,1431,1432,1435,1436,1437,1442,1443,1445,1446,1448,1449,1452,1455,1456,1458,1695,1696,1947,1953,1954,1955,1959,1961,1962,1979,1980,1991,1992,1996,1999,2003,2004,2005,2016,2017,2018,2019,2021,2022,2024,2025,2032,2041,2048,2110,2112,2113,2114,2115,2120,2121,2132,2153,2154,2155,2156,2157,2158,2159,2160,2161,2162,2165,2174,2181,2184,2186,2189,2195,2197,2198,2199,2200,2202,2205,2206,2207,2208,2213,2237,2238,2239,2240,2241,2242,2243,2246,2247,2248,2251,2268,2269,2280,2336,2337,2338,2341,2343,2344,2345,2346,2347,2353,2356,2357,2458,2460,2489,2490,2496,2521,2522,2524,2532,2560,2567,2572,2581,2583,2584,2586,2593,2749,2750,2792,2793,2813,2909,2910,2912,2916,2917,2923,2924,2926,2927,2974,3002,3003,3006,3237,3238,3243,3277,3359,3360,3362,3366,3371,3379,3381,3473,3474,3568,3569,3572,3609,3610,3611,3613,3616,3693,3695,3696,3697,3740,3741,3791,3962,3963,3964,3968,3969,3970,3972,3973,3982,3984,3987]},{"id":"symbol-598f62db34f4eb27","tex":"\\Dom","aliases":"Dom","display":null,"kind":"notation-table","occurrences":[2553,494,500,506,508,534,551,569,617,1282,1284,1352,1353,1426,1429,1435,1437,1449,1470,1474,1475,1490,1496]},{"id":"symbol-2dfbbad49009071f","tex":"\\dot p","aliases":"dot p dot","display":"\\dot p","kind":"notation-table","occurrences":[2549,2019,2022,2037,2049,2063,2064,2098,2112,2115,2158,2171,2176,2181,2239,2241,2243,2338,2345,2346]},{"id":"symbol-81596f18cb9ce128","tex":"\\dot p_b","aliases":"dot p_b dot","display":"\\dot p_b","kind":"notation-table","occurrences":[2549,2019,2022,2112,2115,2158,2181,2239,2241,2243,2338,2345,2346]},{"id":"symbol-381aa4b32306117e","tex":"\\dot\\psi","aliases":"dotψ dot psi","display":"\\dot\\psi","kind":"notation-table","occurrences":[2541,819,2014,2022,2039,2077,2087,2236,2335,2726,2727,2730,2738,2753,2754,2755]},{"id":"symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","occurrences":[2543,2552,2553,2556,325,326,1378,1924,2290,2291,2292,2293,2294,228,229,231,232,235,240,241,242,243,245,248,249,252,253,255,257,260,262,270,274,283,286,287,290,294,295,307,310,311,315,321,349,494,497,500,501,506,507,508,511,513,518,524,526,527,528,534,536,538,539,551,552,554,556,560,562,569,571,577,578,589,591,593,617,618,619,620,622,623,625,629,632,655,669,670,674,677,678,679,681,683,684,685,686,687,689,690,691,694,698,706,780,784,785,809,810,813,814,815,819,820,822,824,825,828,842,861,865,871,873,875,879,880,890,939,952,961,963,966,1003,1010,1015,1041,1063,1064,1083,1095,1277,1281,1282,1284,1287,1291,1294,1295,1300,1301,1355,1425,1426,1428,1432,1442,1443,1445,1446,1448,1454,1455,1467,1468,1469,1478,1496,1498,1499,1502,1561,1562,1564,1568,1572,1582,1585,1588,1589,1591,1592,1601,1603,1604,1606,1611,1612,1614,1615,1619,1625,1626,1629,1634,1638,1640,1645,1646,1647,1651,1654,1657,1660,1663,1684,1687,1694,1696,1701,1761,1766,1773,1775,1776,1778,1842,1846,1899,1903,1904,1905,1925,1927,1928,1929,1934,1935,1941,2006,2008,2029,2030,2032,2035,2089,2109,2110,2113,2114,2120,2121,2124,2172,2191,2212,2296,2298,2300,2303,2304,2306,2307,2308,2309,2310,2315,2317,2320,2321,2324,2329,2333,2335,2336,2338,2339,2340,2341,2343,2344,2346,2351,2352,2353,2355,2358,2374,2375,2376,2388,2389,2390,2392,2397,2398,2403,2407,2447,2448,2459,2460,2461,2462,2463,2465,2467,2468,2469,2472,2474,2477,2511,2725,2726,2727,2728,2729,2753,2756,3160,3161,3163,3184,3185,3187,3188,3192,3196,3197,3200,3205,3206,3214,3215,3216,3239,3245,3433,3434,3436,3439,3440,3456,3458,3459,3460,3462,3473,3474,3627,3628,3631,3637,3643,3647,3721,3725,3777,3837,3980,3981,3983,3984,4139,4140,4149,4151,4152,4154]},{"id":"symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","occurrences":[1944,1418,2562,2564,156,159,758,760,761,1378,1379,1382,1924,4,5,7,9,12,13,14,18,19,29,40,47,52,53,58,59,60,66,67,70,72,74,76,78,96,97,98,100,122,123,126,128,129,130,133,134,135,136,150,151,152,153,162,167,168,173,181,182,184,187,188,189,190,192,194,195,196,197,198,199,202,203,204,205,206,207,209,215,218,219,221,257,258,260,261,262,263,264,265,266,267,268,270,272,273,274,275,276,277,278,280,281,604,605,606,607,663,710,712,714,715,720,738,739,770,771,772,863,940,942,949,958,961,962,968,971,974,975,980,982,984,997,998,999,1000,1001,1012,1014,1026,1138,1140,1383,1391,1395,1397,1398,1403,1410,1411,1421,1535,1539,1540,1541,1569,1570,1571,1572,1574,1576,1577,1580,1581,1582,1584,1589,1590,1592,1640,1641,1642,1648,1684,1686,1687,1688,1691,1694,1696,1701,1703,1705,1706,1707,1708,1718,1719,1720,1721,1722,1723,1724,1725,1726,1727,1729,1730,1731,1733,1734,1736,1738,1739,1740,1741,1742,1743,1744,1745,1746,1749,1752,1753,1755,1758,1761,1764,1765,1766,1773,1774,1775,1778,1780,1807,1809,1815,1825,1826,1828,1850,1859,1863,1866,1868,1897,1899,1900,1902,1903,1905,1907,1910,1922,1926,1928,1930,1933,1935,2497,2498,2499,2558,2559,2561,2567,2576,2577,2579,2582,2583,2584,2587,2592,2594,2595,2596,2597,2601,2616,2617,2626,2716,2723,2725,2726,2727,2728,2729,2730,2732,2734,2735,2740,2744,2746,2750,2751,2753,2755,2756,2758,2764,2772,2775,2778,2781,2782,2785,2786,2790,2791,2794,2796,2799,2800,2802,2845,2848,2850,2851,2852,2855,2856,2858,2860,2861,2862,2863,2864,2865,2866,3070,3072,3074,3075,3077,3078,3080,3082,3089,3091,3092,3093,3095,3096,3102,3107,3113,3114,3117,3164,3214,3216,3217,3509,3521,3522,3525,3527,3528,3533,3534,3535,3537,3538,3546,3550,3587,3589,3590,3592,3593,3595,3598,3613,3667,3668,3669,3672,3674,3675,3677,3678,3679,3680,3681,3682,3683,3684,3688,3689,3690,3691,3693,3698,3699,3700,3701,3702,3706,3707,3718,3719,3720,3721,3722,3723,3724,3725,3726,3727,3729,3730,3732,3733,3735,3736,3737,3740,3742,3745,3748,3750,3751,3753,3754,3766,3767,3769,3770,3773,3775,3776,3777,3778,3779,3786,3788,3789,3791,3793,3795,3796,3797,3799,3811,3819,3821,3823,3826,3827,3828,3829,3934,3940,3959,3964,3965,3967,3969,3972,3973,3977,3984,3985,3987,3993,4040,4042,4050,4118,4127,4138,4149]},{"id":"symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","occurrences":[2543,2556,1378,1805,2503,2821,2822,2823,2824,2826,2827,2828,2829,2831,2832,2833,467,528,531,535,536,542,553,558,623,656,677,684,685,687,693,694,696,814,819,824,829,840,841,844,853,854,855,902,922,923,925,926,928,929,934,936,1026,1205,1280,1283,1284,1291,1295,1300,1323,1355,1413,1513,1522,1542,1544,1546,1554,1555,1562,1565,1566,1580,1581,1583,1603,1612,1651,1654,1663,1671,1673,1678,1755,1756,1764,1775,1776,1842,1846,1847,1848,1849,1850,1855,1857,1858,1871,1873,1875,1878,1880,1881,1882,1883,1884,1885,1886,1889,1984,1985,1986,2029,2033,2036,2037,2089,2109,2115,2122,2139,2164,2165,2166,2167,2168,2172,2212,2335,2366,2375,2377,2378,2384,2388,2389,2449,2451,2458,2481,2505,2506,2507,2514,2515,2516,2517,2524,2690,2694,2697,2723,2726,2727,2730,2732,2753,2755,2758,2762,2772,2778,2780,2783,2790,2792,2799,2800,2801,2802,2988,2989,2990,2991,2994,2995,2996,2997,2998,3006,3010,3013,3051,3065,3070,3071,3073,3074,3075,3076,3077,3078,3079,3080,3081,3083,3084,3085,3089,3091,3092,3097,3099,3113,3153,3154,3155,3156,3157,3159,3161,3162,3163,3220,3231,3232,3233,3237,3238,3239,3240,3243,3275,3279,3280,3281,3282,3298,3305,3333,3342,3345,3346,3370,3371,3372,3373,3374,3376,3377,3386,3387,3388,3393,3395,3399,3402,3403,3407,3415,3502,3505,3506,3587,3588,3590,3593,3594,3595,3598,3609,3613,3729,3730,3731,3735,3736,3737,3738,3739,3744,3745,3746,3747,3748,3749,3751,3752,3754,3761,3762,3763,3765,3769,3770,3773,3774,3775,3776,3779,3781,3786,3789,3831,3985,4091,4092]},{"id":"symbol-d4faea86910a340e","tex":"\\inner","aliases":"inner","display":null,"kind":"notation-table","occurrences":[2534,2218,379,385,422,644,1357,1432,1437,1442,1443,1469,1492,1508,1976,1980,1981,2016,2022,2024,2032,2048,2118,2119,2226,2237,2238,2241,2243,2246,2276,2280,2336,2338,2341,2343,2344,2353,2389,2511,2624,2625,2669,2867,2929,2931,2974,3729,3740,3742,3745,3967,3969,3973,3984]},{"id":"symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"notation-table","occurrences":[2546,2549,762,770,771,772,786,787,796,797,798,799,805,806,807,812,961,962,998,999,1012,1014,1026,1265,1664,1665,1668,1669,1670,1671,2016,2019,2021,2032,2037,2048,2049,2050,2053,2055,2056,2062,2066,2098,2110,2112,2125,2181,2182,2184,2186,2189,2213,2237,2238,2239,2241,2246,2251,2252,2280,2336,2337,2345,2346,2347,2353,2649,2650,2652,2730,2733,2735,2740,2744,2746,2750,2751,2752,2756,3136,3137,3265,3268,3270,3272,3273,3274,3275,3279,3282,3286,3294,3298,3305,3309,3337,3341,3342,3345,3346,3347,3353,3356,3359,3360,3362,3374,3375,3396,3397,3399,3401,3421,3428,3429,3430,3432,3434,3438,3439,3443,3459,3650,3651,3652,3654,3659]},{"id":"symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","occurrences":[1944,766,1093,1225,2821,2822,2823,2825,2826,2827,2828,2830,2831,2832,2833,2834,2835,523,524,580,581,583,770,771,829,830,840,843,848,858,859,1033,1085,1086,1087,1089,1104,1105,1155,1208,1210,1212,1213,1216,1221,1222,1223,1232,1233,1234,1238,1239,1242,1243,1245,1248,1249,1256,1258,1259,1266,1267,1270,1272,1276,1280,1281,1282,1283,1284,1297,1300,1301,1305,1307,1308,1312,1318,1319,1324,1325,1326,1327,1330,1332,1333,1334,1339,1347,1513,1545,1548,1549,1552,1556,1557,1558,1565,1566,1570,1573,1577,1578,1580,1581,1593,1594,1595,1596,1598,1701,1702,1719,1721,1739,1742,1743,1744,1749,1825,1826,1865,2139,2164,2167,2168,2432,2433,2434,2435,2497,2499,2515,2516,2517,2613,2631,2632,2633,2634,2635,2636,2637,2638,2640,2642,2643,2644,2645,2647,2648,2649,2650,2653,2654,2655,2656,2721,2722,2723,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2744,2746,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2774,2775,2802,2810,2837,2838,2839,2881,2882,2883,2884,2885,3036,3037,3038,3041,3042,3043,3045,3078,3079,3080,3083,3084,3085,3089,3090,3091,3092,3099,3228,3229,3230,3237,3238,3243,3371,3373,3374,3427,3517,3519,3520,3521,3522,3523,3524,3525,3526,3527,3528,3531,3532,3533,3534,3535,3536,3539,3541,3546,3549,3550,3661,3662,3664,3668,3675,3729,3730,3731,3735,3736,3737,3738,3739,3748,3749,3751,3761,3762,3765,3769,3770,3775,3776,3781,3934,3938,3939,3940,3943,3952,3962,3975,3979,4118]},{"id":"symbol-1a670656747a1061","tex":"J_{b\\leftarrow a}","aliases":"J_b← a leftarrow","display":"J_{b\\leftarrow a}","kind":"notation-table","occurrences":[2546,2549,2016,2019,2021,2032,2048,2181,2184,2186,2189]},{"id":"symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","occurrences":[2541,2542,2543,2554,2556,1047,1048,1055,2501,3838,3840,638,639,644,646,696,697,816,817,818,925,928,931,937,1026,1042,1043,1395,1398,1619,1620,1621,1622,1624,2014,2015,2016,2022,2024,2029,2039,2040,2048,2077,2078,2079,2081,2087,2092,2095,2210,2211,2212,2236,2237,2238,2243,2246,2278,2280,3195,3197,3199,3200,3201,3220,3221,3222,3223,3228,3229,3236,3242,3244,3246,3247,3250,3251,3253,3254,3272,3279,3327,3328,3329,3341,3345,3346,3360,3368,3369,3372,3374,3397,3400,3402,3403,3418,3419,3475,3478,3554,3556,3558,3563,3567,3569,3572,3573,3576,3830,3831,3832,3833,3834,3835]},{"id":"symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"notation-table","occurrences":[2542,2078,2211]},{"id":"symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","occurrences":[2538,408,409,1225,268,270,378,416,426,490,491,492,495,537,576,603,773,774,864,866,867,868,873,889,890,892,893,1120,1121,1141,1168,1176,1204,1205,1208,1210,1212,1213,1216,1220,1221,1222,1223,1232,1235,1236,1237,1238,1240,1241,1243,1245,1248,1249,1255,1257,1258,1259,1260,1266,1267,1270,1272,1280,1285,1289,1292,1293,1303,1307,1308,1310,1318,1319,1320,1323,1324,1325,1330,1331,1332,1334,1337,1339,1341,1342,1343,1344,1345,1346,1348,1349,1414,1416,1419,1488,1500,1514,1526,1527,1664,1668,1670,1671,1691,1692,1694,1696,1706,1707,1764,1765,1766,1780,1783,1859,1866,1900,1902,1921,1922,1923,1951,1954,1955,1956,1957,1959,1962,2155,2156,2160,2602,2613,3131,3137,3139,3146,3147,3149,3221,3251,3272,3279,3341,3345,3360,3397,3475,3476,3477,3478,3479,3507,3521,3528,3538,3612,3613,3615,3711,3713,3715,3961,3971,3972,3975,3977,3988,4032]},{"id":"symbol-03713cede48e6adf","tex":"L_{ba}","aliases":"L_ba","display":"L_{ba}","kind":"notation-table","occurrences":[2538,1954,1955,2155,2156,2160]},{"id":"symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"notation-table","occurrences":[2545,1973,2221,743,2289,2293,150,153,187,286,349,397,435,438,461,476,479,481,488,494,505,510,511,513,524,527,531,532,547,550,552,558,560,563,567,568,571,577,592,617,623,634,677,681,694,696,698,770,821,906,916,919,920,931,938,1251,1356,1362,1374,1377,1388,1390,1395,1398,1403,1405,1415,1421,1426,1427,1465,1466,1469,1470,1472,1474,1476,1483,1491,1493,1501,1502,1974,1976,2041,2048,2076,2223,2225,2264,2276,2302,2303,2317,2336,2344,2346,2350,2363,2364,2389,2390,2395,2398,2407,2418,2420,2422,2424,2427,2430,2433,2435,2436,2438,2439,2443,2448,2460,2461,2481,2579,2582,2584,2587,2626,2648,2755,2859,2979,3017,3019,3024,3025,3029,3030,3043,3052,3058,3062,3064,3082,3102,3137,3138,3149,3680,3696,3697,3777,3928,3954,3972,3977,3998,3999,4000,4001]},{"id":"symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","occurrences":[2537,2538,2545,1969,1970,1972,3854,2100,2283,2286,2289,2291,2292,2899,2900,3167,3169,3170,3173,3174,226,227,229,230,231,235,237,238,240,241,250,282,283,285,290,295,667,668,672,677,679,681,684,685,687,690,691,692,693,694,695,698,778,779,841,844,854,857,862,889,890,1298,1299,1300,1359,1362,1383,1393,1395,1398,1405,1460,1465,1606,1607,1608,1609,1610,1611,1612,1621,1622,1625,1627,1628,1629,1650,1651,1654,1937,1946,1947,1948,1954,1956,1959,1960,1961,1962,1963,1976,1977,1980,1984,1991,1992,1994,1996,1997,1999,2001,2003,2004,2005,2011,2017,2022,2041,2045,2073,2075,2076,2094,2105,2106,2107,2110,2113,2114,2120,2129,2131,2136,2160,2222,2223,2336,2338,2341,2343,2344,2346,2349,2350,2356,2357,2466,2468,2475,2478,2479,2481,2520,2521,2522,2523,2524,2531,2532,2659,2660,2665,2666,2921,2926,3015,3025,3029,3056,3058,3148,3149,3150,3151,3153,3154,3155,3156,3162,3183,3185,3188,3189,3193,3194,3196,3197,3203,3205,3216,3261,3263,3333,3334,3617,3621,3623,3625,3626,3627,3630,3631,3635,3637,3638,3652,3655,3656,3673,3674,3675,3676,3678,3723,3725,3850,3851,3852,3853,3856,3860,3861,3868,3872,3875,3876,3884,3887,3888,3894,3914,3941,3943,3948,3954,4000,4001,4043,4044,4048,4056,4057,4058,4061,4073,4074]},{"id":"symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","occurrences":[1944,2545,1973,2218,2220,2221,155,157,158,159,324,329,332,334,335,406,407,723,724,725,760,761,764,765,767,768,1092,3,5,6,8,9,10,11,12,13,15,16,17,18,19,21,22,23,24,25,26,27,28,29,30,31,32,33,35,36,37,39,40,47,48,50,52,53,66,67,70,72,74,76,78,121,123,125,126,127,129,132,136,148,150,161,162,165,168,170,171,173,175,176,177,178,179,183,184,185,186,187,188,189,190,192,193,195,196,197,198,200,201,202,203,206,207,209,211,215,216,217,218,219,221,223,224,256,257,259,260,261,262,264,265,266,267,268,269,270,274,275,276,277,281,299,300,302,303,306,307,308,309,310,311,313,318,321,336,337,339,341,342,348,349,351,352,354,355,360,361,363,365,366,368,370,372,373,374,375,376,377,378,379,380,382,384,385,387,389,391,392,393,394,395,396,397,399,402,403,404,405,417,425,427,439,440,443,444,452,461,464,466,467,468,469,471,472,476,477,480,481,484,494,545,546,549,550,557,593,596,597,603,604,605,606,607,609,611,675,703,704,705,708,712,713,715,720,738,739,770,771,772,775,788,789,790,792,794,795,797,801,806,818,833,837,838,841,843,844,845,846,847,849,850,853,854,855,857,859,862,863,940,941,947,949,951,954,956,957,958,969,972,980,983,989,990,991,994,995,997,1004,1006,1008,1009,1029,1030,1042,1061,1062,1063,1064,1066,1067,1068,1069,1070,1071,1072,1074,1076,1078,1079,1080,1084,1085,1087,1088,1090,1091,1095,1102,1103,1104,1105,1106,1107,1110,1111,1112,1114,1116,1137,1139,1140,1165,1166,1167,1170,1172,1174,1183,1185,1186,1189,1190,1192,1194,1195,1198,1302,1303,1305,1307,1309,1310,1311,1313,1314,1315,1316,1317,1318,1320,1321,1330,1331,1332,1334,1339,1468,1469,1470,1472,1474,1491,1493,1494,1495,1521,1522,1560,1561,1562,1564,1566,1570,1571,1580,1587,1588,1591,1593,1594,1595,1598,1601,1602,1632,1633,1701,1702,1719,1721,1737,1738,1739,1742,1743,1744,1749,1814,1825,1826,1862,1941,1987,1988,2019,2021,2036,2048,2053,2055,2065,2096,2097,2105,2108,2135,2136,2143,2145,2157,2158,2160,2161,2170,2171,2175,2176,2179,2180,2184,2185,2186,2188,2189,2196,2200,2209,2223,2224,2251,2264,2332,2336,2578,2579,2581,2582,2586,2587,2588,2590,2951,2952,2957,2958,2959,2960,2961,2962,2963,2970,2971,2976,2978,2979,2983,2984,2994,2995,2996,2997,3057,3058,3059,3070,3074,3075,3076,3077,3085,3089,3091,3095,3096,3101,3113,3442,3444,3448,3450,3456,3473,3474,3613,3791,3793,3794,3795,3796,3797,3799,3814,3824,3836,3867,3875,3877,3882,3884,3889,3890,3895,3896,3904,3913,3914,3952,3954,3955,3956,3957,3958,4042]},{"id":"symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"notation-table","occurrences":[2537,2538,2545,1969,1970,1972,2100,2283,2286,2289,2291,2292,1946,1947,1948,1954,1956,1959,1960,1962,1963,1976,1977,1980,1991,1992,1994,1997,2001,2003,2005,2011,2045,2073,2076,2105,2106,2107,2110,2113,2114,2120,2129,2131,2136,2160,2222,2223,2336,2341,2344,2349,2350,2520,2521,2522,2524,2531,2532]},{"id":"symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"notation-table","occurrences":[2545,1973,2218,2220,2221,2019,2048,2105,2108,2158,2160,2161,2180,2184,2185,2186,2196,2200,2223,2224,2251,2264,3793,3795,3799,3836]},{"id":"symbol-a7abbe02c9f0ed6a","tex":"P_a^*","aliases":"P_a^*","display":"P_a^*","kind":"notation-table","occurrences":[2537,1970]},{"id":"symbol-72bf1da84f9b9c1d","tex":"P_a^2","aliases":"P_a^2","display":"P_a^2","kind":"notation-table","occurrences":[2537,2106]},{"id":"symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"notation-table","occurrences":[2538,1970,2100,1947,1954,1959,1961,1962,1980,1991,1992,1996,1999,2003,2004,2005,2017,2022,2041,2110,2113,2114,2120,2160,2336,2338,2341,2343,2344,2346,2356,2357,2521,2522,2524,2532]},{"id":"symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"notation-table","occurrences":[1944,1566,1570,1580,1594,1595,1701,1702,1719,1721,1739,1742,1743,1744,1749,1825,1826]},{"id":"symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"notation-table","occurrences":[1944,326,329,334,291,369,1074,1084,1087,1131,1192,1195,1196,1200,1295,1570,1575,1580,1612,1616,1636,1709,1710,1712,1714,1715,1716,1739,1826,1845,1848,1849,1856,2036,2037,2093,2095,2097,2098,2943,3109,3113,3115,3779,3786,3985,3990]},{"id":"symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"notation-table","occurrences":[2551,1142,1152,1626,1627,3783,3785]},{"id":"symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","occurrences":[2534,2535,159,407,1047,1048,1050,1051,1053,1054,1093,2287,2288,2289,2293,2294,2889,2890,2891,2892,2893,2894,2900,2901,2902,2903,2904,2905,2906,2,31,32,34,36,45,77,115,116,129,160,162,264,275,276,277,655,656,716,717,719,751,902,904,913,1010,1034,1040,1042,1043,1044,1061,1062,1064,1067,1070,1071,1073,1075,1076,1081,1082,1085,1095,1096,1097,1100,1101,1103,1104,1106,1111,1114,1118,1126,1127,1128,1129,1133,1134,1135,1142,1143,1144,1151,1153,1158,1160,1169,1174,1190,1191,1192,1413,1427,1517,1518,1522,1526,1530,1542,1546,1554,1561,1562,1565,1577,1671,1673,1684,1701,1702,1842,1846,1850,1855,1856,1858,1860,1865,2117,2118,2140,2141,2295,2296,2301,2302,2303,2304,2423,2424,2445,2446,2476,2478,2479,2603,2606,2607,2608,2609,2610,2611,2612,2617,2620,2621,2622,2623,2624,2625,2626,2665,2666,2667,2669,2670,2678,2679,2680,2681,2682,2684,2686,2687,2703,2706,2707,2769,2772,2775,2778,2779,2780,2781,2782,2783,2784,2785,2789,2790,2797,2798,2799,2801,2803,2804,2805,2806,2807,2808,2809,2810,2812,2813,2815,2816,2871,2872,2875,2876,2877,2908,2909,2910,2911,2912,2913,2914,2915,2916,2918,2922,2923,2924,2925,2926,2927,2928,2929,2930,2932,2933,2945,2946,2947,2948,2949,2951,2952,2953,2963,2964,2965,2972,2985,2986,3000,3019,3024,3025,3031,3047,3048,3052,3064,3069,3082,3086,3088,3094,3095,3096,3100,3103,3104,3105,3108,3109,3110,3111,3113,3115,3116,3136,3220,3227,3230,3240,3370,3374,3380,3385,3387,3388,3393,3403,3405,3481,3482,3484,3486,3487,3489,3492,3511,3548,3549,3550,3552,3553,3554,3556,3558,3560,3561,3568,3569,3572,3610,3613,3650,3651,3653,3654,3658,3773,4040,4116,4128,4129]},{"id":"symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"notation-table","occurrences":[2534,2536,407,129,275,276,277,279,818,902,903,904,906,913,916,1263,1264,1265,1266,1267,1271,1272,1427,1542,1544,1546,1554,1561,1562,1565,1701,1702,1842,1846,1856,1858,1865,2258,2309,2310,2312,2314,2610,2611,2871,2873,2875,2876,2878,2879,2945,2946,2947,2948,2949,2951,2952,2954,2963,2964,2965,2972,2985,2986,3001,3047,3048,3052,3064,3069,3082,3086,3087,3094,3095,3096,3100,3103,3104,3106,3109,3110,3111,3113,3115,3116,3388,3393,3481,3482,3487,3488,3495,3497,3658,3659,3661,3663,3664,4042]},{"id":"symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"notation-table","occurrences":[2550,2551,743,744,745,746,103,748,751,816,1207,1209,2103,2104,2105,2107,2108,2109,2110,2111,2112,2113,2114,2116,2119,2121,2122,2134,2135,2136,2139,2142,2155,2156,2157,2160,2167,2168,2171,2172,2177,2178,2648,2650,2653,2654,2734,2737,2750,3214,3216,3491,3492,3580,3582,3585,3586,3590,3593,3595,3604]},{"id":"symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"notation-table","occurrences":[2541,2545,2552,2553,1971,1972,1973,2217,2218,2221,159,740,742,743,2291,2900,162,173,284,286,288,468,474,819,821,822,824,825,1040,1105,1282,1284,1296,1300,1301,1974,1975,1976,1978,1980,1981,1982,1984,1985,1986,1996,2002,2013,2014,2016,2017,2022,2023,2024,2032,2035,2036,2039,2040,2041,2048,2073,2076,2077,2087,2091,2116,2118,2119,2223,2225,2226,2229,2235,2236,2237,2238,2241,2243,2246,2257,2264,2276,2279,2280,2333,2335,2336,2338,2339,2341,2343,2344,2346,2350,2351,2353,2356,2357,2358,2363,2364,2462,2603,2608,2609,2617,2618,2619,2624,2625,2626,2632,2638,2647,2648,2665,2669,2721,2725,2726,2730,2731,2734,2738,2739,2741,2758,2762,2770,2772,2777,2778,2780,2783,2788,2795,2798,2801,2803,2806,2812,2816,2882,2913,2918,2919,2920,2926,2931,2974,2979,2993,3006,3016,3017,3019,3025,3030,3056,3135,3136,3137,3271,3275,3279,3298,3305,3311,3313,3342,3345,3346,3383,3387,3390,3391,3392,3396,3414]},{"id":"symbol-a86ef458cc61e377","tex":"\\hbar","aliases":"ℏ hbar","display":"\\hbar","kind":"notation-table","occurrences":[2544,2555,297,791,1515,2031,2888,3500,3501,3502,3505,3506]},{"id":"symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"notation-table","occurrences":[2546,2549,822,2016,2019,2021,2032,2037,2048,2049,2050,2056,2062,2098,2110,2112,2125,2158,2181,2184,2186,2189,2199,2202,2610,2611,2871,2875,2876]},{"id":"symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"notation-table","occurrences":[2553,1969,1970,2216,157,158,406,2283,2286,2288,2289,2294,2831,12,72,74,76,78,192,338,417,426,434,476,477,480,494,500,506,508,546,549,551,564,569,603,611,635,640,643,717,748,828,874,944,948,969,972,1011,1141,1217,1246,1247,1284,1307,1356,1361,1399,1412,1422,1426,1435,1437,1441,1447,1449,1456,1470,1471,1474,1490,1501,1520,1524,1529,1530,1531,1532,1560,1587,1670,1684,1691,1694,1696,1706,1707,1766,1780,1900,1902,1915,1925,1933,1946,1947,1959,1960,1961,1962,1975,1990,2021,2045,2048,2151,2174,2267,2306,2309,2313,2333,2346,2349,2350,2353,2366,2374,2376,2385,2394,2396,2397,2446,2462,2476,2531,2532,2558,2565,2566,2607,2631,2632,2633,2634,2637,2686,2717,2721,2722,2803,2804,2805,2844,2853,2945,2974,2992,3019,3024,3055,3133,3136,3149,3228,3420,3619,3773,3865,3923]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","occurrences":[2549,1970,2216,2221,743,744,745,746,1227,2286,2289,129,136,150,152,153,165,170,173,186,187,188,192,209,211,214,257,262,274,276,300,301,307,312,314,388,392,397,401,453,461,468,469,472,476,479,494,505,526,528,531,532,542,552,553,556,566,571,592,593,617,622,635,720,748,799,806,807,812,816,817,818,819,822,823,824,861,862,897,898,949,970,972,978,989,997,1164,1206,1208,1210,1212,1216,1243,1249,1254,1263,1266,1270,1283,1284,1300,1318,1324,1327,1329,1330,1334,1339,1356,1357,1362,1417,1426,1432,1470,1472,1474,1476,1483,1485,1491,1493,1501,1502,1513,1525,1530,1531,1565,1581,1593,1598,1603,1604,1670,1684,1691,1694,1696,1701,1702,1706,1707,1743,1749,1766,1768,1772,1780,1802,1865,1874,1889,1900,1902,1915,1925,1933,1947,1955,1959,1960,1961,1962,1963,1976,1977,1996,1999,2001,2003,2011,2019,2021,2022,2045,2048,2049,2108,2112,2115,2135,2139,2156,2158,2165,2174,2176,2181,2184,2185,2186,2188,2189,2200,2202,2239,2241,2245,2246,2248,2250,2251,2255,2258,2269,2306,2316,2317,2319,2344,2345,2346,2349,2350,2353,2357,2386,2401,2407,2433,2523,2524,2531,2532,2576,2617,2624,2625,2669,2686,2725,2726,2734,2750,2751,2754,2756,2813,2929,2974,3019,3025,3041,3043,3124,3126,3137,3138,3151,3153,3220,3227,3229,3240,3319,3521,3527,3528,3541,3546,3590,3598,3650,3661,3662,3675,3729,3740,3773,3777,3778,3786,3789,3793,3795,3799,3836]},{"id":"symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"notation-table","occurrences":[2549,2221,2289,187,388,397,401,635,1963,1976,1977,1996,2001,2011,2019,2022,2108,2112,2115,2135,2158,2181,2185,2188,2200,2239,2241,2245,2246,2255,2258,2344,2345,3137,3138,3793,3795,3799,3836]},{"id":"symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"notation-table","occurrences":[2550,2216,2220,2563,758,759,1046,1227,4,16,122,136,142,191,194,413,436,449,465,468,476,478,484,492,493,494,526,528,566,571,593,676,709,710,792,793,904,925,942,1097,1121,1144,1165,1169,1184,1204,1233,1239,1253,1264,1266,1306,1329,1337,1338,2277,2299,2326,2330,2362,2401,2464,2580,3121,3508,3517,3649,3660,3666,3696,3700,3777]},{"id":"symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","occurrences":[2564,3854,159,324,329,407,745,746,761,765,767,1228,1382,2100,2293,2504,2822,2827,2831,2900,3844,5,7,12,16,17,19,21,24,27,36,37,47,54,56,59,67,72,76,88,89,96,101,108,112,113,116,143,153,168,184,185,186,188,189,190,192,197,203,206,207,240,260,261,267,270,276,286,307,322,377,392,393,394,395,468,481,484,486,519,530,550,552,558,560,568,573,585,593,639,641,644,646,648,653,655,656,660,663,674,694,698,748,751,753,757,770,772,797,801,827,828,836,839,841,847,850,857,858,859,860,902,904,913,922,923,925,938,946,952,953,954,958,974,975,982,997,1000,1002,1010,1016,1033,1039,1040,1058,1059,1062,1064,1070,1074,1083,1085,1104,1106,1143,1154,1155,1210,1216,1218,1221,1223,1245,1251,1253,1254,1266,1286,1307,1308,1316,1318,1319,1349,1351,1388,1389,1398,1427,1466,1468,1469,1470,1474,1491,1493,1494,1495,1501,1502,1513,1539,1549,1561,1562,1565,1577,1581,1588,1591,1592,1593,1594,1595,1598,1602,1606,1609,1610,1612,1621,1622,1628,1629,1632,1645,1651,1654,1730,1742,1743,1749,1761,1769,1833,1836,1842,1843,1850,1855,1856,1857,1858,1859,1861,1865,1866,1920,2083,2087,2122,2144,2155,2167,2168,2170,2171,2177,2229,2235,2241,2281,2282,2317,2325,2328,2360,2361,2368,2421,2424,2444,2453,2455,2458,2459,2460,2461,2462,2479,2484,2496,2497,2572,2576,2581,2582,2584,2585,2586,2590,2594,2595,2596,2600,2601,2603,2606,2609,2612,2617,2618,2620,2625,2626,2636,2638,2639,2644,2647,2649,2650,2652,2653,2654,2659,2660,2665,2666,2678,2680,2706,2707,2708,2725,2726,2730,2731,2734,2735,2746,2758,2764,2772,2775,2778,2780,2785,2790,2800,2806,2809,2837,2852,2858,2859,2882,2883,2885,2913,2921,2926,2929,2985,3015,3025,3029,3039,3041,3052,3064,3067,3070,3074,3075,3076,3077,3082,3089,3091,3095,3096,3100,3113,3121,3144,3162,3179,3185,3186,3187,3188,3190,3195,3196,3197,3198,3200,3205,3209,3216,3221,3237,3245,3248,3258,3261,3263,3272,3275,3278,3282,3283,3297,3298,3301,3304,3305,3319,3337,3338,3340,3342,3344,3349,3352,3353,3363,3381,3392,3398,3403,3405,3408,3424,3449,3450,3456,3474,3481,3482,3484,3485,3486,3488,3489,3496,3501,3503,3511,3517,3520,3521,3522,3525,3526,3531,3533,3534,3535,3541,3550,3553,3554,3557,3558,3563,3573,3574,3586,3590,3593,3595,3598,3607,3608,3609,3610,3613,3621,3623,3627,3631,3637,3643,3651,3654,3661,3664,3668,3674,3675,3678,3680,3689,3690,3691,3711,3713,3735,3737,3740,3745,3748,3751,3762,3769,3770,3773,3777,3778,3781,3789,3795,3812,3820,3821,3823,3832,3850,3852,3853,3856,3860,3865,3868,3869,3873,3874,3887,3888,3894,3901,3917,3920,3925,3927,3928,3934,3943,3949,3954,3962,3964,3972,3973,3977,3981,3990,4040,4044,4052,4059,4061,4063,4070,4071,4075,4077,4078,4082,4087,4090,4093,4095,4096,4097,4101,4103,4105,4108,4112,4118,4119,4121,4130,4131,4149,4174]},{"id":"symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","occurrences":[2215,2216,1804,2283,2285,2896,2897,2899,2900,3844,110,114,117,118,120,312,315,427,429,614,617,618,619,620,622,623,625,627,629,631,632,635,747,748,749,865,870,922,925,1010,1096,1097,1100,1103,1158,1246,1248,1249,1251,1255,1256,1257,1259,1261,1262,1658,1659,1791,1793,1795,1798,1802,1810,1812,1819,1832,1848,1850,1851,1869,1871,1876,1877,1878,1880,1881,1882,1884,1885,1886,1887,1891,1893,1895,1903,1913,1946,1947,1959,1960,1962,1984,1985,1987,2007,2008,2009,2044,2045,2047,2048,2049,2050,2051,2052,2058,2062,2063,2127,2133,2134,2135,2136,2139,2142,2143,2145,2146,2173,2174,2175,2176,2249,2274,2275,2348,2439,2459,2460,2461,2581,2582,2587,2591,2651,2652,2657,2658,2659,2660,2661,2662,2665,2666,2671,2722,2723,2724,2729,2732,2756,2758,2767,2772,2773,2774,2775,2776,2778,2780,2783,2787,2790,2797,2799,2801,2802,2814,2844,2845,2846,2852,2854,2858,2920,2940,2942,3006,3015,3021,3024,3069,3070,3071,3073,3074,3076,3077,3079,3083,3085,3087,3089,3108,3109,3115,3116,3150,3151,3153,3154,3207,3208,3209,3231,3237,3240,3245,3248,3258,3337,3339,3342,3352,3353,3367,3380,3388,3393,3402,3403,3406,3652,3739,3740,3772,3773,3875,3964,3969,3984,4070,4072,4081,4087,4089,4090,4131,4133,4134]},{"id":"symbol-1aa9a70355d1f987","tex":"b^{m+1}","aliases":"b^m+1","display":"b^{m+1}","kind":"definition-passage","occurrences":[2564,2572,2584,2586]},{"id":"symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","occurrences":[2216,2218,2219,743,744,745,746,1225,1226,1231,2285,2286,2287,2289,2294,3165,3166,3167,3168,3170,3173,3174,77,78,83,215,216,525,526,527,621,622,633,634,748,829,1206,1208,1210,1214,1249,1263,1267,1880,1881,1882,1883,2158,2199,2202,2203,2222,2225,2226,2227,2229,2235,2237,2238,2239,2240,2241,2242,2243,2245,2246,2250,2251,2252,2264,2266,2276,2277,2280,2281,2282,2295,2296,2339,2346,2576,2577,2582,2698,2700,2702,2704,2706,2707,2851,2852,2855,2856,2864,3036,3038,3039,3041,3042,3043,3078,3079,3080,3082,3099,3110,3151,3179,3185,3187,3188,3196,3200,3209,3210,3211,3216,3237,3240,3272,3279,3286,3289,3290,3291,3293,3294,3340,3423,3424,3425,3481,3482,3483,3502,3503,3505,3506,3525,3526,3541,3766,3836,3837,3870,3871,3873,3874,3875,3877,3878,3879,3880,3882,3884,3887,3891,3892,3895,3896,3897,3900,3905,3914,3915,3990,4038]},{"id":"symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"definition-passage","occurrences":[1418,1414,1415,1417,1420,1421,1496,1498,1503,1507,1508,1509,1510,1511]},{"id":"symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","occurrences":[2562,2564,406,407,129,150,153,184,185,187,225,228,229,231,235,240,241,242,243,244,245,248,249,253,254,264,283,286,287,290,294,295,379,397,398,400,401,403,415,417,419,422,426,427,430,431,435,438,461,476,477,479,480,481,494,498,499,508,509,510,511,513,526,527,531,532,536,546,547,549,550,551,552,553,558,560,566,567,568,569,571,592,607,608,617,622,623,630,631,634,639,640,643,644,880,882,883,1015,1063,1064,1083,1163,1164,1167,1173,1174,1175,1280,1281,1282,1283,1284,1286,1297,1300,1301,1415,1421,1422,1496,1497,1498,1502,1503,1526,1530,1532,1577,1684,1691,1694,1696,1764,1765,1766,1780,1850,1855,1859,1866,1900,1902,1921,1922,2043,2044,2052,2155,2183,2184,2447,2448,2449,2459,2484,2497,2499,2558,2561,2576,2577,2579,2589,2594,2595,2596,2693,2694,2695,2780,2781,2782,2783,2784,2785,2789,3006,3060,3062,3063,3066,3067,3120,3121,3122,3123,3124,3126,3129,3149,3228,3230,3668,3672,3682,3683,3793,3812,3821,3822,3823,3901,3927,3928,3931]},{"id":"symbol-f1ceb5c186bbf16a","tex":"\\F^{-1}","aliases":"F^-1 F","display":null,"kind":"definition-passage","occurrences":[1418,1415,1417,1420,1496,1507]},{"id":"symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"definition-passage","occurrences":[2216,2218,2219,83,2222,2225,2227,2237,2238,2245,2250,2264,2276,2277,2280]},{"id":"symbol-a67dba11d23b584e","tex":"F_a^*","aliases":"F_a^*","display":"F_a^*","kind":"definition-passage","occurrences":[2216]},{"id":"symbol-7937103467a5b8d0","tex":"\\HH","aliases":"HH","display":null,"kind":"definition-passage","occurrences":[2214,406,725,2288,2290,378,382,396,399,417,439,440,461,476,477,494,545,546,557,603,616,678,679,680,681,864,866,868,893,1246,1414,1415,1421,1422,1500,1502,1964,1966,1967,1968,1975,1983,1990,1993,2102,2127,2128,2138,2254,2256,2265,2267,2297,2334,2473,2476,2606,2612,3019]},{"id":"symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","occurrences":[1969,1970,2216,2219,325,326,728,729,2284,3168,3169,3170,3171,3173,3174,107,110,113,114,117,118,133,136,153,189,201,202,214,215,218,227,239,268,317,438,462,479,481,482,487,488,529,547,558,560,568,577,604,606,609,611,623,634,635,668,696,737,739,770,777,779,798,799,812,817,818,828,841,844,857,879,892,904,911,1012,1013,1014,1027,1062,1063,1064,1070,1083,1143,1144,1151,1210,1240,1255,1257,1327,1420,1468,1545,1548,1553,1557,1588,1591,1592,1597,1598,1606,1610,1612,1615,1620,1621,1622,1623,1624,1628,1635,1645,1646,1652,1657,1725,1800,1886,1890,1907,1947,1963,1965,2020,2028,2043,2046,2151,2245,2255,2259,2271,2273,2274,2276,2282,2411,2421,2424,2434,2436,2442,2443,2459,2462,2523,2606,2658,2659,2661,2815,2840,2849,2858,2859,2860,2913,2920,2931,2935,2936,2940,2974,2993,2998,3006,3008,3009,3032,3035,3038,3039,3041,3043,3044,3046,3183,3185,3188,3189,3190,3192,3194,3195,3196,3197,3202,3205,3214,3216,3218,3251,3265,3267,3268,3270,3272,3273,3274,3275,3279,3281,3282,3286,3294,3298,3305,3309,3335,3337,3341,3342,3344,3345,3346,3347,3359,3360,3361,3362,3363,3374,3375,3396,3397,3401,3475,3478,3563,3618,3621,3622,3632]},{"id":"symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id","display":null,"kind":"definition-passage","occurrences":[1970,1977,1999,2001,2071,2072,2082,2133,2136,2146,3058]},{"id":"symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","occurrences":[3854,3855,3856,3859,3860,3883]},{"id":"symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","occurrences":[2563,2564,407,410,2902,2904,2906,43,44,45,46,151,152,153,154,196,197,215,218,377,381,382,384,385,386,387,389,390,391,392,393,394,395,397,401,403,404,405,411,412,413,414,415,419,420,422,423,424,432,436,438,439,441,442,443,444,446,448,449,450,451,453,454,455,456,457,458,459,461,464,465,466,468,470,471,473,476,479,483,484,493,494,496,499,502,503,505,514,515,517,519,521,526,527,528,531,532,542,552,553,555,556,564,566,571,577,579,580,581,586,592,593,594,597,600,782,783,1010,1011,1015,1016,1019,1117,1118,1120,1122,1125,1126,1128,1131,1133,1134,1135,1170,1171,1184,1185,1303,1304,1324,1355,1426,1429,1432,1433,1434,1435,1436,1437,1449,1450,1452,1458,1460,1462,1465,1466,1468,1469,1470,1473,1474,1475,1476,1477,1490,1491,1492,1493,1495,1496,1498,1499,1502,1508,1513,1535,1536,1538,1561,1562,1564,1565,1572,1581,1586,1588,1590,1591,1592,1593,1594,1598,1601,1602,1632,1633,1643,1648,1688,1699,1736,1737,1751,1757,1759,1808,1867,1897,1941,2567,2568,2569,2570,2571,2572,2574,2576,2577,2581,2582,2584,2585,2586,2587,2590,2591,2717,2725,2730,2912,2913,2925,2929,3125,3126,3128,3243,3245,3279,3281,3282,3283,3284,3345,3422,3480,3481,3484,3485,3488,3496,3517,3543,3609,3650,3665,3666,3755,4044,4051,4058,4059,4077,4080,4088,4092,4115,4116,4124,4136,4151,4152]},{"id":"symbol-5ab8627907d416c2","tex":"P_\\Lambda","aliases":"P_Λ Lambda","display":"P_\\Lambda","kind":"definition-passage","occurrences":[3854,3850,3852,3853,3856,3860,3875,3887,3888,3894,3941,3954,4000,4048,4058,4073]},{"id":"symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","occurrences":[3854,3170,3173,3174,227,229,230,231,232,233,236,240,241,243,283,286,294,295,668,672,677,678,679,680,681,684,685,687,690,691,692,693,694,695,698,706,779,785,1094,1095,1250,1251,1253,1254,1324,1325,1326,1327,1330,1349,2045,2047,2048,2130,2131,2349,2350,2351,2352,2353,2358,2363,2531,2532,2652,3060,3061,3062,3063,3066,3067,3214,3215,3218,3261,3263,3405,3617,3621,3623,3624,3625,3626,3627,3630,3631,3636,3637,3638,3648,3673,3674,3675,3676,3678,3724,3725,3850,3851,3852,3856,3860,3864,3866,3867,3869,3875,3880,3882,3885,3894,3895,3897,3898,3899,3901,3902,3903,3906,3908,3909,3912,3917,3919,3926,3936,3939,3940,3941,3943,3948,3949,3950,3951,3952,3953,3962,3976,3992,4045]},{"id":"symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"definition-passage","occurrences":[3854,3852,3860,3866,3867,3869,3880,3882,3894,3895,3901,3903,3912,3917,3926,3939,3940,3943,3948,3949,3950,3951,3952,3962,3976]},{"id":"symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","occurrences":[3854,329,335,1092,2835,3842,3844,3845,209,212,213,214,300,310,311,313,318,368,731,732,958,961,964,965,966,1034,1062,1064,1076,1081,1101,1104,1106,1111,1112,1118,1126,1128,1133,1135,1152,1166,1167,1174,1189,1190,1192,1201,1202,1203,1298,1299,1300,1770,1771,1772,1777,1872,1874,2051,2053,2055,2131,2132,2135,2259,2260,2272,2274,2276,2472,2475,2478,2479,2480,2481,2576,2577,2579,2584,2587,2813,2815,2836,2837,2840,2913,2931,2935,2936,3272,3279,3340,3482,3489,3490,3492,3556,3558,3569,3570,3571,3586,3589,3850,3851,3852,3853,3856,3857,3860,3861,3862,3866,3867,3868,3869,3870,3873,3875,3876,3881,3884,3887,3888,3889,3890,3894,3898,3899,3901,3904,3907,3912,3914,3916,3917,3922,3923,3931,3938,3939,3940,3941,3942,3943,3944,3945,3947,3948,3949,3950,3951,3952,3954,3955,3962,3964,3972,3974,3975,3976,3979,3998,4001,4044,4046,4056,4057,4061]},{"id":"symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"definition-passage","occurrences":[3854,3851,3852,3853,3856,3857,3860,3862,3881,3887,3894,3944]},{"id":"symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","occurrences":[2562,2564,760,761,1804,1811,635,636,712,715,751,753,940,958,980,1033,1034,1035,1062,1064,1070,1071,1072,1083,1107,1118,1131,1134,1153,1169,1395,1396,1398,1400,1403,1788,1789,1793,1795,1796,1797,1798,1800,1801,1802,1803,1842,1901,1902,1909,2567,2577,2582,2584,2587,2594,2596,2598,2610,2611,2634,2638,2644,2647,2766,2870,2871,2875,2876,3397,3425,3427,3428,3429,3432,3439,3460,3714,3715,3742,3745,3777,3781,3981,3984]},{"id":"symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","occurrences":[1418,326,329,334,335,1378,1379,1381,1382,1805,2100,3844,12,47,49,62,68,246,249,274,276,286,291,368,528,529,530,531,535,536,541,542,547,548,552,553,558,563,623,656,658,663,665,677,684,685,688,693,694,698,699,701,702,812,814,925,926,929,935,936,938,1039,1118,1130,1131,1132,1180,1247,1248,1249,1251,1259,1261,1262,1279,1280,1283,1284,1286,1290,1291,1294,1295,1296,1297,1300,1355,1383,1391,1392,1395,1396,1398,1403,1421,1535,1603,1604,1612,1615,1616,1636,1637,1644,1645,1646,1647,1648,1651,1653,1654,1655,1657,1660,1663,1684,1686,1687,1688,1691,1694,1696,1755,1756,1759,1761,1764,1765,1766,1775,1776,1777,1778,1779,1780,1781,1844,1845,1846,1847,1848,1849,1850,1854,1856,1859,1865,1866,1870,1871,1879,1880,1881,1882,1883,1884,1892,1893,1894,1895,1897,1899,1900,1902,1905,1908,1921,1922,1930,1934,2013,2014,2015,2016,2019,2021,2036,2037,2038,2039,2040,2070,2073,2076,2091,2093,2168,2171,2180,2184,2200,2240,2279,2333,2363,2364,2366,2367,2368,2369,2372,2375,2377,2384,2389,2390,2393,2395,2396,2398,2449,2454,2455,2456,2457,2726,2750,3063,3064,3067,3070,3074,3076,3081,3082,3097,3098,3102,3260,3261,3262,3264,3336,3337,3340,3343,3434,3439,3440,3459,3464,3469,3472,3511,3519,3528,3538,3541,3542,3552,3553,3554,3556,3558,3559,3561,3568,3569,3572,3607,3609,3613,3733,3734,3735,3737,3740,3742,3745,3748,3749,3759,3775,3850,3868,3870,3872,3873,3874,3887,3891,3892,3896,3912,3922,3972,3977,3987,3990,3993,4040,4044,4052,4079,4080,4087,4092,4108,4139,4149,4151,4153,4154,4164,4167]},{"id":"symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"definition-passage","occurrences":[2562]},{"id":"symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"definition-passage","occurrences":[2564,1118,1131,1134,2567,2577,2582,2584,2587]},{"id":"symbol-db2bda42ac0870c5","tex":"T_{m+1}","aliases":"T_m+1","display":"T_{m+1}","kind":"definition-passage","occurrences":[2564,2584]},{"id":"symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","occurrences":[1418,1047,1048,1054,1379,1382,2099,2893,2894,2895,5,7,13,14,53,66,78,274,276,528,529,530,531,536,541,542,547,552,558,563,738,739,770,771,772,811,812,1036,1037,1038,1039,1061,1062,1064,1067,1068,1069,1070,1071,1072,1082,1084,1095,1097,1099,1118,1126,1131,1134,1144,1145,1153,1383,1391,1392,1395,1398,1403,1421,1603,1612,1615,1645,1646,1647,1651,1654,1657,1660,1684,1686,1687,1694,1696,1761,1766,1778,1846,1899,1905,1930,1934,2070,2071,2072,2073,2079,2082,2083,2085,2086,2087,2089,2093,2101,2366,2367,2368,2369,2372,2375,2390,2393,2395,2398,2606,2609,2612,2615,2617,2618,2619,2622,2625,2626,2627,2628,2633,2634,2637,2638,2644,2647,2664,2675,2676,2678,2683,2685,2686,2688,2689,2723,2725,2730,2731,2741,2743,2759,2772,2802,2804,2805,2806,2818,2819,2853,2881,2882,2885,2886,2887,2992,3023,3055,3209,3300,3477,3478,3557,3572,3670,3673,3674,3675,3678,3679,3680,3687,3689,3692,3717,3742,3745]},{"id":"symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"definition-passage","occurrences":[1418,74,75,78,384,385,387,391,394,444,452,604,1355,1421]},{"id":"symbol-8a830c0eacd44ea9","tex":"U_h","aliases":"U_h","display":"U_h","kind":"definition-passage","occurrences":[1418,1379,1382,1383,1391,1395,1398,1403,1421,2772,3742,3745]},{"id":"symbol-6908b76a39ebf92a","tex":"\\mathcal U_h","aliases":"U_h mathcal","display":"\\mathcal U_h","kind":"definition-passage","occurrences":[1418,1421]},{"id":"symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"definition-passage","occurrences":[1970,324,329,331,2503,2504,136,170,211,212,323,385,453,472,798,812,1180,1182,1185,1210,1326,1402,1403,1406,1701,1705,1706,1721,1723,1725,1726,1727,1899,1900,1905,1907,1947,2132,2511,2512,2514,2517,2524,2530,2532,2559,2579,3024,3027,3030,3051,3052,3064,3103,3179,3181,3186,3187,3188,3189,3190,3192,3194,3195,3197,3198,3199,3200,3202,3205,3214,3216,3218,3226,3230,3237,3243,3245,3300,3342,3371,3374,3378,3380,3387,3388,3392,3393,3403,3406,3407,3728,3729,3735,3736,3737,3738,3739,3740,3748,3749,3751,3752,3761,3762,3765,3769,3770,3771,3775,3776,3777,3778,3781,3791,3831,3832,3833,3834,3928,3954,3998,3999,4000,4001]},{"id":"symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"definition-passage","occurrences":[1970,385,1947,2132,3791]},{"id":"symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","occurrences":[3854,3843,3844,3846,3152,3153,3154,3155,3156,3157,3159,3161,3162,3163,3164,3237,3279,3345,3682,3683,3781,3850,3851,3852,3853,3856,3857,3860,3862,3874,3875,3881,3884,3887,3888,3894,3898,3899,3941,3943,3947,3948,3954,3956,3999,4000,4044,4047,4048,4057,4058,4060,4073,4074,4076]},{"id":"symbol-ef65869303e9087f","tex":"\\psi^2","aliases":"ψ^2 psi","display":"\\psi^2","kind":"definition-passage","occurrences":[2221]},{"id":"symbol-d80f8057d7f2fe60","tex":"\\psi_a","aliases":"ψ_a psi","display":"\\psi_a","kind":"definition-passage","occurrences":[1972,1973,1976,1996,2016,2022,2024,2032,3135,3136,3137]},{"id":"symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"definition-passage","occurrences":[2216,1947]},{"id":"symbol-2b67385de1e4af35","tex":"\\sum_{a\\in I}","aliases":"∑_a∈ I sum in","display":"\\sum_{a\\in I}","kind":"definition-passage","occurrences":[1970]},{"id":"symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","occurrences":[2219,156,128,134,137,144,151,182,184,190,197,368,369,377,384,438,442,454,468,481,513,527,535,556,558,560,581,677,681,694,696,700,754,795,797,904,906,916,931,935,984,1085,1155,1180,1208,1223,1254,1297,1305,1311,1317,1318,1339,2316,2344,2346,2378,2390,2422,2439,2481,2570,2574,2579,2582,2584,2587,3024,3043,3052,3062,3064,3081,3082,3085,3089,3091,3096,3102,3107,3116,3121,3124,3126,3713,3714]},{"id":"symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","occurrences":[1225,1230,3844,197,202,215,800,806,814,815,818,831,1223,1249,1267,1784,1788,1792,2042,2043,3100,3102,3107,3112,3118]},{"id":"symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"introduction-passage","occurrences":[2896,865,1659,2657,2662]},{"id":"symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"introduction-passage","occurrences":[2897,110,114,2658,2661]},{"id":"symbol-e7851cacbe3babd8","tex":"\\mathcal A_{3,1}","aliases":"A_3,1 mathcal","display":"\\mathcal A_{3,1}","kind":"introduction-passage","occurrences":[1230]},{"id":"symbol-a0e924755e68f546","tex":"\\mathcal A_{L,j}","aliases":"A_L,j mathcal","display":"\\mathcal A_{L,j}","kind":"introduction-passage","occurrences":[1225,1223,1249,1267]},{"id":"symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","occurrences":[1379,1206,1211,1356,1357,1360,1362,1366,1374,1417,1424,1426,1432,1440,1455,1470,1472,1474,1476,1481,1483,1491,1493,1529,1530,1531,1567,1568,1670,1682,1684,1691,1694,1696,1706,1707,1766,1780,1900,1902,1905,1915,1925,1929,1933,1935,2331]},{"id":"symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"introduction-passage","occurrences":[3844,3875]},{"id":"symbol-4d8140b92561cfb1","tex":"\\mathrm{arc}","aliases":"arc mathrm","display":"\\mathrm{arc}","kind":"introduction-passage","occurrences":[725]},{"id":"symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","occurrences":[2502,2503,2504,283,286,289,296,377,384,385,390,391,393,401,468,649,651,652,655,671,672,674,677,681,682,685,687,690,691,692,693,694,695,698,865,866,870,877,879,881,898,899,904,908,919,922,923,925,926,931,936,1013,1021,1024,1263,1266,1267,1270,1272,1803,1869,1871,1873,1874,1875,1880,1881,1883,1884,1891,1893,1895,1904,1961,1962,1984,1986,1988,2007,2008,2010,2048,2049,2050,2059,2062,2064,2128,2133,2134,2135,2136,2138,2139,2142,2143,2145,2146,2173,2174,2175,2176,2249,2274,2275,2470,2475,2479,2481,2505,2506,2508,2509,2510,2511,2512,2513,2514,2515,2516,2517,2522,2524,2527,2528,2530,2532,2814,2844,2845,2847,2852,2854,2858,3085,3086,3089,3091,3092,3096,3100,3111,3209,3341,3342,3346,3374,3388,3393,3531,3533,3534,3535,3738,3740,3850,3981,3984,4071,4072,4082,4087,4089,4090]},{"id":"symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"introduction-passage","occurrences":[2502,649,671,865,870,877,879,881,898,899,904,908,919,923,925,926,1021,2508]},{"id":"symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"introduction-passage","occurrences":[1226,650,652,655,656]},{"id":"symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"introduction-passage","occurrences":[1227,1228]},{"id":"symbol-10c299c3911c0c85","tex":"b_s","aliases":"b_s","display":"b_s","kind":"introduction-passage","occurrences":[1226]},{"id":"symbol-d7d42b2ecde077f4","tex":"\\bar s","aliases":"bar s bar","display":"\\bar s","kind":"introduction-passage","occurrences":[3170,3174,3175]},{"id":"symbol-87909045a7273a12","tex":"\\bar s_C","aliases":"bar s_C bar","display":"\\bar s_C","kind":"introduction-passage","occurrences":[3170,3175]},{"id":"symbol-8aba1c70aa95f18d","tex":"\\bar s_C^2","aliases":"bar s_C^2 bar","display":"\\bar s_C^2","kind":"introduction-passage","occurrences":[3174]},{"id":"symbol-1dd2fc5c9ef2e084","tex":"\\mathrm{bright}","aliases":"bright mathrm","display":"\\mathrm{bright}","kind":"introduction-passage","occurrences":[325,326,321]},{"id":"symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","occurrences":[742,760,761,3167,3168,3169,3170,3174,95,100,512,513,581,582,585,711,715,732,738,739,751,878,879,881,1029,1031,1237,1238,1240,1241,1702,1705,1707,1723,1738,1778,1780,1899,1900,1902,1905,1912,1913,1914,1915,1933,1935,1960,1961,1962,2020,2021,2026,2027,2028,2044,2045,2048,2174,2348,2349,2350,2353,2359,2400,2401,2405,2406,2407,2531,2532,2579,2580,2582,2587,2637,2638,2705,2768,2771,2852,2863,3037,3038,3039,3041,3042,3043,3118,3121,3124,3126,3130,3134,3136,3137,3175,3178,3182,3183,3185,3188,3189,3190,3191,3194,3195,3197,3202,3204,3205,3209,3212,3213,3214,3216,3353,3397,3398,3489,3490,3491,3492,3501,3503,3505,3506,3507,3675,3678,3682,3683,3684,3689,3690,3691,3693,3702,3706,3707,3718,3735,3745,3747,3752,3753,3754,3755,3757,3761,3769,3773,3943]},{"id":"symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"introduction-passage","occurrences":[725,2500,2898,1,66,74,123,338,378,525,796,804,958,969,1032,1119,1141,1217,1247,1356,1365,1414,1441,1500,1983,2007,2034,2102,2152,2228,2482,2483,2494,2495,2533,2605,2632,2674,2677,2720,2721,2803,3019,3133,3149,3802]},{"id":"symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"introduction-passage","occurrences":[3165,3167,3168,3182,3344,3345,3347]},{"id":"symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","occurrences":[328,333,3841,21,22,24,36,38,225,228,229,231,235,240,242,245,249,252,304,305,307,310,311,316,318,319,321,322,323,337,371,388,389,391,403,594,595,596,659,660,661,662,670,677,861,862,969,970,972,973,978,987,1436,1437,1442,1443,1445,1446,1447,1515,1844,1846,1856,1857,1864,1865,1959,1996,2003,2067,2068,2155,2197,2198,2199,2200,2206,2207,2251,2306,2316,2317,2319,2323,2324,2353,2366,2370,2372,2374,2375,2376,2377,2379,2380,2384,2386,2390,2394,2395,2397,2398,2400,2406,2407,2523,2524,2525,2567,2568,2569,2693,2703,2706,2707,2974,2992,2993,3061,3062,3063,3067,3207,3209,3238,3302,3304,3305,3353,3354,3355,3360,3366,3371,3379,3381,3515,3517,3521,3522,3525,3531,3545,3548,3550,3582,3605,3607,3608,3609,3613,3616,3773,4040,4041,4096,4097,4116,4128,4129]},{"id":"symbol-da5f510477c944c3","tex":"\\C^4","aliases":"C^4 C","display":null,"kind":"introduction-passage","occurrences":[2898,1356,1365,1414,1441,1500,1983,2007,2677,3019,3133,3149]},{"id":"symbol-ace7df14d6591de3","tex":"\\C^{\\tau_p}","aliases":"C^τ_p C tau","display":null,"kind":"introduction-passage","occurrences":[725]},{"id":"symbol-c8691383f78796e0","tex":"\\mathcal C_C","aliases":"C_C mathcal","display":"\\mathcal C_C","kind":"introduction-passage","occurrences":[3167,3168,3182]},{"id":"symbol-6bb415dfb5cb7b14","tex":"\\mathcal C_F","aliases":"C_F mathcal","display":"\\mathcal C_F","kind":"introduction-passage","occurrences":[3165,3167,3168]},{"id":"symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"introduction-passage","occurrences":[760,761,711,715,751,3041,3043]},{"id":"symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"introduction-passage","occurrences":[326,658,663,847,860,1073,1074,1084,1085,1106,1107,1111,1114,1115,1127,1128,1130,1161,1164,1174,1191,1286,1294,1339,1343,1612,1710,1711,1712,1714,1844,1848,1849,1856,2036,2037,2093,2095,2096,2691,2708,3110,3652]},{"id":"symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","occurrences":[2290,2292,2294,134,135,136,143,146,150,152,153,167,173,181,189,190,218,221,257,258,273,274,451,864,865,866,869,870,873,874,877,878,886,894,895,896,897,901,904,907,908,919,923,925,926,929,930,933,936,1010,1012,1014,1015,1021,1577,1580,1581,1810,1812,1822,1835,1849,1850,1851,1914,2200,2306,2308,2309,2315,2324,2333,2352,2374,2376,2397,2403,2462,2467,2469,2474,2483,2484,2485,2488,2491,2492,2493,2496,2497,2499,2617,2618,2625,2772,2778,2979,3006,3038,3039,3041,3043,3045,3063,3064,3067,3082,3102,3153,3154,3155,3156,3157,3159,3161,3162,3163,3239,3245,3248,3296,3340,3346,3352,3484,3491,3499,3501,3503,3504,3505,3506,3507,3574,3668,3669,3674,3675,3677,3678,3722,3725,3754,3756,3757,3769,3792,3793,3795,3796,3797,3798,3799,3800,3801,3804,3805,3807,3809,3812,3816,3817,3818,3819,3821,3823,3826,3934,3938,3940,3943,3945,3947,3950,3951,3952,3954,3962,3964,3966,3969,3972,3973,3978,3979,3981,3984,4118,4119,4120,4121]},{"id":"symbol-62bae077a0709320","tex":"\\mathrm D","aliases":"D mathrm","display":"\\mathrm D","kind":"introduction-passage","occurrences":[3411,3412,3278,3279,3344,3345,3387,3393,3429,3430,3432,3434,3438,3439,3459]},{"id":"symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","occurrences":[159,407,726,727,763,765,767,1051,1052,1053,2100,3844,41,42,160,165,170,184,186,187,188,197,203,204,206,207,208,209,211,212,213,214,379,391,392,393,427,463,464,468,469,472,473,685,751,753,925,926,928,934,970,972,978,1060,1160,1164,1174,1190,1271,1282,1296,1324,1517,1518,1526,1671,1710,1714,1875,1884,1916,1917,1918,1920,2019,2021,2184,2200,2279,2375,2383,2384,2385,2389,2398,2698,2726,2750,2772,2775,2780,3070,3074,3097,3121,3124,3126,3144,3366,3486,3487,3489,3492,3498,3511,3541,3553,3554,3556,3558,3561,3573,3574,3576,3582,3586,3607,3613,3621,3634,3639,3643,3663,3680,3681,3682,3683,3700,3745,3850,3868,3873,3926,3927,3928,3930,3952,3954,3958,4052,4092]},{"id":"symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"introduction-passage","occurrences":[1052,203,206,463,3700]},{"id":"symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"introduction-passage","occurrences":[1053,203]},{"id":"symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"introduction-passage","occurrences":[1051,1060,2384]},{"id":"symbol-4144475705f1d2e6","tex":"d_{n-1}","aliases":"d_n-1","display":"d_{n-1}","kind":"introduction-passage","occurrences":[1051]},{"id":"symbol-30c3820e382d42d4","tex":"d_{n-2}","aliases":"d_n-2","display":"d_{n-2}","kind":"introduction-passage","occurrences":[1051]},{"id":"symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"introduction-passage","occurrences":[159,160]},{"id":"symbol-d3b6e2eaef5fb02a","tex":"d_v","aliases":"d_v","display":"d_v","kind":"introduction-passage","occurrences":[726]},{"id":"symbol-052240bfc014845d","tex":"d_v^{-1/2}","aliases":"d_v^-1/2","display":"d_v^{-1/2}","kind":"introduction-passage","occurrences":[727]},{"id":"symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"introduction-passage","occurrences":[1047,873,894,1015,1095,3237,3238,3277,3362,3366,3370]},{"id":"symbol-35679b068dabbae2","tex":"\\dot P","aliases":"dot P dot","display":"\\dot P","kind":"introduction-passage","occurrences":[2100,3670,3674,3894,4076]},{"id":"symbol-e4b951c58979abc7","tex":"\\dot P_a","aliases":"dot P_a dot","display":"\\dot P_a","kind":"introduction-passage","occurrences":[2100]},{"id":"symbol-d1997e4b5359f675","tex":"\\dot U","aliases":"dot U dot","display":"\\dot U","kind":"introduction-passage","occurrences":[2099,2073,2083,2086,2087,2101,3674,3723]},{"id":"symbol-913b6583a9281098","tex":"\\dot X","aliases":"dot X dot","display":"\\dot X","kind":"introduction-passage","occurrences":[2100]},{"id":"symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","occurrences":[333,1225,1231,110,111,112,113,119,129,130,133,134,140,141,143,145,147,149,152,153,189,190,272,275,319,321,322,323,337,781,847,874,876,877,885,886,897,898,899,907,908,910,911,919,921,1016,1018,1216,1232,1235,1236,1243,1248,1249,1259,1261,1262,1278,1280,1282,1296,1298,1318,1330,1537,1591,1592,1602,1605,1606,1611,1612,1613,1616,1617,1618,1619,1625,1629,1631,1636,1641,1642,1649,1651,1653,1654,1656,1700,1738,1741,1743,1747,1748,1749,1750,1752,1754,1755,1773,1774,1777,1778,1780,1827,1828,1843,1844,1845,1856,1938,1939,2557,2558,2566,2722,2725,2726,2728,2730,2734,2751,2756,2758,2765,3232,3237,3238,3240,3243,3257,3258,3290,3291,3292,3337,3341,3345,3355,3359,3360,3361,3363,3367,3370,3374,3399,3527,3773]},{"id":"symbol-5a3fc062502fca22","tex":"\\mathsf E","aliases":"E mathsf","display":"\\mathsf E","kind":"introduction-passage","occurrences":[407,411,414,415,419,422,438,444,447,448,453,455,456,458,460,479,481,482,487,488,509,510,511,542,547,556,558,560,568,577,606,607,612,622,623,634,782,783]},{"id":"symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","occurrences":[326,766,1378,1811,2901,2904,2905,3844,3849,210,211,212,249,255,274,276,286,383,384,385,388,392,528,531,535,542,552,553,558,591,592,593,623,656,677,684,685,687,694,696,770,771,814,829,840,841,844,853,854,855,925,926,928,929,932,934,936,959,966,1026,1063,1064,1083,1205,1215,1240,1241,1243,1249,1251,1252,1283,1284,1291,1296,1297,1300,1323,1329,1330,1335,1339,1342,1355,1413,1417,1522,1530,1566,1567,1568,1577,1578,1579,1580,1603,1612,1651,1654,1665,1671,1673,1678,1684,1706,1707,1755,1756,1772,1778,1780,1795,1812,1842,1846,1847,1850,1855,1871,1873,1874,1875,1878,1880,1881,1882,1883,1884,1885,1886,1940,1941,2091,2137,2141,2144,2148,2149,2151,2155,2168,2171,2300,2301,2304,2311,2316,2329,2366,2377,2378,2384,2392,2432,2433,2434,2435,2444,2449,2454,2455,2486,2487,2488,2491,2492,2690,2694,2697,2723,2758,2790,2792,2799,2800,2802,2845,2911,2913,2922,2929,2974,3006,3070,3074,3075,3076,3080,3081,3089,3091,3095,3097,3220,3260,3261,3267,3272,3279,3286,3287,3290,3291,3294,3330,3337,3370,3371,3381,3382,3385,3387,3388,3389,3390,3391,3392,3394,3397,3398,3399,3401,3413,3414,3415,3416,3417,3500,3505,3506,3525,3527,3539,3541,3553,3556,3557,3558,3563,3565,3566,3569,3570,3572,3575,3577,3582,3604,3605,3609,3620,3621,3623,3625,3627,3631,3637,3652,3653,3674,3675,3678,3712,3713,3715,3716,3725,3803,3805,3807,3813,3814,3815,3850,3875,3897,3899,3972,3974,3975,3977,4040,4116,4128,4129,4131]},{"id":"symbol-cc613c8c38547509","tex":"e^{-\\mathrm{i}tH_{\\mathrm{bright}}}","aliases":"e^-itH_bright mathrm mathrm","display":"e^{-\\mathrm{i}tH_{\\mathrm{bright}}}","kind":"introduction-passage","occurrences":[326]},{"id":"symbol-7abee99367ff24ec","tex":"e^{-itH_{h,n}}","aliases":"e^-itH_h,n","display":"e^{-itH_{h,n}}","kind":"introduction-passage","occurrences":[1378]},{"id":"symbol-41db5006fbb2b3f4","tex":"e^{-itH_n}","aliases":"e^-itH_n","display":"e^{-itH_n}","kind":"introduction-passage","occurrences":[1378]},{"id":"symbol-acc086f034dde581","tex":"e^{-\\mathrm{i}t\\overline E}","aliases":"e^-itoverline E mathrm overline","display":"e^{-\\mathrm{i}t\\overline E}","kind":"introduction-passage","occurrences":[326]},{"id":"symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"introduction-passage","occurrences":[1811,841,1240,1241,1812,1842,1846,1847,1850,1855,1940,1941,2144,2148,3716,3813]},{"id":"symbol-58f890b6f450ab6a","tex":"E_{3,1}","aliases":"E_3,1","display":"E_{3,1}","kind":"introduction-passage","occurrences":[1231]},{"id":"symbol-ad3caf981ac30812","tex":"E_c","aliases":"E_c","display":"E_c","kind":"introduction-passage","occurrences":[333,319,321,322,323,337]},{"id":"symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"introduction-passage","occurrences":[766,770,771,1577,1578,2432,2433,2434,2435,3525,3527,3539,3541,3975]},{"id":"symbol-bb785fdcd708a453","tex":"E_{L,j}","aliases":"E_L,j","display":"E_{L,j}","kind":"introduction-passage","occurrences":[1225,1216,1243,1248,1249,1259,1280,1318,1330]},{"id":"symbol-d8c9f25b4e1a222a","tex":"\\mathsf E_m","aliases":"E_m mathsf","display":"\\mathsf E_m","kind":"introduction-passage","occurrences":[407,411,414,415,419,422,438,444,448,455,456,458,782,783]},{"id":"symbol-d4c39e5e1f0a0028","tex":"e_x","aliases":"e_x","display":"e_x","kind":"introduction-passage","occurrences":[2901,2904,2905,2911,2913,2922,2929,3095]},{"id":"symbol-28d5a6a46f63be54","tex":"e_{\\tau x}","aliases":"e_τ x tau","display":"e_{\\tau x}","kind":"introduction-passage","occurrences":[2904,2905]},{"id":"symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"introduction-passage","occurrences":[155,77,88,94,467,2772,2780,3113,3557,3558,3577]},{"id":"symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"introduction-passage","occurrences":[1231,78,3039]},{"id":"symbol-f43c89d5480f3fa1","tex":"F_{L-1}","aliases":"F_L-1","display":"F_{L-1}","kind":"introduction-passage","occurrences":[1225,1249]},{"id":"symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"introduction-passage","occurrences":[1226]},{"id":"symbol-5ac97b405fd2d8eb","tex":"F_{u\\to v}^n","aliases":"F_u→ v^n to","display":"F_{u\\to v}^n","kind":"introduction-passage","occurrences":[745,746,748]},{"id":"symbol-5b2cee62bbda2690","tex":"F_{v\\to u}^n","aliases":"F_v→ u^n to","display":"F_{v\\to u}^n","kind":"introduction-passage","occurrences":[743,744,746,748]},{"id":"symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","occurrences":[324,329,332,334,728,3841,123,309,310,321,336,341,348,349,351,352,732,2250,2647,2671,2853,2857,3151,3239,3240,3245,3248,3348,3349,3353,3426,3428,3429,3430,3437,3438,3460,3461,3572,3575,3579,3582,3585,3586,3590,3593,3595,3598,3605,3606,3610,3613,3616,3621,3627,3628,3629,3631,3634,3637,3639,3641,3643,3645,3647,3656,3679,3682,3721,3725,3766,3836,4041]},{"id":"symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","occurrences":[2892,188,338,339,341,354,355,357,358,360,361,364,368,370,372,373,379,418,422,434,435,437,438,500,501,503,505,506,624,625,627,642,644,654,655,656,658,663,665,999,1000,1003,1007,1008,1307,1504,1508,1509,1511,1512,1526,2035,2036,2037,2124,2126,2484,2499,2579,2582,2589,2607,2608,2609,2611,2612,2618,2619,2621,2622,2623,2643,2644,2647,2656,2670,2671,2682,2686,2687,2688,2805,2807,2808,2813,2874,2875,2885,2914,2916,2917,2918,3309,3310,3311,3312,3313,3314,3315,3678,3679,3680,3692,3695,3696,3697,3709,3717,3793,3822,3823,3962,3964,3971,3972,3975,3976,3977]},{"id":"symbol-81a99880d3bfaeb5","tex":"G_{p,w}","aliases":"G_p,w","display":"G_{p,w}","kind":"introduction-passage","occurrences":[324,329,332,334,309,310,321,336,341,348,349,351,352]},{"id":"symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"introduction-passage","occurrences":[728,732]},{"id":"symbol-6ec2d03508e15ce2","tex":"H_{\\mathrm{bright}}","aliases":"H_bright mathrm","display":"H_{\\mathrm{bright}}","kind":"introduction-passage","occurrences":[325,326,321]},{"id":"symbol-51a7901d5bb6ee54","tex":"H_{h,n}","aliases":"H_h,n","display":"H_{h,n}","kind":"introduction-passage","occurrences":[1378,1572,1582,1589,1640,1684,1687,1694,1696,1701,1761,1766,1773,1775,1778,1899,1903,1905,1928,1935]},{"id":"symbol-5e246b7f4f9bbfa9","tex":"H_{h,S}","aliases":"H_h,S","display":"H_{h,S}","kind":"introduction-passage","occurrences":[1924]},{"id":"symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"introduction-passage","occurrences":[1378,1425,1426,1428,1432,1442,1443,1445,1446,1448,1454,1455,1467,1469,1564,1568,1585,1589,1684,1687,1701,1761,1766,1773,1776,1778,1899,1904,1905,1925,1927,1929,1935]},{"id":"symbol-2e77d5fad40985b0","tex":"H_S","aliases":"H_S","display":"H_S","kind":"introduction-passage","occurrences":[1924]},{"id":"symbol-32945ba3eade4c63","tex":"\\HH_{\\mathrm{arc}}","aliases":"HH_arc HH mathrm","display":null,"kind":"introduction-passage","occurrences":[725]},{"id":"symbol-45418eea5d6e493e","tex":"\\HH_p","aliases":"HH_p HH","display":null,"kind":"introduction-passage","occurrences":[406,378,396,399,417,440,461,476,477,494,545,546,557]},{"id":"symbol-510e2db8852c553e","tex":"\\mathrm{i}","aliases":"i mathrm","display":"\\mathrm{i}","kind":"introduction-passage","occurrences":[155,326,77,88,94,249,274,276,286]},{"id":"symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"introduction-passage","occurrences":[3169,3170,3172,3173,3185,3187,3188,3189,3194,3196,3200,3216,3332,3333,3384]},{"id":"symbol-8a8b98766d675837","tex":"\\mathrm{id}_C","aliases":"id_C mathrm","display":"\\mathrm{id}_C","kind":"introduction-passage","occurrences":[3169,3185,3189,3194]},{"id":"symbol-71b9c5b82b8f477d","tex":"\\mathrm{id}_F","aliases":"id_F mathrm","display":"\\mathrm{id}_F","kind":"introduction-passage","occurrences":[3170,3173,3185,3187,3188,3196,3200,3216]},{"id":"symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"introduction-passage","occurrences":[1047,1048,1043]},{"id":"symbol-3ab3f094fe0f10d8","tex":"K_{n-1}","aliases":"K_n-1","display":"K_{n-1}","kind":"introduction-passage","occurrences":[1055]},{"id":"symbol-309607a909e10c97","tex":"\\KK","aliases":"KK","display":null,"kind":"introduction-passage","occurrences":[1380,602,603,604,606,610,611,615,616,626,1356,1363,1367,1372,1376,1399,1412,1414,1415,1421,1426,1430,1447,1453,1456,1471,1474,2253,2254,2256,2262,2265,2273]},{"id":"symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"introduction-passage","occurrences":[409]},{"id":"symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"introduction-passage","occurrences":[408,378,416,426,490,495,537,576,603,773,774,1168,1176,1289,1292,1345,1414,1416,1419,1488,1500,1514,1526,1527,1664,1668,1670,1671,1691,1692,1694,1696,1706,1707,1764,1765,1766,1780,1783,1859,1866,1900,1902,1921,1922,1923,2602,3131,3137,3139,3146,3147,3149,3613,3961,3988,4032]},{"id":"symbol-318bb55567cfc011","tex":"L^L","aliases":"L^L","display":"L^L","kind":"introduction-passage","occurrences":[1225,1249]},{"id":"symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","occurrences":[2500,72,73,74,76,78,79,207,209,434,436,442,443,444,445,451,452,453,478,479,481,482,483,484,485,486,487,488,489,492,509,510,511,532,533,538,540,542,543,544,547,550,552,553,556,558,560,561,563,568,573,577,587,638,639,640,641,643,644,646,775,793,794,796,797,798,812,829,830,843,849,858,859,1029,1032,1033,1062,1065,1095,1114,1115,1141,1142,1143,1144,1147,1149,1150,1151,1152,1154,1155,1157,1158,1244,1245,1732,1733,1958,1959,2119,2120,2188,2191,2192,2193,2938,2940,2999,3049,3050,3251,3335,3484,3582,3588,3589,3590,3591,3597,3598,3604,3605,3606,3610,3613,3616,3627,3629,3631,3634,3637,3639,3641,3643,3645,3656,3713,3715,3721,3725,3863,3866,3867,3869,3875,3880,3882,3883,3884,3889,3890,3895,3896,3901,3902,3903,3906,3907,3908,3910,3912,3914,3917,3918,3926,3931,3935,3939,3940,3943,3946,3948,3949,3950,3951,3952,3953,3954,3955,3956,3957,3958,3962,3976,3991,4141]},{"id":"symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"introduction-passage","occurrences":[2500]},{"id":"symbol-6e16a4907811a989","tex":"m_x","aliases":"m_x","display":"m_x","kind":"introduction-passage","occurrences":[2902,2904,2906,2912,2913,2925,2929]},{"id":"symbol-a6a28ba806797f76","tex":"m_{\\tau x}","aliases":"m_τ x tau","display":"m_{\\tau x}","kind":"introduction-passage","occurrences":[2906]},{"id":"symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","occurrences":[759,760,761,2500,3839,3840,3844,3847,3848,81,83,88,94,195,709,711,715,717,720,738,739,751,770,771,772,794,796,799,804,827,860,863,1120,1164,1802,1925,1932,2301,2302,2303,2304,2316,2317,2395,2398,2405,2407,2408,2409,2410,2413,2416,2418,2420,2421,2422,2424,2428,2430,2434,2436,2437,2440,2441,2443,2444,2445,2448,2453,2455,2459,2460,2461,2462,2482,2483,2494,2533,2630,2631,2715,2734,2937,2938,2939,2940,2941,2942,3040,3041,3043,3160,3161,3163,3224,3225,3227,3229,3230,3239,3240,3241,3244,3245,3247,3252,3254,3329,3369,3418,3419,3508,3510,3511,3512,3513,3517,3518,3667,3802,3810,3829,3850,3853,3874,3887,3888,3893,3894,3898,3899,3963,3997,4023,4027,4035,4039,4048,4137,4150,4163,4168]},{"id":"symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","occurrences":[740,742,743,744,745,746,1046,1047,1048,1049,1051,1054,1055,1378,1379,34,35,36,37,77,82,84,85,88,94,299,300,301,309,413,414,453,454,455,457,460,748,751,754,755,756,757,815,827,1034,1035,1036,1039,1040,1043,1056,1058,1059,1060,1097,1098,1101,1117,1118,1120,1123,1125,1126,1128,1131,1134,1135,1144,1146,1148,1152,1153,1155,1169,1170,1178,1180,1181,1183,1184,1185,1206,1211,1212,1213,1241,1254,1263,1273,1274,1322,1324,1329,1350,1351,1356,1357,1360,1361,1362,1364,1365,1366,1368,1369,1373,1374,1413,1417,1423,1424,1425,1426,1428,1432,1438,1440,1442,1443,1445,1446,1448,1454,1455,1467,1468,1469,1470,1472,1474,1476,1479,1481,1483,1491,1493,1494,1495,1501,1502,1520,1521,1522,1529,1530,1531,1563,1564,1566,1567,1568,1571,1572,1579,1580,1582,1585,1589,1591,1592,1640,1641,1642,1664,1665,1666,1668,1669,1670,1671,1680,1681,1682,1684,1687,1691,1694,1696,1700,1701,1702,1704,1705,1706,1707,1719,1721,1723,1724,1725,1726,1727,1736,1737,1738,1741,1742,1743,1747,1748,1749,1750,1751,1752,1754,1755,1760,1761,1764,1766,1773,1774,1775,1776,1777,1778,1780,1810,1812,1814,1815,1816,1819,1822,1827,1828,1829,1832,1835,1862,1863,1898,1899,1900,1902,1903,1904,1905,1907,1912,1913,1914,1915,1925,1927,1928,1929,1933,1935,2102,2152,2300,2305,2306,2311,2312,2316,2317,2318,2319,2322,2323,2325,2326,2327,2328,2329,2330,2331,2360,2361,2362,2365,2366,2377,2379,2380,2384,2386,2391,2392,2399,2400,2401,2406,2407,2448,2450,2453,2454,2458,2460,2461,2464,2632,2633,2634,2635,2636,2637,2638,2643,2644,2646,2647,2648,2649,2650,2652,2653,2654,2656,3006,3117,3118,3119,3120,3121,3122,3124,3125,3126,3127,3129,3130,3140,3141,3142,3143,3145,3148,3149,3235,3239,3243,3245,3420,3474,3480,3481,3484,3485,3488,3496,3582,3605,3649,3652,3653,3655,3685,3689,3690,3691,3693,3712,3713,3714,3743,3744,3745,3746,3748,3751,3753,3755,3757,3762,3765,3769,3773,3774,3912,3922]},{"id":"symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"introduction-passage","occurrences":[3844,2938,2940]},{"id":"symbol-ebcd9b4bdca5cfcd","tex":"N^r","aliases":"N^r","display":"N^r","kind":"introduction-passage","occurrences":[3844,3848,3850,3853,3874,3887,3888,3894,3963,4048]},{"id":"symbol-0ff3814053977f1f","tex":"\\overline E","aliases":"overline E overline","display":"\\overline E","kind":"introduction-passage","occurrences":[325,326,322]},{"id":"symbol-a71d099153a6ba75","tex":"p^{-(h-r+1)/2}","aliases":"p^-(h-r+1)/2","display":"p^{-(h-r+1)/2}","kind":"introduction-passage","occurrences":[159]},{"id":"symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"introduction-passage","occurrences":[407,377,382,384,385,392,393,466]},{"id":"symbol-da8a2616906ad75b","tex":"p^{r-1}","aliases":"p^r-1","display":"p^{r-1}","kind":"introduction-passage","occurrences":[157,159,136,148,161,171,173,179,276,983,989,991,995,997]},{"id":"symbol-95c96988a1a8fc97","tex":"p^w","aliases":"p^w","display":"p^w","kind":"introduction-passage","occurrences":[335,299,355]},{"id":"symbol-a825e050f8755dad","tex":"P_+","aliases":"P_+","display":"P_+","kind":"introduction-passage","occurrences":[2900,1606,1609,1610,1612,1622,1628,1629,1651,1654,2921,2926,3015,3025,3029]},{"id":"symbol-cec3c913d3e4fd13","tex":"P_-","aliases":"P_-","display":"P_-","kind":"introduction-passage","occurrences":[2900,1606,1609,1610,1612,1622,1628,1629,1651,1654,1937,2921,2926,3015,3025,3029,3334]},{"id":"symbol-4f0aaf3fe5bb5cd6","tex":"P_\\pm","aliases":"P_± pm","display":"P_\\pm","kind":"introduction-passage","occurrences":[2899,1607,1608,1611,1621,1625,1627,1650,3333]},{"id":"symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","occurrences":[159,12,67,827,1033,1213,1310,1349,3664]},{"id":"symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"introduction-passage","occurrences":[329,335,368]},{"id":"symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","occurrences":[158,159,1227,1228,165,166,169,170,173,174,222,223,468,469,472,565,566,567,568,569,570,571,572,573,574,575,609,845,847,850,852,853,854,855,856,858,859,1106,1155,1210,1216,1218,1221,1224,1245,1266,1268,1306,1307,1308,1310,1316,1570,1571,1572,1573,1592,1701,1736,1738,1742,1815,1863,1864,1865,2174,2175,2176,2187,2188,2189,2190,2191,2192,2193,2458,2461,2698,2699,2706,2707,2753,2754,2755,2812,2813,2915,2916,2918,2977,2978,2979,2980,2981,3057,3058,3133,3135,3138,3275,3276,3277,3298,3305,3346,3359,3362,3366,3381,3405,3407,3442,3444,3445,3446,3447,3449,3453,3456,3661,3791,3793,3794,3795,3796,3797,3799,3801,3813,3816,3818,3824,3825,3826,3836]},{"id":"symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","occurrences":[156,157,159,1227,1228,3844,3848,124,125,126,128,129,130,131,132,133,134,135,136,137,139,140,141,142,143,144,145,147,148,150,152,153,160,161,163,164,167,168,171,172,173,179,180,181,182,190,197,220,221,256,257,258,259,262,271,272,273,274,275,276,611,720,781,795,796,797,798,799,801,802,805,806,807,810,812,819,820,821,822,823,824,825,826,835,836,837,841,851,853,943,944,945,946,947,948,949,950,951,952,953,954,955,957,959,960,966,981,982,983,984,989,991,993,995,996,997,998,999,1000,1002,1005,1007,1008,1087,1088,1090,1221,1222,1223,1308,1311,1316,1349,1451,1452,1455,1525,1672,1673,1676,1677,1678,1731,1733,1768,1769,1786,1788,1841,1842,1843,1844,1845,1846,1856,1858,1862,1864,1865,1889,2155,2165,2167,2168,2249,2250,2251,2565,2567,2569,2570,2572,2573,2574,2575,2576,2605,2780,2973,2974,2975,2992,2993,2994,2997,2998,3214,3216,3459,3460,3461,3462,3540,3541,3580,3581,3582,3583,3584,3585,3586,3587,3590,3593,3595,3601,3602,3603,3604,3605,3613,3660,3661,3662,3664,3723,3724,3733,3734,3735,3737,3740,3742,3748,3749,3759,3775,3850,3853,3870,3871,3873,3874,3875,3878,3884,3887,3888,3891,3892,3894,3896,3897,3899,3903,3905,3912,3914,3915,3920,3921,3922,3925,3927,3928,3963,3990,4040,4046,4048,4051,4052,4053,4058,4059,4061,4062,4063,4068,4072,4074,4075,4077,4078,4079,4080,4082,4087,4088,4089,4090,4091,4092,4095,4108,4109,4111,4112,4115,4116,4117,4124,4128,4129,4130,4131,4132,4135,4136,4137,4140,4149,4150,4151,4152,4163,4168]},{"id":"symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"introduction-passage","occurrences":[3844,2259,3852,3860,3868,3869,3888,3890,3894,3898,3899,3901,3904,3949,3976,4044]},{"id":"symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"introduction-passage","occurrences":[2835,2837,3938,3939,3940,3943,3952,3962,3975,3979]},{"id":"symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"introduction-passage","occurrences":[1092,1062,1064,1076,1104,1106,1111,1112,1166,1167,1174,1189]},{"id":"symbol-76599921b0fd3fb7","tex":"R_{p,w}","aliases":"R_p,w","display":"R_{p,w}","kind":"introduction-passage","occurrences":[329,335,300,310,311,313,318,368]},{"id":"symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"introduction-passage","occurrences":[3412,3403,3405]},{"id":"symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","occurrences":[1924,2286,2287,2289,2294,2900,264,267,268,663,733,736,737,738,739,827,828,834,861,1141,1358,1359,1361,1362,1383,1393,1394,1395,1398,1404,1405,1409,1460,1464,1465,1528,1530,1531,1532,1533,1670,1684,1685,1686,1689,1691,1693,1694,1696,1697,1706,1707,1762,1766,1780,1781,1810,1900,1902,1906,1915,1925,1930,1931,1933,2054,2055,2185,2295,2296,2316,2317,2339,2346,2395,2398,2407,2410,2413,2416,2418,2420,2421,2422,2424,2428,2430,2434,2436,2437,2440,2443,2445,2448,2459,2462,2511,2665,2666,2669,2671,2676,2685,2686,2688,2689,2698,2804,2805,2818,2819,2853,2887,2920,2940,2942,2998,3006,3015,3019,3020,3023,3024,3025,3224,3230,3243,3245,3248,3278,3279,3294,3319,3321,3323,3324,3326,3366,3481,3483,3489,3490,3500,3501,3502,3505,3506,3586,3590,3592,3613,3621,3745,3850,3972,3977]},{"id":"symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","occurrences":[727,728,730,1226,1382,3165,3166,3170,3174,3844,12,64,95,133,136,137,138,140,141,144,145,475,476,480,481,484,530,549,550,685,687,698,798,805,806,810,812,819,820,822,824,825,925,926,935,971,972,976,979,985,1349,1481,1483,1484,1485,1486,1487,1489,1644,1645,1651,1655,1756,1845,1846,1856,1857,1864,1865,1872,1873,1875,1880,1881,1884,2241,2247,2323,2324,2331,2368,2390,2720,2721,2772,2775,2780,2943,2992,2993,2996,2997,3097,3178,3179,3180,3182,3183,3185,3186,3188,3190,3191,3195,3196,3197,3198,3202,3204,3205,3211,3216,3303,3304,3305,3310,3311,3312,3313,3314,3319,3321,3322,3325,3424,3511,3525,3526,3528,3536,3540,3541,3553,3582,3607,3609,3643,3868,4042,4052]},{"id":"symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"introduction-passage","occurrences":[3844,1857,3313,3322,3325,3511,3553,3582,3607,3868,4052]},{"id":"symbol-8fb5b936111102da","tex":"S_{A,\\tau,u}","aliases":"S_A,τ,u tau","display":"S_{A,\\tau,u}","kind":"introduction-passage","occurrences":[2900,2665,2671,2920,2940,3006,3024]},{"id":"symbol-a9e11db6243bd912","tex":"S_F","aliases":"S_F","display":"S_F","kind":"introduction-passage","occurrences":[2286,2287,2289,2294,2295,2296,2339,2346,3481,3483,3502,3505,3506]},{"id":"symbol-26736bc4d2f28e27","tex":"s_F","aliases":"s_F","display":"s_F","kind":"introduction-passage","occurrences":[3165,3170,3174,3179,3211]},{"id":"symbol-a8c7172f7f354bea","tex":"s_F^2","aliases":"s_F^2","display":"s_F^2","kind":"introduction-passage","occurrences":[3166]},{"id":"symbol-1d641eb69668a366","tex":"s_v","aliases":"s_v","display":"s_v","kind":"introduction-passage","occurrences":[727,728,730]},{"id":"symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"introduction-passage","occurrences":[329,334,291,1712,1856,2036,2097,3786,3990]},{"id":"symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"introduction-passage","occurrences":[760,761,1806,715,1559,1599,1639,1662,1683,1698,1728,1782,1896,1911,1936,2698,3558,3569,3578]},{"id":"symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"introduction-passage","occurrences":[1804,1793,1795,1798,1800,1801,1802,1803]},{"id":"symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","occurrences":[3844,286,368,694,699,701,702,3511,3553,3568,3607,4052]},{"id":"symbol-1e83927f5099f8d9","tex":"T_{p,h}","aliases":"T_p,h","display":"T_{p,h}","kind":"introduction-passage","occurrences":[760,712,715,940,958,980]},{"id":"symbol-b1dcc31cf92718a0","tex":"T_{p,h+1}","aliases":"T_p,h+1","display":"T_{p,h+1}","kind":"introduction-passage","occurrences":[761]},{"id":"symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"introduction-passage","occurrences":[2504,103,593,2104,2105,2107,2108,2110,2113,2114,2119,2135,2136,2510,2511,2686,2814,3151,3153,3160,3162,3163,3834]},{"id":"symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","occurrences":[740,743,744,745,746,2889,2892,2893,2900,21,23,75,369,523,524,564,566,583,663,696,734,735,748,876,877,880,881,882,884,888,909,911,915,916,919,920,922,923,924,925,926,938,967,972,973,977,986,992,1015,1017,1021,1023,1174,1198,1200,1709,1710,1712,1714,1846,1847,1848,1849,1850,1851,1852,1853,1856,1865,1866,1878,1885,1886,1889,2196,2199,2200,2202,2205,2208,2378,2612,2614,2629,2664,2665,2666,2670,2671,2910,2920,2923,2924,2926,2927,2940,2974,2996,2997,3005,3006,3011,3023,3024,3025,3026,3031,3048,3051,3052,3063,3065,3066,3067,3068,3070,3075,3295,3297,3298,3301,3353,3405,3406,3407,3584,3585,3586,3590,3591,3593,3594,3595,3600,3601,3912,3922,3934,3964,3967,3969,3973,3984,4118]},{"id":"symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"introduction-passage","occurrences":[2099,812,2071,2072,2079,2082,2083,2085,2086,2087,2089,2101]},{"id":"symbol-35ecd39772b3647e","tex":"u^g","aliases":"u^g","display":"u^g","kind":"introduction-passage","occurrences":[2892,2670,2671]},{"id":"symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"introduction-passage","occurrences":[1047,1048,1054,1036,1039,1097]},{"id":"symbol-f221f487fed33512","tex":"U_{n-1}","aliases":"U_n-1","display":"U_{n-1}","kind":"introduction-passage","occurrences":[1048,1039,1097,1118,1131,1134,1144,1153]},{"id":"symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","occurrences":[1378,1379,2099,2100,184,187,188,189,190,192,218,225,226,229,231,234,235,238,239,240,242,243,245,249,251,260,267,392,394,439,443,444,609,666,667,670,677,682,776,777,778,780,1323,1324,1328,1365,1366,1368,1370,1374,1375,1377,1384,1388,1389,1391,1945,1951,2254,2255,2256,2257,2258,2259,2261,2264,2266,2268,2269,2270,2271,2274,2276,2278,2279,2280,2604,2607,2624,2625,2663,2664,3239,3245,3248,3300,3456,3457,3671,3673,3674,3675,3678,3679,3680,3687,3689,3692,3717,3727,3732]},{"id":"symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","occurrences":[726,727,728,730,740,741,743,744,745,746,1804,16,54,277,716,731,732,734,735,748,750,852,854,856,877,880,881,887,909,919,922,923,924,925,926,929,936,938,949,1021,1024,1025,1237,1238,1245,1440,1441,1442,1443,1444,1445,1469,1481,1482,1524,1525,1544,1647,1659,1660,1661,1665,1667,1668,1671,1681,1710,1711,1726,1786,1788,1789,1790,1948,1949,2566,2567,2576,2867,2868,2911,2912,2920,2940,2942,2973,2974,2975,2994,2997,3003,3004,3005,3006,3007,3012,3023,3024,3025,3026,3047,3051,3052,3063,3065,3067,3132,3135,3136,3137,3138,3301,3516,3517,3521,3522,3525,3596,3597,3598,3599,3694,3695,3696,3697,3867,3868,3874,3887,3890,4048,4051,4052,4055,4059,4061,4062,4063,4065,4068,4069,4070,4071,4074,4076,4077,4078,4086,4088,4089,4090,4095,4097,4098,4099,4100,4101,4102,4103,4104,4107,4108,4112,4113,4114,4122,4123,4125,4126,4162,4173]},{"id":"symbol-a0402d5bad7ad318","tex":"V_{h,n}","aliases":"V_h,n","display":"V_{h,n}","kind":"introduction-passage","occurrences":[1378,1379]},{"id":"symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"introduction-passage","occurrences":[1378,1379,1365,1366,1368,1374]},{"id":"symbol-bb5783d1e38c64af","tex":"\\vec E","aliases":"vec E vec","display":"\\vec E","kind":"introduction-passage","occurrences":[721,725]},{"id":"symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","occurrences":[2831,2832,2835,1369,1371,1385,1388,1389,1392,2678,2682,2686,2688,2692,2693,2694,2697,2707,2817,2836,2845,2849,2851,2858,2860,3052,3063,3066,3067,3068,3558,3569,3578,3960,3989]},{"id":"symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","occurrences":[324,329,332,334,335,1226,1227,1228,1229,191,192,193,298,299,300,302,303,306,307,308,309,310,311,313,318,321,336,337,339,341,342,343,345,348,349,351,352,354,355,360,361,363,365,366,368,370,372,373,675,676,703,704,705,708,821,823,826,965,966,972,1154,1155,1494,1495,1524,1525,1647,1659,1660,1661,1667,1668,1671,1681,2387,2388,2389,2867,2869,3514,3517,3521,3522,3523,3525,3544,3723,3724,3967,4093,4094,4095,4096,4101,4103,4129,4130]},{"id":"symbol-65d03882bd3591c3","tex":"w^q","aliases":"w^q","display":"w^q","kind":"introduction-passage","occurrences":[1227]},{"id":"symbol-6b00da79051237b2","tex":"W_{0j}","aliases":"W_0j","display":"W_{0j}","kind":"introduction-passage","occurrences":[2835]},{"id":"symbol-107faad68a0eb5b2","tex":"W_{1j}","aliases":"W_1j","display":"W_{1j}","kind":"introduction-passage","occurrences":[2835]},{"id":"symbol-f504f709b964471b","tex":"W_{ij}","aliases":"W_ij","display":"W_{ij}","kind":"introduction-passage","occurrences":[2831,2832]},{"id":"symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","occurrences":[2100,2503,2504,2821,2831,2835,0,1,803,805,806,808,1767,1768,1769,1770,1771,1772,1775,1990,1991,1992,1995,1996,1998,2000,2003,2005,2011,2012,2267,2268,2269,2270,2412,2413,2414,2415,2416,2417,2418,2419,2420,2421,2424,2425,2427,2433,2434,2435,2436,2438,2442,2505,2506,2510,2511,2513,2514,2517,2518,2519,2521,2524,2526,2529,2530,2532,2836,2837,2838,2840,2841,2842,2957,2958,2959,2960,2961,2962,2963,2966,2967,2968,2969,2970,2971,2983,2984,2999,3035,3037,3050,3053,3059,3334,3729,3782,3783,3831,3832,3833,3834]},{"id":"symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"introduction-passage","occurrences":[2504,1996,1998,2005,2438,2511,3833]},{"id":"symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"introduction-passage","occurrences":[2289,2293,1081,1160,3511,3553,3568,3610,3650,3651,3654,3658]},{"id":"symbol-b16b8b5c9ec900ff","tex":"X_{0j}","aliases":"X_0j","display":"X_{0j}","kind":"introduction-passage","occurrences":[2835,2837,2838]},{"id":"symbol-f26f4a25faf10ece","tex":"X_{0j}^{-1}","aliases":"X_0j^-1","display":"X_{0j}^{-1}","kind":"introduction-passage","occurrences":[2835,2837,2838]},{"id":"symbol-61836166435158f9","tex":"x_H^2","aliases":"x_H^2","display":"x_H^2","kind":"introduction-passage","occurrences":[2293]},{"id":"symbol-4b93e02a435f89a0","tex":"X_{i,j+1}^{-1}","aliases":"X_i,j+1^-1","display":"X_{i,j+1}^{-1}","kind":"introduction-passage","occurrences":[2831]},{"id":"symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"introduction-passage","occurrences":[2821,2831]},{"id":"symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"introduction-passage","occurrences":[1093,1085,2775,3549,3550]},{"id":"symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","occurrences":[2504,2826,2831,1767,1768,1769,1770,1771,1772,1776,1995,1996,2003,2426,2427,2429,2430,2439,2505,2506,2510,2511,2836,2837,2839,2840,2957,2958,2959,2960,2961,2962,2963,2966,2967,2968,2969,2970,2971,2983,2984,2999,3035,3036,3050,3054,3059,3729,3832]},{"id":"symbol-3e697fdc5343c7a2","tex":"Y_{i+1,j}","aliases":"Y_i+1,j","display":"Y_{i+1,j}","kind":"introduction-passage","occurrences":[2831]},{"id":"symbol-25a00e69dec9ace2","tex":"Y_{ij}","aliases":"Y_ij","display":"Y_{ij}","kind":"introduction-passage","occurrences":[2826]},{"id":"symbol-8a25f3ab0c5819b4","tex":"Y_{ij}^{-1}","aliases":"Y_ij^-1","display":"Y_{ij}^{-1}","kind":"introduction-passage","occurrences":[2831]},{"id":"symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","occurrences":[407,5,7,11,12,39,50,52,66,74,123,375,377,378,382,425,603,717,1307,1356,1358,1501,2631,2696,3619]},{"id":"symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","occurrences":[1225,1231,1805,248,249,293,295,336,337,367,516,517,518,519,520,522,524,528,559,560,577,584,585,588,678,679,681,784,785,786,787,869,870,871,872,873,878,879,880,882,890,894,896,898,901,902,904,905,906,913,916,917,927,928,932,934,999,1000,1002,1003,1010,1014,1015,1034,1041,1062,1063,1064,1070,1076,1077,1083,1085,1089,1095,1124,1126,1133,1135,1136,1206,1207,1208,1209,1210,1211,1215,1216,1240,1243,1318,1336,1505,1508,1510,1543,1546,1554,1555,1561,1562,1565,1575,1701,1702,1713,1714,1715,1716,1717,1719,1794,1795,1802,1842,1846,1856,1865,2033,2113,2114,2115,2310,2312,2313,2471,2472,2475,2477,2478,2479,2480,2481,2731,2732,2760,2806,2810,2876,2880,2945,2946,2947,2948,2950,2951,2952,2953,2954,2963,2964,2985,3047,3048,3052,3055,3056,3064,3069,3082,3100,3103,3105,3106,3108,3109,3420,3424,3425,3434,3436,3437,3439,3440,3442,3445,3459,3460,3461,3462,3470,3471,3473,3474,3481,3488,3493,3672,3673,3675,3685,3689,3690,3691,3693,3699]},{"id":"symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"introduction-passage","occurrences":[1231,1715]},{"id":"symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"introduction-passage","occurrences":[1225]},{"id":"symbol-4e5172da9206de94","tex":"z^L","aliases":"z^L","display":"z^L","kind":"introduction-passage","occurrences":[1225]},{"id":"symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z","display":null,"kind":"introduction-passage","occurrences":[407,375,377,378,425,603]},{"id":"symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","occurrences":[324,328,329,334,728,730,1228,2,53,75,82,95,99,101,102,104,105,106,107,173,185,186,187,192,197,207,218,231,286,291,294,300,304,305,307,310,311,312,313,315,316,318,336,341,348,349,351,352,354,355,357,358,360,361,367,368,369,370,371,373,391,397,444,494,496,505,515,527,531,535,552,560,577,579,581,585,586,646,650,651,652,655,656,663,665,677,678,679,681,688,694,696,698,700,701,753,754,797,801,802,827,892,916,949,952,960,972,1013,1033,1175,1180,1190,1193,1195,1200,1207,1209,1251,1282,1296,1297,1322,1324,1329,1339,1350,1351,1396,1403,1423,1468,1469,1470,1472,1474,1491,1493,1494,1495,1501,1502,1525,1588,1591,1592,1602,1604,1632,1702,1705,1706,1709,1711,1712,1715,1716,1720,1721,1723,1725,1726,1727,1738,1739,1744,1745,1755,1777,1778,1780,1786,1843,1857,1858,1871,1872,1899,1900,1902,1905,1907,1908,1921,1933,1935,2116,2155,2306,2317,2319,2324,2344,2346,2378,2379,2380,2386,2390,2448,2460,2559,2579,2582,2587,2601,2649,2653,2654,2792,2850,2979,2981,2985,3024,3025,3026,3058,3081,3082,3085,3089,3091,3092,3093,3096,3098,3116,3294,3462,3607,3613,3696,3773,3781,3789,3902,3906,3908,3928,3971,3975,3987,4040]},{"id":"symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"introduction-passage","occurrences":[324,329,187,336,341,352,355,358,361,368,373,391,397,494,505,531,552,681,1175,1282,1329,1468,1469,1470,1472,1474,1491,1493,1494,1495,1501,1502,1525,1588,1591,1592,1602,1632,1706,1725,1726,1780,1843,1857,1858,1900,1907,2306,2317,2319,2380,2386,2649,2653,2654,2979,3025,3091,3773,3781,3789]},{"id":"symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"introduction-passage","occurrences":[2899,110,111,119,348,1156,1607,1608,1611,1621,1624,1625,1627,1650,1755,1756,3281,3333,3903,4108]},{"id":"symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","occurrences":[3411,1413,1522,1583,1671,2840,3211,3220,3444,3682,3683,3695]},{"id":"symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"introduction-passage","occurrences":[324,329,3027,3030,3198,3729,3735,3737,3748,3762,3765,3769,3771,3776,3778]},{"id":"symbol-8474abafe881bcf4","tex":"\\delta_B","aliases":"δ_B delta","display":"\\delta_B","kind":"introduction-passage","occurrences":[2503,2504,2511,2512,2514,2524,2530,2532]},{"id":"symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"introduction-passage","occurrences":[333,3841,256,257,259,261,262,274,276,317,320,321,322,323,336,367,493,494,496,497,500,501,503,505,506,507,508,511,513,515,517,518,519,521,524,526,527,528,531,534,535,536,538,539,542,551,552,553,554,555,556,560,562,566,569,571,577,578,579,581,586,589,591,592,593,594,597,635,944,952,962,974,1000,1002,1003,1005,3237,3238,3243,3266,3272,3279,3341,3345,3360,3371,3373,3374,3379,3381,3397,3932,3933]},{"id":"symbol-890e777d8bea7959","tex":"\\epsilon_0","aliases":"ϵ_0 epsilon","display":"\\epsilon_0","kind":"introduction-passage","occurrences":[3841,593,962,1003]},{"id":"symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"introduction-passage","occurrences":[3410,3412,867,873,889,890,892,3352,3353,3355]},{"id":"symbol-ca62cb8c93dad62f","tex":"\\iota^*","aliases":"ι^* iota","display":"\\iota^*","kind":"introduction-passage","occurrences":[3412]},{"id":"symbol-03cd5a90c4468e52","tex":"\\Lambda^2","aliases":"Λ^2 Lambda","display":"\\Lambda^2","kind":"introduction-passage","occurrences":[3844,3852,3860,3894]},{"id":"symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","occurrences":[407,376,377,378,379,385,391,427,603,816,817,818,828,841,847,857,860,1034,1064,1076,1078,1083,1085,1126,1133,1135,1329,1330,1335,1336,1338,1339,1342,2130,2131,2132,2135,2143,2145,2606,2609,2612,2617,2618,2622,2625,2626,2678,2682,2686,2698,2700,2701,2706,2707,2804,2806,2810,3243,3245,3301,3304,3348,3366,3485,3486,3489,3494,3689,3691]},{"id":"symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"introduction-passage","occurrences":[407,376,377,378,379,385,391,427,603,1064,1076,1078]},{"id":"symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","occurrences":[155,3841,77,88,94,467,665,829,830,847,850,859,860,1085,1087,1106,1109,1114,1115,1160,1162,1164,1174,1177,1197,1199,1205,1323,1339,1343,1345,1413,1485,1501,1502,1521,1522,1671,1673,1678,1820,1823,1825,1833,1836,1840,1850,1852,1853,1855,1866,1913,1914,1915,1916,1921,2194,2195,2196,2197,2200,2201,2209,2696,3219,3230,3349,3582,3585,3586,3590,3593,3595,3604,3605,3606,3619,3621,3634,3667,3731,3734,3735,3742,3745,3748,3749,3751,3759,3761,3766,3768,3769,3775,3776,3777,3778,3990,4042,4109]},{"id":"symbol-21e99a4d2b6a614c","tex":"\\rho_v^n","aliases":"ρ_v^n rho","display":"\\rho_v^n","kind":"introduction-passage","occurrences":[743,744,746,748]},{"id":"symbol-3628d2a1d0d8cb5d","tex":"\\rho_v^{n+1}","aliases":"ρ_v^n+1 rho","display":"\\rho_v^{n+1}","kind":"introduction-passage","occurrences":[745,746,748]},{"id":"symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"introduction-passage","occurrences":[3412,295,336,337,679,681,870,898,900,904,908,911,918,921,928,1001,1002,1003,1014,3272,3279,3335,3336,3338,3339,3340,3342,3345,3346,3352,3353,3356,3357,3358,3374,3380,3387,3393,3403,3405]},{"id":"symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","occurrences":[2904,2905,2906,2907,248,497,528,678,681,869,874,895,907,1010,1014,1015,1016,1018,1545,1546,1552,1554,1555,1556,1558,2142,2811,2812,2813,2920,2940,2942,2988,2990,2994,2997,3006,3341,3342,3345,3346,3385,3386,3604,3605,3606,4048,4052,4054,4061,4062,4063,4072,4074,4075,4077,4078,4081,4082,4087,4089,4090,4095,4097,4101,4108,4117]},{"id":"symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"introduction-passage","occurrences":[2905,2906,2811,3604,3605,4052]},{"id":"symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","occurrences":[722,723,725,2891,2892,2900,2903,2904,2905,2906,925,928,931,932,934,1026,2142,2663,2665,2666,2668,2670,2671,2916,2920,2923,2926,2940,2989,2991,3006,3014,3022,3024,3025,3028,3227,3230,3239,3259,3260,3620,3621,3623,3625,3627,3628,3631,3635,3636,3637,3639,3642,3643,3646,3652,3653,3654,3656,3672,3699,3703,3704,3705,3708,3710,3711,3713,3714,3715,3725]},{"id":"symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"introduction-passage","occurrences":[2900,2665,2666,2923,2926,3025]},{"id":"symbol-e675956848d7751d","tex":"\\tau_2","aliases":"τ_2 tau","display":"\\tau_2","kind":"introduction-passage","occurrences":[722]},{"id":"symbol-96a7e03674728790","tex":"\\tau_p","aliases":"τ_p tau","display":"\\tau_p","kind":"introduction-passage","occurrences":[723,725]},{"id":"symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"introduction-passage","occurrences":[3844,3849,816,2929,2974,3422,3427,3436,3437,3444,3451,3460,3461,3581,3582,3583,3584,3585,3605,3607,3608,3609,3610,3613,3850,3875,3897,3899,4048,4057,4059,4061,4062,4063,4064,4067,4068,4070,4071,4075,4077,4078,4085,4087,4090,4093,4094,4105,4131,4161]},{"id":"symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"introduction-passage","occurrences":[742,743,1169,1170,1172,1178,1183,1185,1187,1975,1976,1980,1981,1982,2073,2074,2076,2079,2080,2087,2095,2624,2666,2669,2931,3551,3553,3554,3556,3558,3559,3560,3568,3569,3570,4042]},{"id":"symbol-ea3431124039103d","tex":"\\phi^n","aliases":"ϕ^n phi","display":"\\phi^n","kind":"introduction-passage","occurrences":[742,743]},{"id":"symbol-12e16dc509b4020a","tex":"\\Phi_e","aliases":"Φ_e Phi","display":"\\Phi_e","kind":"introduction-passage","occurrences":[3844,3849,3850,3875]},{"id":"symbol-d28947f5acd9f188","tex":"\\psi^n","aliases":"ψ^n psi","display":"\\psi^n","kind":"introduction-passage","occurrences":[740,742,743]},{"id":"symbol-5c498202a7960764","tex":"\\psi_{r,a,q}^{(h)}","aliases":"ψ_r,a,q^(h) psi","display":"\\psi_{r,a,q}^{(h)}","kind":"introduction-passage","occurrences":[159,173]},{"id":"symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"introduction-passage","occurrences":[324,325,326,327,330,3412,3844,2952,2955,2956,2963,2964,2972,2978,2979,2982,2985,3057,3058,3080,3096,3113,3209,3210,3211,3212,3213,3337,3338,3340,3342,3346,3349,3350,3352,3353,3357,3358,3367,3372,3373,3374,3556,3868,4052]},{"id":"symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"introduction-passage","occurrences":[155,159,165,170,207,209,211,467,468,469,472,1205,1208,1212,1243,1249,1324,1330,1342,3268,3272,3275,3296,3331,3337,3340,3398,3554,3555,3556,3557,3573,3574,3576,3577,3786,3789]},{"id":"symbol-23a05885bf7340b9","tex":"\\Omega^2","aliases":"Ω^2 Omega","display":"\\Omega^2","kind":"introduction-passage","occurrences":[325,3844,3868,4052]},{"id":"symbol-dc7236f880f8be15","tex":"\\omega_p","aliases":"ω_p omega","display":"\\omega_p","kind":"introduction-passage","occurrences":[155,467]},{"id":"symbol-a791b3585eff1a50","tex":"\\omega_p^{q d_r(x)}","aliases":"ω_p^q d_r(x) omega","display":"\\omega_p^{q d_r(x)}","kind":"introduction-passage","occurrences":[159]},{"id":"symbol-dadabf5f8288039e","tex":"\\Omega_{\\mathrm{red}}","aliases":"Ω_red Omega mathrm","display":"\\Omega_{\\mathrm{red}}","kind":"introduction-passage","occurrences":[3412]},{"id":"symbol-d7aecc42f2bbbfbb","tex":"\\Omega_\\Sigma","aliases":"Ω_Σ Omega Sigma","display":"\\Omega_\\Sigma","kind":"introduction-passage","occurrences":[3412,3357,3358]},{"id":"symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","occurrences":[157,158,727,1226,192,217,845,949,969,971,972,974,975,1029,1088,1236,1257,3040,3422,3540,3588,3661,3929,3986,3987]},{"id":"symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","occurrences":[725,1,66,74,123,209,211,382,861,862,958,980,1031,1208,1212,1243,1249,1324,1330,1339,1340,1342,1356,2299,2306,2313,2324,2366,2371,2373,2381,2382,2385,2394,2402,2404,2446,2946,2947,2948,2951,2974,2985,3018,3019,3062,3064,4108,4110,4111]},{"id":"symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"introduction-passage","occurrences":[725,1,66,74,123,382,958,980,1031,1356,2299,2306,2313,2324,2366,2371,2373,2381,2382,2385,2394,2402,2404,2446,3018,3019]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","occurrences":[743,744,745,746,2287,2290,2903,3167,3168,225,415,443,444,482,510,511,524,525,536,579,586,607,622,734,735,748,796,866,958,1181,1182,1365,1451,1452,1539,1575,1664,1730,1761,1765,1785,1910,1951,1992,2043,2159,2249,2254,2256,2295,2296,2302,2339,2416,2418,2420,2430,2436,2445,2483,2663,2664,2778,3023,3035,3036,3037,3119,3128,3251,3475,3478,3572,3615,3788]},{"id":"symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","occurrences":[1804,103,185,1756,2372,2583,2758,3575,3770]},{"id":"symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","occurrences":[1227,1228,72,76,1221,1308,2567,2581,2693,3230]},{"id":"symbol-0fcd1056f96dbe36","tex":"\\prod_r","aliases":"∏_r prod","display":"\\prod_r","kind":"introduction-passage","occurrences":[1227,1228]},{"id":"symbol-07961363fdf94e98","tex":"\\sum_{a\\in F}","aliases":"∑_a∈ F sum in","display":"\\sum_{a\\in F}","kind":"introduction-passage","occurrences":[2286,2289,2346]},{"id":"symbol-eb07f8d7b26ea999","tex":"\\sum_{q\\geq0}","aliases":"∑_q≥0 sum geq","display":"\\sum_{q\\geq0}","kind":"introduction-passage","occurrences":[1227,1266]},{"id":"symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"introduction-passage","occurrences":[744,745]},{"id":"symbol-93b73f4100ccda1a","tex":"\\sum_{u\\sim v}","aliases":"∑_u∼ v sum sim","display":"\\sum_{u\\sim v}","kind":"introduction-passage","occurrences":[743,746]},{"id":"symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"introduction-passage","occurrences":[407,379,391,427,685,925,926,928,934,1164,1174,1190,1282,1296,1518,1526,1671,1710,1714,1884,2698,2726,2772,2775,2780,3070,3074,3097,3137,3230,3279,3294,3319,3342,3374,3380,3387,3393,3401,3403,3405,3424,3505,3506,3557,3558,3577,3582,3586,3613,3621,3634,3639,3643,3745,3850]},{"id":"symbol-edf83e015e4dd5ae","tex":"\\int_{x+p^m\\Z_p}","aliases":"∫_x+p^mZ_p int Z","display":null,"kind":"introduction-passage","occurrences":[407]},{"id":"symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","occurrences":[743,746,748]},{"id":"symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"introduction-passage","occurrences":[159,67,129,275,276,1213,1310,1324,1349]},{"id":"symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"introduction-passage","occurrences":[2285,2290,2292,147,637,1358,1528,2352,2693,3221]},{"id":"symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","occurrences":[725,66,74,78,82,83,107,185,197,202,215,218,603,604,606,609,611,635,655,656,732,770,771,802,892,1012,1013,2131,2133,2136,2141,2142,2144,2146,2148,2149,2677]},{"id":"symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","occurrences":[728,730,106,107,173,218,307,822,824,825,911,916,919,966,987,999,1003,1013,1014,1026,1185,1282,1296,1324,1330,1525,1526,1647,1659,1660,1668,1671,1696,1726,1925,1929,1934,1935,1941,2116,2155,3287,3319,3349,3751,3753,3754,3766,3778]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","occurrences":[328,728,730,2,75,82,95,99,101,102,104,105,106,107,173,185,186,192,197,207,218,300,304,305,307,310,311,312,313,315,316,318,371,650,651,652,655,656,797,801,802,822,824,825,827,892,911,916,919,949,952,960,966,972,987,999,1003,1014,1026,1033,1185,1282,1296,1322,1324,1329,1330,1350,1351,1525,1526,1647,1659,1660,1668,1671,1696,1726,1925,1929,1934,1935,1941,2116,2155,3287,3319,3349,3751,3753,3754,3766,3778]},{"id":"symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"introduction-passage","occurrences":[2294,74,531,1164,1300,1414,1788,2317,2453,2460,2461,2462,2606,2612,3149]},{"id":"symbol-2a3551ffdc934706","tex":"+_p","aliases":"+_p","display":"+_p","kind":"use","occurrences":[185,186]},{"id":"symbol-b0fd131ff163229e","tex":"+_x","aliases":"+_x","display":"+_x","kind":"use","occurrences":[655,656]},{"id":"symbol-23aa8a462b6e6789","tex":"\\boldsymbol1","aliases":"1 boldsymbol","display":"\\boldsymbol1","kind":"use","occurrences":[2053]},{"id":"symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","occurrences":[275,276,384,401,468]},{"id":"symbol-256011eff0fc0b21","tex":"\\boldsymbol1_{\\mathrm c}","aliases":"1_ c boldsymbol mathrm","display":"\\boldsymbol1_{\\mathrm c}","kind":"use","occurrences":[2053]},{"id":"symbol-9022dcc510c5116d","tex":"\\boldsymbol1_{\\mathrm f}","aliases":"1_ f boldsymbol mathrm","display":"\\boldsymbol1_{\\mathrm f}","kind":"use","occurrences":[2053]},{"id":"symbol-a107946ab1522f43","tex":"\\mathbf1_{B_{m,a}}","aliases":"1_B_m,a mathbf","display":"\\mathbf1_{B_{m,a}}","kind":"use","occurrences":[384,401]},{"id":"symbol-df2fdc1df922b224","tex":"\\mathbf1_{B_{m,a+d p^{m-1}}}","aliases":"1_B_m,a+d p^m-1 mathbf","display":"\\mathbf1_{B_{m,a+d p^{m-1}}}","kind":"use","occurrences":[468]},{"id":"symbol-ee4af272d467ef2b","tex":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}","aliases":"1_x≡ y (mod p^r) mathbf equiv mathrm","display":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}","kind":"use","occurrences":[276]},{"id":"symbol-572d3cc9f9fb28e2","tex":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}","aliases":"1_x≡ y (mod p^r-1) mathbf equiv mathrm","display":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}","kind":"use","occurrences":[276]},{"id":"symbol-033181fac41ece68","tex":"\\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}","aliases":"1_{x≡ y (mod p^r)} mathbf equiv mathrm","display":"\\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}","kind":"use","occurrences":[275]},{"id":"symbol-02cdc56fb77a61c8","tex":"A^{\\mathcal H}","aliases":"A^ H mathcal","display":"A^{\\mathcal H}","kind":"use","occurrences":[1793,1795,1802]},{"id":"symbol-44f43aa87edf0679","tex":"a^{(r)}","aliases":"a^(r)","display":"a^{(r)}","kind":"use","occurrences":[1451,1452]},{"id":"symbol-00ae19a4939e8cce","tex":"\\mathcal A^*","aliases":"A^* mathcal","display":"\\mathcal A^*","kind":"use","occurrences":[806,814,815,818]},{"id":"symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","occurrences":[82,87]},{"id":"symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","occurrences":[2709,3546]},{"id":"symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","occurrences":[2713]},{"id":"symbol-e046b0ebe8054038","tex":"a^4","aliases":"a^4","display":"a^4","kind":"use","occurrences":[2706,2710]},{"id":"symbol-09acaea0cd848de9","tex":"a^5","aliases":"a^5","display":"a^5","kind":"use","occurrences":[2714]},{"id":"symbol-79f1bb0d0b74b664","tex":"a^6","aliases":"a^6","display":"a^6","kind":"use","occurrences":[2707,2711]},{"id":"symbol-b51192cce36ea46d","tex":"a^8","aliases":"a^8","display":"a^8","kind":"use","occurrences":[2712]},{"id":"symbol-c44dac86d60fd4be","tex":"\\mathcal A^{\\mathrm{ad}}_{p,h}","aliases":"A^ad_p,h mathcal mathrm","display":"\\mathcal A^{\\mathrm{ad}}_{p,h}","kind":"use","occurrences":[197,202,215]},{"id":"symbol-be96b81185b4c8b5","tex":"a^g","aliases":"a^g","display":"a^g","kind":"use","occurrences":[2916,2917,2918]},{"id":"symbol-e0424812d1c195ce","tex":"A^k","aliases":"A^k","display":"A^k","kind":"use","occurrences":[1798,1802,1887]},{"id":"symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","occurrences":[1254]},{"id":"symbol-c60967b39bed49c7","tex":"a^n","aliases":"a^n","display":"a^n","kind":"use","occurrences":[1254]},{"id":"symbol-1d7c8ce7a043e522","tex":"a^{Q+1}","aliases":"a^Q+1","display":"a^{Q+1}","kind":"use","occurrences":[1251,1254]},{"id":"symbol-b2c8f1aafb4a6058","tex":"\\mathcal A^{\\mathrm{std}}_{p,h}","aliases":"A^std_p,h mathcal mathrm","display":"\\mathcal A^{\\mathrm{std}}_{p,h}","kind":"use","occurrences":[197,202]},{"id":"symbol-004a43f65c127c71","tex":"a^\\dagger","aliases":"a^† dagger","display":"a^\\dagger","kind":"use","occurrences":[1349,1351,1353]},{"id":"symbol-60168b3ae1d1b815","tex":"a_{\\mathbf d}","aliases":"a_ d mathbf","display":"a_{\\mathbf d}","kind":"use","occurrences":[312,314]},{"id":"symbol-8200d9e2af5f6cdb","tex":"A_{\\mathbf r}","aliases":"A_ r mathbf","display":"A_{\\mathbf r}","kind":"use","occurrences":[614,617,622,627,631,635]},{"id":"symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","occurrences":[1246,1248,1249,1251,1257,1259,1261,1262]},{"id":"symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","occurrences":[3117]},{"id":"symbol-826262fb5a1181ed","tex":"a_0^2","aliases":"a_0^2","display":"a_0^2","kind":"use","occurrences":[3118]},{"id":"symbol-ed1a1979512421da","tex":"A_0^j","aliases":"A_0^j","display":"A_0^j","kind":"use","occurrences":[1256]},{"id":"symbol-6906ba14ccd2485d","tex":"A_0^L","aliases":"A_0^L","display":"A_0^L","kind":"use","occurrences":[1255]},{"id":"symbol-93103c835d52db50","tex":"A_0^{L-1}","aliases":"A_0^L-1","display":"A_0^{L-1}","kind":"use","occurrences":[1257]},{"id":"symbol-e95e07807750445a","tex":"a_{2n+1,2n}","aliases":"a_2n+1,2n","display":"a_{2n+1,2n}","kind":"use","occurrences":[2361]},{"id":"symbol-126841ea628e2845","tex":"a_{2n,2n+1}","aliases":"a_2n,2n+1","display":"a_{2n,2n+1}","kind":"use","occurrences":[2360]},{"id":"symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","occurrences":[3245,3248,3380,3402,3403,3406]},{"id":"symbol-c57dad5f04c4ba4a","tex":"A_a^i","aliases":"A_a^i","display":"A_a^i","kind":"use","occurrences":[3231,3237,3240]},{"id":"symbol-c2c0df3352740364","tex":"A_a^j","aliases":"A_a^j","display":"A_a^j","kind":"use","occurrences":[3237]},{"id":"symbol-ac0ad444a337d8a7","tex":"A_b^i","aliases":"A_b^i","display":"A_b^i","kind":"use","occurrences":[3237]},{"id":"symbol-69ddbdb9d32d9cf1","tex":"A_b^k","aliases":"A_b^k","display":"A_b^k","kind":"use","occurrences":[3237]},{"id":"symbol-8da2f1b2bafe729c","tex":"A_h","aliases":"A_h","display":"A_h","kind":"use","occurrences":[3964,3969,3984]},{"id":"symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","occurrences":[3070,3071,3073,3074,3076,3077,3079]},{"id":"symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","occurrences":[3738,3739]},{"id":"symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","occurrences":[2722,2723,2729,2732,2756,2758,2774,2802,3079,3083,3085,3089]},{"id":"symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","occurrences":[3517]},{"id":"symbol-622bd8519fe153e7","tex":"a_{j+1/2}","aliases":"a_j+1/2","display":"a_{j+1/2}","kind":"use","occurrences":[3517]},{"id":"symbol-28ff8ef535373c67","tex":"a_{Lq+j}","aliases":"a_Lq+j","display":"a_{Lq+j}","kind":"use","occurrences":[1210]},{"id":"symbol-f327e0098ac5621a","tex":"A_{m,0}","aliases":"A_m,0","display":"A_{m,0}","kind":"use","occurrences":[2582,2591]},{"id":"symbol-290d99ad3e8de4dc","tex":"A_{m,p}","aliases":"A_m,p","display":"A_{m,p}","kind":"use","occurrences":[2581,2582,2587]},{"id":"symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","occurrences":[2459,2460,2461]},{"id":"symbol-1dc1a9b01a1e0c65","tex":"a_n^{(r)}","aliases":"a_n^(r)","display":"a_n^{(r)}","kind":"use","occurrences":[1455]},{"id":"symbol-612a17fd14dd2159","tex":"a_{n_0}","aliases":"a_n_0","display":"a_{n_0}","kind":"use","occurrences":[1812]},{"id":"symbol-2fc3b3c3f3fa024a","tex":"a_{n_A}","aliases":"a_n_A","display":"a_{n_A}","kind":"use","occurrences":[1812]},{"id":"symbol-a7522c9b01a1bd98","tex":"a_{n_D}","aliases":"a_n_D","display":"a_{n_D}","kind":"use","occurrences":[1812]},{"id":"symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","occurrences":[3583]},{"id":"symbol-073c74444893e767","tex":"a_{\\rm tr}","aliases":"a_rm tr rm","display":"a_{\\rm tr}","kind":"use","occurrences":[3740]},{"id":"symbol-895b4c69e53e4df6","tex":"A_x","aliases":"A_x","display":"A_x","kind":"use","occurrences":[3069,3108]},{"id":"symbol-a9f66cfc2078279a","tex":"A_y","aliases":"A_y","display":"A_y","kind":"use","occurrences":[3069,3087,3109,3115,3116]},{"id":"symbol-2e74d3cc58dddca4","tex":"A_z","aliases":"A_z","display":"A_z","kind":"use","occurrences":[3069,3108]},{"id":"symbol-06b5cc3e240ef745","tex":"a_\\mu","aliases":"a_μ mu","display":"a_\\mu","kind":"use","occurrences":[2701,2706]},{"id":"symbol-4d9af95cd8418d57","tex":"a_\\mu^2","aliases":"a_μ^2 mu","display":"a_\\mu^2","kind":"use","occurrences":[2707]},{"id":"symbol-39dbcce24a8d9bcb","tex":"a_\\nu","aliases":"a_ν nu","display":"a_\\nu","kind":"use","occurrences":[2701,2706]},{"id":"symbol-1c9cbd53094797df","tex":"a_\\nu^2","aliases":"a_ν^2 nu","display":"a_\\nu^2","kind":"use","occurrences":[2707]},{"id":"symbol-5ac775657d62befa","tex":"\\mathcal A_{\\partial}","aliases":"A_∂ mathcal partial","display":"\\mathcal A_{\\partial}","kind":"use","occurrences":[3100,3102,3107,3112,3118]},{"id":"symbol-f6c4b36b119e9415","tex":"a_\\perp","aliases":"a_⊥ perp","display":"a_\\perp","kind":"use","occurrences":[3337,3340]},{"id":"symbol-357745da2a118b68","tex":"\\mathrm{ad}","aliases":"ad mathrm","display":"\\mathrm{ad}","kind":"use","occurrences":[197,202,215]},{"id":"symbol-6ea43fbb8cbfe05b","tex":"\\operatorname{area}","aliases":"area operatorname","display":"\\operatorname{area}","kind":"use","occurrences":[3569]},{"id":"symbol-9544dc0281e02973","tex":"\\operatorname{asinh}","aliases":"asinh operatorname","display":"\\operatorname{asinh}","kind":"use","occurrences":[3613]},{"id":"symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","occurrences":[1246,1966,1990,2267,3381,3659,3661,3662,3664]},{"id":"symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","occurrences":[82,86]},{"id":"symbol-a80a840f9ada088f","tex":"b^{-r}","aliases":"b^-r","display":"b^{-r}","kind":"use","occurrences":[2572]},{"id":"symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","occurrences":[3610,3613]},{"id":"symbol-a7ffa2898efa8288","tex":"\\mathcal B^a","aliases":"B^a mathcal","display":"\\mathcal B^a","kind":"use","occurrences":[3381]},{"id":"symbol-ad0ae9f47e95e104","tex":"\\mathcal B^b","aliases":"B^b mathcal","display":"\\mathcal B^b","kind":"use","occurrences":[3381]},{"id":"symbol-97a0df13c9cd3826","tex":"b^g","aliases":"b^g","display":"b^g","kind":"use","occurrences":[2916,2917]},{"id":"symbol-c4a00e5b81eeab04","tex":"b^k","aliases":"b^k","display":"b^k","kind":"use","occurrences":[2567,2581]},{"id":"symbol-d260ef111195a9e3","tex":"b^{k-p}","aliases":"b^k-p","display":"b^{k-p}","kind":"use","occurrences":[2581]},{"id":"symbol-8b15ec8eb6abf2f6","tex":"b^{k-r}","aliases":"b^k-r","display":"b^{k-r}","kind":"use","occurrences":[2567]},{"id":"symbol-cea555c0ffd5ad40","tex":"b^{m+1-p}","aliases":"b^m+1-p","display":"b^{m+1-p}","kind":"use","occurrences":[2586]},{"id":"symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","occurrences":[650,651,655,656]},{"id":"symbol-cf1be6e890041939","tex":"B_{0i}","aliases":"B_0i","display":"B_{0i}","kind":"use","occurrences":[3374]},{"id":"symbol-fa1a68eae3317067","tex":"B_{ba}","aliases":"B_ba","display":"B_{ba}","kind":"use","occurrences":[2522]},{"id":"symbol-ffbdc00380cf8d6d","tex":"B_{bc}","aliases":"B_bc","display":"B_{bc}","kind":"use","occurrences":[2524]},{"id":"symbol-b5384093c6e0cfdb","tex":"B_{ca}","aliases":"B_ca","display":"B_{ca}","kind":"use","occurrences":[2524]},{"id":"symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","occurrences":[2515,2516,2517,3738]},{"id":"symbol-28c8ff44f8fbfe84","tex":"B_{IJ}","aliases":"B_IJ","display":"B_{IJ}","kind":"use","occurrences":[3341,3342,3346,3374]},{"id":"symbol-2ff1187fcdc685de","tex":"B_{ij}","aliases":"B_ij","display":"B_{ij}","kind":"use","occurrences":[3374]},{"id":"symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","occurrences":[2515,2516,2517,3531,3533,3534,3535]},{"id":"symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","occurrences":[3962]},{"id":"symbol-89304b2594ebe322","tex":"B_{j-1}","aliases":"B_j-1","display":"B_{j-1}","kind":"use","occurrences":[3533]},{"id":"symbol-2fc9f3d3fd680082","tex":"B_{j\\mid i}","aliases":"B_jmid i mid","display":null,"kind":"use","occurrences":[3085,3089,3091,3092]},{"id":"symbol-8184a87901449233","tex":"B_{L,j}","aliases":"B_L,j","display":"B_{L,j}","kind":"use","occurrences":[1266,1267,1270,1272]},{"id":"symbol-be137cd5e9ecea3a","tex":"B_{L,j-1\\bmod L}","aliases":"B_L,j-1bmod L bmod","display":"B_{L,j-1\\bmod L}","kind":"use","occurrences":[1272]},{"id":"symbol-9277d882004917b6","tex":"B_{m+1,a+d p^m}","aliases":"B_m+1,a+d p^m","display":"B_{m+1,a+d p^m}","kind":"use","occurrences":[393]},{"id":"symbol-76b84f0a7edc903f","tex":"B_{m,a}","aliases":"B_m,a","display":"B_{m,a}","kind":"use","occurrences":[377,384,385,390,391,393,401]},{"id":"symbol-3b4beaedece992ae","tex":"B_{m,a+d p^{m-1}}","aliases":"B_m,a+d p^m-1","display":"B_{m,a+d p^{m-1}}","kind":"use","occurrences":[468]},{"id":"symbol-f7dc3c6a96166d73","tex":"B_{m,b}","aliases":"B_m,b","display":"B_{m,b}","kind":"use","occurrences":[385]},{"id":"symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","occurrences":[1357,1432,1442,1443,1445,1446,1448,1455,1696]},{"id":"symbol-82d5eaeb539c2f07","tex":"\\mathcal B_{r,j}","aliases":"B_r,j mathcal","display":"\\mathcal B_{r,j}","kind":"use","occurrences":[3661,3662,3664]},{"id":"symbol-8cdcafa2076b5d63","tex":"\\mathcal B_{r,j-1}","aliases":"B_r,j-1 mathcal","display":"\\mathcal B_{r,j-1}","kind":"use","occurrences":[3664]},{"id":"symbol-4b94a4e9d2f773e0","tex":"b_{\\rm tr}","aliases":"b_rm tr rm","display":"b_{\\rm tr}","kind":"use","occurrences":[3740]},{"id":"symbol-4a3dd5cb1c2ae296","tex":"B_{x\\mid y}","aliases":"B_xmid y mid","display":null,"kind":"use","occurrences":[3096,3100]},{"id":"symbol-d93a18a8f7a52a97","tex":"B_{x\\mid z}","aliases":"B_xmid z mid","display":null,"kind":"use","occurrences":[3100]},{"id":"symbol-123a2e3b9d146b7e","tex":"B_{y\\mid x}","aliases":"B_ymid x mid","display":null,"kind":"use","occurrences":[3086,3096,3100,3111]},{"id":"symbol-cb32484d4f016b34","tex":"B_{y\\mid z}","aliases":"B_ymid z mid","display":null,"kind":"use","occurrences":[3100]},{"id":"symbol-75c7c82addb66e22","tex":"B_{z\\mid x}","aliases":"B_zmid x mid","display":null,"kind":"use","occurrences":[3100]},{"id":"symbol-a0327f04e760f963","tex":"B_{z\\mid y}","aliases":"B_zmid y mid","display":null,"kind":"use","occurrences":[3100]},{"id":"symbol-093efc86fbc935a9","tex":"B_\\partial","aliases":"B_∂ partial","display":"B_\\partial","kind":"use","occurrences":[3850]},{"id":"symbol-cf72cfc588b3df8b","tex":"\\bar p","aliases":"bar p bar","display":"\\bar p","kind":"use","occurrences":[3450,3451]},{"id":"symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","occurrences":[3449,3451,3454,3455]},{"id":"symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"use","occurrences":[2734,2747]},{"id":"symbol-a09b010e18ba8e9a","tex":"\\bar\\chi","aliases":"barχ bar chi","display":"\\bar\\chi","kind":"use","occurrences":[3388]},{"id":"symbol-40c6a191c91db0c5","tex":"\\bar\\chi_B","aliases":"barχ_B bar chi","display":"\\bar\\chi_B","kind":"use","occurrences":[3388]},{"id":"symbol-9c3ecc4b70c12695","tex":"\\bar\\psi","aliases":"barψ bar psi","display":"\\bar\\psi","kind":"use","occurrences":[3271,3275,3279,3299,3305,3307,3308,3312,3313,3314,3315,3342,3345,3346,3383,3396]},{"id":"symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","occurrences":[393,2348]},{"id":"symbol-15ef91d0a0f35fa0","tex":"\\bigsqcup_d","aliases":"bigsqcup_d bigsqcup","display":null,"kind":"use","occurrences":[393]},{"id":"symbol-f805aa5d780b628b","tex":"\\bigsqcup_{\\alpha}","aliases":"bigsqcup_α bigsqcup alpha","display":null,"kind":"use","occurrences":[2348]},{"id":"symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","occurrences":[82,84,85,86,87,1241,1272]},{"id":"symbol-f157f71e0163a897","tex":"\\Box","aliases":"Box","display":"\\Box","kind":"use","occurrences":[3282,3283,3284]},{"id":"symbol-c0d312a637b5f115","tex":"\\mathrm{BV}","aliases":"BV mathrm","display":"\\mathrm{BV}","kind":"use","occurrences":[3319,3323]},{"id":"symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","occurrences":[1524,1527,1529,1531,1600,1664,1764,1785]},{"id":"symbol-ac286c55ce5c92f8","tex":"\\mathrm c","aliases":"c mathrm","display":"\\mathrm c","kind":"use","occurrences":[2048,2049,2050,2053,2062,2063,2064,2065,2066,2175,2176]},{"id":"symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","occurrences":[987]},{"id":"symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","occurrences":[739]},{"id":"symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","occurrences":[879,881]},{"id":"symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","occurrences":[2407,2705,2768]},{"id":"symbol-a620c9c352cae951","tex":"\\C^2","aliases":"C^2 C","display":null,"kind":"use","occurrences":[2034,2228,2632,2677,2803]},{"id":"symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","occurrences":[662,1857]},{"id":"symbol-da615aa5778187eb","tex":"\\C^2_A","aliases":"C^2_A C","display":null,"kind":"use","occurrences":[2007]},{"id":"symbol-071a19e037d78134","tex":"\\C^2_B","aliases":"C^2_B C","display":null,"kind":"use","occurrences":[2007]},{"id":"symbol-7de172a5e1aa4414","tex":"\\C^2_{\\mathrm{sign}}","aliases":"C^2_sign C mathrm","display":null,"kind":"use","occurrences":[66,74]},{"id":"symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","occurrences":[2771]},{"id":"symbol-5a1581d983eceed7","tex":"\\mathbb C^4","aliases":"C^4 mathbb","display":"\\mathbb C^4","kind":"use","occurrences":[1524,1527,1529,1531,1600,1664,1764]},{"id":"symbol-082b7a73628d3cfa","tex":"c^A","aliases":"c^A","display":"c^A","kind":"use","occurrences":[3207,3209]},{"id":"symbol-0330172c3e4700bc","tex":"c^B","aliases":"c^B","display":"c^B","kind":"use","occurrences":[3209]},{"id":"symbol-84321f2846a02874","tex":"\\C^{h+1}","aliases":"C^h+1 C","display":null,"kind":"use","occurrences":[958]},{"id":"symbol-a3740fcff7b45731","tex":"C^{IJ}","aliases":"C^IJ","display":"C^{IJ}","kind":"use","occurrences":[3397]},{"id":"symbol-d8416f1ffda0d03b","tex":"\\C^M","aliases":"C^M C","display":null,"kind":"use","occurrences":[796,1032]},{"id":"symbol-2ff8af0060364d2c","tex":"\\C^N","aliases":"C^N C","display":null,"kind":"use","occurrences":[796,804,2482,2483,2494,2533,3802]},{"id":"symbol-06e2221a9a4e2168","tex":"\\C^n","aliases":"C^n C","display":null,"kind":"use","occurrences":[2102,2152]},{"id":"symbol-d14ca8f6e8828e5e","tex":"\\C^p","aliases":"C^p C","display":null,"kind":"use","occurrences":[969]},{"id":"symbol-1756adec424805d9","tex":"\\C^{p-1}_{\\mathrm{tors}}","aliases":"C^p-1_tors C mathrm","display":null,"kind":"use","occurrences":[66,74]},{"id":"symbol-fcd04e249acdc68d","tex":"\\C^r","aliases":"C^r C","display":null,"kind":"use","occurrences":[2605]},{"id":"symbol-6ab8119bd804b07e","tex":"\\C^s","aliases":"C^s C","display":null,"kind":"use","occurrences":[2720,2721]},{"id":"symbol-0fa59d916aa991b8","tex":"\\C^\\times","aliases":"C^× C times","display":null,"kind":"use","occurrences":[2674]},{"id":"symbol-7ce18b1e2aa002e0","tex":"c^\\mu","aliases":"c^μ mu","display":"c^\\mu","kind":"use","occurrences":[3304]},{"id":"symbol-16df97f80ced5579","tex":"c^\\nu","aliases":"c^ν nu","display":"c^\\nu","kind":"use","occurrences":[3304]},{"id":"symbol-78a649bb99dbf34d","tex":"c_+","aliases":"c_+","display":"c_+","kind":"use","occurrences":[4097]},{"id":"symbol-83dae4cbc4840c66","tex":"c_-","aliases":"c_-","display":"c_-","kind":"use","occurrences":[4097]},{"id":"symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","occurrences":[969]},{"id":"symbol-936cd180d8ae5af5","tex":"c_0^2","aliases":"c_0^2","display":"c_0^2","kind":"use","occurrences":[3607,3608,3609,3613,3616]},{"id":"symbol-cf2b03123f263521","tex":"c_0^3","aliases":"c_0^3","display":"c_0^3","kind":"use","occurrences":[3613]},{"id":"symbol-158ec65eae30561d","tex":"c_0^4","aliases":"c_0^4","display":"c_0^4","kind":"use","occurrences":[3609]},{"id":"symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","occurrences":[3039,3041,3043]},{"id":"symbol-94104e5692ada178","tex":"C_{1,j}^n","aliases":"C_1,j^n","display":"C_{1,j}^n","kind":"use","occurrences":[2637,2638]},{"id":"symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","occurrences":[3039]},{"id":"symbol-3080f350b29a38e7","tex":"C_{2,j}^n","aliases":"C_2,j^n","display":"C_{2,j}^n","kind":"use","occurrences":[2637,2638]},{"id":"symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","occurrences":[2852,2863]},{"id":"symbol-99dbb247eeb251a3","tex":"C_A","aliases":"C_A","display":"C_A","kind":"use","occurrences":[2044,2045,2048,2174,3209]},{"id":"symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","occurrences":[3134,3136,3137]},{"id":"symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","occurrences":[388,389,391,403]},{"id":"symbol-89de6f3841051f82","tex":"c_{ab}","aliases":"c_ab","display":"c_{ab}","kind":"use","occurrences":[2198]},{"id":"symbol-a717968c751b4646","tex":"C_B","aliases":"C_B","display":"C_B","kind":"use","occurrences":[2048,2174]},{"id":"symbol-389556078e08449f","tex":"c_{ba}","aliases":"c_ba","display":"c_{ba}","kind":"use","occurrences":[2197,2198,2199,2200,2206,2207]},{"id":"symbol-227ceb7baf501818","tex":"c_d","aliases":"c_d","display":"c_d","kind":"use","occurrences":[970,972,978]},{"id":"symbol-fd685bf246955178","tex":"C_h","aliases":"C_h","display":"C_h","kind":"use","occurrences":[3675,3678,3682,3683,3684,3689,3690,3691,3693,3702,3706,3707,3718]},{"id":"symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","occurrences":[3735,3745,3747,3752,3754,3761]},{"id":"symbol-d17bd79161d5e884","tex":"\\mathcal C_I","aliases":"C_I mathcal","display":"\\mathcal C_I","kind":"use","occurrences":[3344,3347]},{"id":"symbol-de72a57fe0143ff0","tex":"C_{i,x}","aliases":"C_i,x","display":"C_{i,x}","kind":"use","occurrences":[3773]},{"id":"symbol-ba0b2b1b046df4d1","tex":"\\mathcal C_I^{\\mathrm D}","aliases":"C_I^ D mathcal mathrm","display":"\\mathcal C_I^{\\mathrm D}","kind":"use","occurrences":[3344,3345]},{"id":"symbol-94e1dd125c9c41b7","tex":"\\mathcal C_I^{\\mathrm G}","aliases":"C_I^ G mathcal mathrm","display":"\\mathcal C_I^{\\mathrm G}","kind":"use","occurrences":[3344,3345]},{"id":"symbol-56824ebc9521ba33","tex":"\\mathcal C_I^{\\mathrm M}","aliases":"C_I^ M mathcal mathrm","display":"\\mathcal C_I^{\\mathrm M}","kind":"use","occurrences":[3344,3345]},{"id":"symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"use","occurrences":[3037,3038,3042]},{"id":"symbol-24211da935b930d3","tex":"c_{j+1/2}","aliases":"c_j+1/2","display":"c_{j+1/2}","kind":"use","occurrences":[3517,3521,3522,3525,3531]},{"id":"symbol-88f8e11606761017","tex":"C_{j-1}","aliases":"C_j-1","display":"C_{j-1}","kind":"use","occurrences":[3041]},{"id":"symbol-120b20c73e5f2961","tex":"c_{j-1/2}","aliases":"c_j-1/2","display":"c_{j-1/2}","kind":"use","occurrences":[3522,3525]},{"id":"symbol-a8e8fbaa4e63d973","tex":"C_L","aliases":"C_L","display":"C_L","kind":"use","occurrences":[1237,1238,1240]},{"id":"symbol-56b847776dc4e558","tex":"C_L^L","aliases":"C_L^L","display":"C_L^L","kind":"use","occurrences":[1240]},{"id":"symbol-93c7b645d58c2453","tex":"C_L^n","aliases":"C_L^n","display":"C_L^n","kind":"use","occurrences":[1241]},{"id":"symbol-aa77d76fccb29372","tex":"c_{m,0}","aliases":"c_m,0","display":"c_{m,0}","kind":"use","occurrences":[2568]},{"id":"symbol-127e9660aac8b912","tex":"C_{M,j}","aliases":"C_M,j","display":"C_{M,j}","kind":"use","occurrences":[3943]},{"id":"symbol-542cae4028b5c3e4","tex":"c_{m,r}","aliases":"c_m,r","display":"c_{m,r}","kind":"use","occurrences":[2567,2569]},{"id":"symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","occurrences":[1442,1443,1445,1446,2306,2316,2317,2319,2323,2366,2377,2379,2380,2384,2386,2400,2406]},{"id":"symbol-3a505891c8ed892e","tex":"C_{P,j}","aliases":"C_P,j","display":"C_{P,j}","kind":"use","occurrences":[3943]},{"id":"symbol-c6bc925728d707f0","tex":"c_{p-1}","aliases":"c_p-1","display":"c_{p-1}","kind":"use","occurrences":[969]},{"id":"symbol-9980fbd02a4085b1","tex":"C_{\\mathrm{phase}}","aliases":"C_phase mathrm","display":"C_{\\mathrm{phase}}","kind":"use","occurrences":[95,100]},{"id":"symbol-e8b9f48bd6d09aed","tex":"c_q","aliases":"c_q","display":"c_q","kind":"use","occurrences":[1864,1865]},{"id":"symbol-1e2542ffdcecdf00","tex":"C_{Q,j}","aliases":"C_Q,j","display":"C_{Q,j}","kind":"use","occurrences":[3943]},{"id":"symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","occurrences":[1844,1846,1856,1864,1865]},{"id":"symbol-a6ff7660eca84d81","tex":"C_x","aliases":"C_x","display":"C_x","kind":"use","occurrences":[3773]},{"id":"symbol-637c0e5aa42a2ea2","tex":"C_\\alpha","aliases":"C_α alpha","display":"C_\\alpha","kind":"use","occurrences":[2348,2349,2350,2353,2359,2531,2532]},{"id":"symbol-581321bc41f1865f","tex":"C_\\beta","aliases":"C_β beta","display":"C_\\beta","kind":"use","occurrences":[2353,2359,2532]},{"id":"symbol-1b598868f1055551","tex":"C_{\\Lambda,j}","aliases":"C_Λ,j Lambda","display":"C_{\\Lambda,j}","kind":"use","occurrences":[3943]},{"id":"symbol-8ec5cf28b1acacd8","tex":"c_\\ell","aliases":"c_ℓ ell","display":"c_\\ell","kind":"use","occurrences":[861,862]},{"id":"symbol-67c1622d52b0e019","tex":"\\mathrm{coarse}","aliases":"coarse mathrm","display":"\\mathrm{coarse}","kind":"use","occurrences":[2998]},{"id":"symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","occurrences":[1856,2036,2096]},{"id":"symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","occurrences":[1261,1286,3563]},{"id":"symbol-b7c63d9340937698","tex":"\\mathrm{CRT}","aliases":"CRT mathrm","display":"\\mathrm{CRT}","kind":"use","occurrences":[74,75,78]},{"id":"symbol-eae54bb0ec4239ae","tex":"\\mathbf d","aliases":"d mathbf","display":"\\mathbf d","kind":"use","occurrences":[192,300,301,307,312,314,716,718,751,753,754]},{"id":"symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","occurrences":[2155,2156,3237,3239,3249,3263,3275,3279,3345,3793,3795,3796,3797,3799,3819,3850,3851,3856,3858,3859,4044]},{"id":"symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","occurrences":[3230,3272,3275,3297,3336,3337,3338,3339,3340,3343,3346,3352,3370,3374,3380,3385,3387,3393,3403,3405,3424]},{"id":"symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"use","occurrences":[450]},{"id":"symbol-cf5b0623c1cc32bf","tex":"D^{\\mathsf T}","aliases":"D^ T mathsf","display":"D^{\\mathsf T}","kind":"use","occurrences":[3798]},{"id":"symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","occurrences":[865]},{"id":"symbol-d4ef1713d263e2f3","tex":"d^*","aliases":"d^*","display":"d^*","kind":"use","occurrences":[3952,3954,3958]},{"id":"symbol-5a84c9ff6c23f102","tex":"D^+_{\\mu,U}","aliases":"D^+_μ,U mu","display":"D^+_{\\mu,U}","kind":"use","occurrences":[2617,2618,2625]},{"id":"symbol-96f254806d6d6b55","tex":"D^+_{\\mu,U^g}","aliases":"D^+_μ,U^g mu","display":"D^+_{\\mu,U^g}","kind":"use","occurrences":[2618]},{"id":"symbol-6311982fa9099aea","tex":"D^{-1/2}","aliases":"D^-1/2","display":"D^{-1/2}","kind":"use","occurrences":[3491,3499]},{"id":"symbol-a182cf4276219221","tex":"d^{-1/2}","aliases":"d^-1/2","display":"d^{-1/2}","kind":"use","occurrences":[3498]},{"id":"symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","occurrences":[3801]},{"id":"symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","occurrences":[1920]},{"id":"symbol-7b5634dbd71abc53","tex":"\\mathrm d^3","aliases":"d^3 mathrm","display":"\\mathrm d^3","kind":"use","occurrences":[3230,3374,3380,3385,3387,3393,3403,3405]},{"id":"symbol-d8ae000fbd2c1692","tex":"D^T","aliases":"D^T","display":"D^T","kind":"use","occurrences":[3981]},{"id":"symbol-e9d8746172823369","tex":"d_*","aliases":"d_*","display":"d_*","kind":"use","occurrences":[3952]},{"id":"symbol-70ffa9c8bbca2ebd","tex":"D_+","aliases":"D_+","display":"D_+","kind":"use","occurrences":[4118,4119,4121]},{"id":"symbol-ff76dfe8b26b12f2","tex":"D_-","aliases":"D_-","display":"D_-","kind":"use","occurrences":[4118,4120]},{"id":"symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","occurrences":[146]},{"id":"symbol-18df6123ecfe74ae","tex":"\\mathbf d_0","aliases":"d_0 mathbf","display":"\\mathbf d_0","kind":"use","occurrences":[751,753,754]},{"id":"symbol-589f69b8f3dcb451","tex":"D_0^{(h)}","aliases":"D_0^(h)","display":"D_0^{(h)}","kind":"use","occurrences":[134,136]},{"id":"symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","occurrences":[3039]},{"id":"symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","occurrences":[3039]},{"id":"symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","occurrences":[3934,3938,3940,3943,3945,3947,3950,3951,3952,3954,3962,3964,3978,3979]},{"id":"symbol-42dbc6c2397b687f","tex":"\\mathcal D_a","aliases":"D_a mathcal","display":"\\mathcal D_a","kind":"use","occurrences":[3237,3239,3249]},{"id":"symbol-9bf608c22091e074","tex":"d_{C_N}","aliases":"d_C_N","display":"d_{C_N}","kind":"use","occurrences":[751]},{"id":"symbol-fdadfd14bd665b99","tex":"D_h","aliases":"D_h","display":"D_h","kind":"use","occurrences":[2497,2499,3668,3669,3674,3675,3677,3678,3722,3725,3821,3823,3826]},{"id":"symbol-a7b7b693578bf211","tex":"\\mathcal D_h","aliases":"D_h mathcal","display":"\\mathcal D_h","kind":"use","occurrences":[3793,3795,3796,3797,3799,3819]},{"id":"symbol-bf1b37e9c699a1d1","tex":"d_h","aliases":"d_h","display":"d_h","kind":"use","occurrences":[3680,3681,3682,3683,3700]},{"id":"symbol-0bac9a55ad46a04a","tex":"D_{h+1}^{(h+1)}","aliases":"D_h+1^(h+1)","display":"D_{h+1}^{(h+1)}","kind":"use","occurrences":[189,190]},{"id":"symbol-fa19135e19c1fe52","tex":"D_{h,j}","aliases":"D_h,j","display":"D_{h,j}","kind":"use","occurrences":[1577,1580,1581]},{"id":"symbol-15102a51e263a437","tex":"d_{h-1}","aliases":"d_h-1","display":"d_{h-1}","kind":"use","occurrences":[203,204]},{"id":"symbol-843cae1be4629d0b","tex":"d_{h-2}","aliases":"d_h-2","display":"d_{h-2}","kind":"use","occurrences":[206]},{"id":"symbol-cd12d05aa24858ef","tex":"D_h^{\\mathsf T}","aliases":"D_h^ T mathsf","display":"D_h^{\\mathsf T}","kind":"use","occurrences":[3669]},{"id":"symbol-88af6b8b39907d56","tex":"D_h^{(h)}","aliases":"D_h^(h)","display":"D_h^{(h)}","kind":"use","occurrences":[218]},{"id":"symbol-243cb287b7b23ea4","tex":"D_h^A","aliases":"D_h^A","display":"D_h^A","kind":"use","occurrences":[2772,2778]},{"id":"symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","occurrences":[3038,3041,3043,3045]},{"id":"symbol-476e3927be5169c0","tex":"\\mathcal D_L","aliases":"D_L mathcal","display":"\\mathcal D_L","kind":"use","occurrences":[2155]},{"id":"symbol-7c4ec459e6d82d41","tex":"\\mathcal D_{L_{ba}}","aliases":"D_L_ba mathcal","display":"\\mathcal D_{L_{ba}}","kind":"use","occurrences":[2156]},{"id":"symbol-f738a79642fbf617","tex":"\\mathsf D_m","aliases":"D_m mathsf","display":"\\mathsf D_m","kind":"use","occurrences":[450]},{"id":"symbol-688ebedc522ae64c","tex":"d_m","aliases":"d_m","display":"d_m","kind":"use","occurrences":[464,473]},{"id":"symbol-5f2dbbf05d954b28","tex":"D_m^{(M)}","aliases":"D_m^(M)","display":"D_m^{(M)}","kind":"use","occurrences":[451]},{"id":"symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","occurrences":[143]},{"id":"symbol-505efcf9f8d360c3","tex":"D_r^{(h)}","aliases":"D_r^(h)","display":"D_r^{(h)}","kind":"use","occurrences":[134,135,136,150,152,153,167,173,181,190,221,257,258,273,274]},{"id":"symbol-d260dde0cb637baf","tex":"D_r^{(h+1)}","aliases":"D_r^(h+1)","display":"D_r^{(h+1)}","kind":"use","occurrences":[190]},{"id":"symbol-a90caebfafd4c4a6","tex":"D_{\\rm fine}","aliases":"D_rm fine rm","display":"D_{\\rm fine}","kind":"use","occurrences":[3006]},{"id":"symbol-eb0eb9d6a46d10ef","tex":"\\mathcal D_{\\rm matt}","aliases":"D_rm matt mathcal rm","display":"\\mathcal D_{\\rm matt}","kind":"use","occurrences":[4044]},{"id":"symbol-1864dd3f4ea54a01","tex":"D_{\\rm total}","aliases":"D_rm total rm","display":"D_{\\rm total}","kind":"use","occurrences":[3063,3064,3067,3082,3102]},{"id":"symbol-3393ff249d1d0525","tex":"D_s^{(h)}","aliases":"D_s^(h)","display":"D_s^{(h)}","kind":"use","occurrences":[136]},{"id":"symbol-d57ec070569225e9","tex":"d_T","aliases":"d_T","display":"d_T","kind":"use","occurrences":[751,753]},{"id":"symbol-cbc718b2c6c7ed2a","tex":"\\mathcal D_\\Sigma","aliases":"D_Σ mathcal Sigma","display":"\\mathcal D_\\Sigma","kind":"use","occurrences":[3345]},{"id":"symbol-84f59126b5493abd","tex":"\\mathrm d_\\Sigma","aliases":"d_Σ mathrm Sigma","display":"\\mathrm d_\\Sigma","kind":"use","occurrences":[3338,3339,3340,3346,3352]},{"id":"symbol-525495744ec84b4c","tex":"D_\\Omega","aliases":"D_Ω Omega","display":"D_\\Omega","kind":"use","occurrences":[3340,3346,3352]},{"id":"symbol-28127f9473d827e7","tex":"D_\\omega","aliases":"D_ω omega","display":"D_\\omega","kind":"use","occurrences":[3296]},{"id":"symbol-fa5a09d88f21bc1a","tex":"\\ddagger","aliases":"ddagger","display":null,"kind":"use","occurrences":[3317,3319]},{"id":"symbol-35b0e63593ef49a1","tex":"\\ddot h","aliases":"ddot h ddot","display":"\\ddot h","kind":"use","occurrences":[3776]},{"id":"symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","occurrences":[962]},{"id":"symbol-237a2ee814c86660","tex":"\\ddot\\varphi","aliases":"ddotφ ddot varphi","display":"\\ddot\\varphi","kind":"use","occurrences":[3522]},{"id":"symbol-a83e49235f2fc701","tex":"\\ddot\\varphi_j","aliases":"ddotφ_j ddot varphi","display":"\\ddot\\varphi_j","kind":"use","occurrences":[3522]},{"id":"symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","occurrences":[1984,2281,3265,3564,3984]},{"id":"symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","occurrences":[100]},{"id":"symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","occurrences":[681]},{"id":"symbol-8f42a8e27ac6c866","tex":"\\operatorname{div}","aliases":"div operatorname","display":"\\operatorname{div}","kind":"use","occurrences":[3492]},{"id":"symbol-4e181058d0a68294","tex":"\\dot A","aliases":"dot A dot","display":"\\dot A","kind":"use","occurrences":[2726,2728,2730,2756,2763,3340,3850,3899]},{"id":"symbol-6f9f83efac22f7f6","tex":"\\dot a","aliases":"dot a dot","display":"\\dot a","kind":"use","occurrences":[3546]},{"id":"symbol-fd4704b404cc6811","tex":"\\dot A_j","aliases":"dot A_j dot","display":"\\dot A_j","kind":"use","occurrences":[2726,2728,2730,2756,2763]},{"id":"symbol-30f684141cc7fd3b","tex":"\\dot A_r","aliases":"dot A_r dot","display":"\\dot A_r","kind":"use","occurrences":[3850,3899]},{"id":"symbol-e25dc6c80c9dc8ef","tex":"\\dot C","aliases":"dot C dot","display":"\\dot C","kind":"use","occurrences":[3752,3760,3763]},{"id":"symbol-c6ef1a371ad0858a","tex":"\\dot c","aliases":"dot c dot","display":"\\dot c","kind":"use","occurrences":[3525,3534]},{"id":"symbol-8b8fe0baf483f067","tex":"\\dot C_i","aliases":"dot C_i dot","display":"\\dot C_i","kind":"use","occurrences":[3752,3763]},{"id":"symbol-dc603732425b2013","tex":"\\dot c_{j+1/2}","aliases":"dot c_j+1/2 dot","display":"\\dot c_{j+1/2}","kind":"use","occurrences":[3525,3534]},{"id":"symbol-3ea553ff75be986b","tex":"\\dot c_{j-1/2}","aliases":"dot c_j-1/2 dot","display":"\\dot c_{j-1/2}","kind":"use","occurrences":[3525]},{"id":"symbol-ca7356d8caa1a365","tex":"\\dot E","aliases":"dot E dot","display":"\\dot E","kind":"use","occurrences":[2729,2730,2733,2746,2752,3528,3546]},{"id":"symbol-0f5f8f38d148f71b","tex":"\\dot e","aliases":"dot e dot","display":"\\dot e","kind":"use","occurrences":[3526]},{"id":"symbol-67df9db4a9358f3a","tex":"\\dot E_h","aliases":"dot E_h dot","display":"\\dot E_h","kind":"use","occurrences":[3528,3546]},{"id":"symbol-5cf1005dd53a7fe5","tex":"\\dot E_j","aliases":"dot E_j dot","display":"\\dot E_j","kind":"use","occurrences":[2729,2730,2733,2746,2752]},{"id":"symbol-6d96070594af5331","tex":"\\dot e_j","aliases":"dot e_j dot","display":"\\dot e_j","kind":"use","occurrences":[3526]},{"id":"symbol-f40edbe0536a4283","tex":"\\dot E_{j-1}","aliases":"dot E_j-1 dot","display":"\\dot E_{j-1}","kind":"use","occurrences":[2746]},{"id":"symbol-5c59246a191bf113","tex":"\\dot F","aliases":"dot F dot","display":"\\dot F","kind":"use","occurrences":[2241,2248,3837]},{"id":"symbol-7deecd1c9f42c06a","tex":"\\dot F_b","aliases":"dot F_b dot","display":"\\dot F_b","kind":"use","occurrences":[2241,2248]},{"id":"symbol-0c809511415d8469","tex":"\\dot{\\mathcal G}","aliases":"dot G dot mathcal","display":"\\dot{\\mathcal G}","kind":"use","occurrences":[2735,2746]},{"id":"symbol-3f95e562bedcba2c","tex":"\\dot{\\mathcal G}_j","aliases":"dot G_j dot mathcal","display":"\\dot{\\mathcal G}_j","kind":"use","occurrences":[2735,2746]},{"id":"symbol-d576368deebdc90f","tex":"\\dot H","aliases":"dot H dot","display":"\\dot H","kind":"use","occurrences":[2192,2751]},{"id":"symbol-f0d1701d0d18bb49","tex":"\\dot h","aliases":"dot h dot","display":"\\dot h","kind":"use","occurrences":[3745,3748,3758,3776]},{"id":"symbol-217fb857dc45993a","tex":"\\dot H_{\\mathrm m}","aliases":"dot H_ m dot mathrm","display":"\\dot H_{\\mathrm m}","kind":"use","occurrences":[2751]},{"id":"symbol-2d1b19f5284ca29c","tex":"\\dot H_E","aliases":"dot H_E dot","display":"\\dot H_E","kind":"use","occurrences":[2751]},{"id":"symbol-cfe2edb8994f8ac8","tex":"\\dot H_h","aliases":"dot H_h dot","display":"\\dot H_h","kind":"use","occurrences":[2751]},{"id":"symbol-9b6d09980d703f8f","tex":"\\dot h_{ij}","aliases":"dot h_ij dot","display":"\\dot h_{ij}","kind":"use","occurrences":[3748]},{"id":"symbol-ca30477e01c3f93b","tex":"\\dot H_q","aliases":"dot H_q dot","display":"\\dot H_q","kind":"use","occurrences":[2192]},{"id":"symbol-efe523b3236f6750","tex":"\\dot M","aliases":"dot M dot","display":"\\dot M","kind":"use","occurrences":[2189,2192]},{"id":"symbol-b48a467a636fcefb","tex":"\\dot m","aliases":"dot m dot","display":"\\dot m","kind":"use","occurrences":[4072,4076,4077,4078]},{"id":"symbol-6ffe398ee826021a","tex":"\\dot M_q","aliases":"dot M_q dot","display":"\\dot M_q","kind":"use","occurrences":[2189,2192]},{"id":"symbol-8e546668fa5b487a","tex":"\\dot p_1","aliases":"dot p_1 dot","display":"\\dot p_1","kind":"use","occurrences":[2171]},{"id":"symbol-974902f9939d7ec2","tex":"\\dot p_2","aliases":"dot p_2 dot","display":"\\dot p_2","kind":"use","occurrences":[2037,2098]},{"id":"symbol-8970ef006a53c2b6","tex":"\\dot p_A^{\\mathrm c}","aliases":"dot p_A^ c dot mathrm","display":"\\dot p_A^{\\mathrm c}","kind":"use","occurrences":[2063]},{"id":"symbol-ae88b8db41308a09","tex":"\\dot p_B^{\\mathrm c}","aliases":"dot p_B^ c dot mathrm","display":"\\dot p_B^{\\mathrm c}","kind":"use","occurrences":[2049,2064,2176]},{"id":"symbol-a4a1cd55f634f90d","tex":"\\dot P_R","aliases":"dot P_R dot","display":"\\dot P_R","kind":"use","occurrences":[3894]},{"id":"symbol-7ab430cdfa9f3016","tex":"\\dot P_\\Lambda","aliases":"dot P_Λ dot Lambda","display":"\\dot P_\\Lambda","kind":"use","occurrences":[3894,4076]},{"id":"symbol-db307e9343ea80e8","tex":"\\dot Q","aliases":"dot Q dot","display":"\\dot Q","kind":"use","occurrences":[3671,3674]},{"id":"symbol-c0667a78ae67ed93","tex":"\\dot R","aliases":"dot R dot","display":"\\dot R","kind":"use","occurrences":[3850,3853,3887,3962,3964]},{"id":"symbol-ea1e67564418c94f","tex":"\\dot R_j","aliases":"dot R_j dot","display":"\\dot R_j","kind":"use","occurrences":[3962]},{"id":"symbol-aacc3fef7c69d461","tex":"\\dot S","aliases":"dot S dot","display":"\\dot S","kind":"use","occurrences":[2186]},{"id":"symbol-51f3bbd1503295c5","tex":"\\dot u","aliases":"dot u dot","display":"\\dot u","kind":"use","occurrences":[922,925]},{"id":"symbol-bddb44ee5b7358ac","tex":"\\dot U^*","aliases":"dot U^* dot","display":"\\dot U^*","kind":"use","occurrences":[2083,2086,2087]},{"id":"symbol-c63fc0870cbf6ad7","tex":"\\dot U_{\\rm raw}","aliases":"dot U_rm raw dot rm","display":"\\dot U_{\\rm raw}","kind":"use","occurrences":[3723]},{"id":"symbol-ae6eb7cbea0e8357","tex":"\\dot V","aliases":"dot V dot","display":"\\dot V","kind":"use","occurrences":[3674,3724]},{"id":"symbol-3ec2563613dd0ed8","tex":"\\dot v","aliases":"dot v dot","display":"\\dot v","kind":"use","occurrences":[923,3525,3533,4087,4089]},{"id":"symbol-458452ed36c03a28","tex":"\\dot v_j","aliases":"dot v_j dot","display":"\\dot v_j","kind":"use","occurrences":[3525,3533]},{"id":"symbol-ef190f09437c0f51","tex":"\\dot V_{\\rm raw}","aliases":"dot V_rm raw dot rm","display":"\\dot V_{\\rm raw}","kind":"use","occurrences":[3724]},{"id":"symbol-60e5e5cbe336ff2a","tex":"\\dot w","aliases":"dot w dot","display":"\\dot w","kind":"use","occurrences":[822,824,2388,3522,3524,3525,3533,4095,4096]},{"id":"symbol-cb751fbbac757a0e","tex":"\\dot w_+","aliases":"dot w_+ dot","display":"\\dot w_+","kind":"use","occurrences":[4095]},{"id":"symbol-121f6cefc8878e5a","tex":"\\dot w_-","aliases":"dot w_- dot","display":"\\dot w_-","kind":"use","occurrences":[4095]},{"id":"symbol-caed6451c7e16f37","tex":"\\dot w_j","aliases":"dot w_j dot","display":"\\dot w_j","kind":"use","occurrences":[3522,3524,3525,3533]},{"id":"symbol-30ea3aa7299df5c5","tex":"\\dot w_r","aliases":"dot w_r dot","display":"\\dot w_r","kind":"use","occurrences":[822,824]},{"id":"symbol-825f9bf753046ea5","tex":"\\dot z","aliases":"dot z dot","display":"\\dot z","kind":"use","occurrences":[3699]},{"id":"symbol-c9e52621fb9f724d","tex":"\\dot{\\bar\\rho}","aliases":"dotbarρ dot bar rho","display":"\\dot{\\bar\\rho}","kind":"use","occurrences":[2735,2745,2746]},{"id":"symbol-74d7eef59b015f03","tex":"\\dot\\theta","aliases":"dotθ dot theta","display":"\\dot\\theta","kind":"use","occurrences":[2095,2098]},{"id":"symbol-a741df871091194a","tex":"\\dot\\Lambda","aliases":"dotΛ dot Lambda","display":"\\dot\\Lambda","kind":"use","occurrences":[3850,3853,3888,3892]},{"id":"symbol-93c44eb7ba707095","tex":"\\dot\\Pi","aliases":"dotΠ dot Pi","display":"\\dot\\Pi","kind":"use","occurrences":[4063,4082]},{"id":"symbol-ba309cf054813f32","tex":"\\dot\\pi","aliases":"dotπ dot pi","display":"\\dot\\pi","kind":"use","occurrences":[3748,3776]},{"id":"symbol-f3cfdbc357b7c1cb","tex":"\\dot\\pi_{ij}","aliases":"dotπ_ij dot pi","display":"\\dot\\pi_{ij}","kind":"use","occurrences":[3748]},{"id":"symbol-826d719bf280601a","tex":"\\dot\\rho","aliases":"dotρ dot rho","display":"\\dot\\rho","kind":"use","occurrences":[2109,2156,2167,2735,2738,2746]},{"id":"symbol-b72c01f71a7b828c","tex":"\\dot\\rho_{ij}","aliases":"dotρ_ij dot rho","display":"\\dot\\rho_{ij}","kind":"use","occurrences":[2167]},{"id":"symbol-3d8a0fef306e4e52","tex":"\\dot\\rho_j","aliases":"dotρ_j dot rho","display":"\\dot\\rho_j","kind":"use","occurrences":[2735,2738,2746]},{"id":"symbol-befcf78b7c16b7f6","tex":"\\dot\\Phi","aliases":"dotΦ dot Phi","display":"\\dot\\Phi","kind":"use","occurrences":[4063,4081]},{"id":"symbol-5c8b9d31659a8bf5","tex":"\\dot\\phi","aliases":"dotϕ dot phi","display":"\\dot\\phi","kind":"use","occurrences":[2079,2087]},{"id":"symbol-4512c0db0b524163","tex":"\\dot\\varphi","aliases":"dotφ dot varphi","display":"\\dot\\varphi","kind":"use","occurrences":[3521,3522,3523,3524,3525,3530,3533,3534,3535,3546,4063]},{"id":"symbol-9a8c0cc0f1e0e9e6","tex":"\\dot\\varphi_j","aliases":"dotφ_j dot varphi","display":"\\dot\\varphi_j","kind":"use","occurrences":[3522,3523,3524,3525,3533,3534,3535]},{"id":"symbol-d706b990909bd88e","tex":"\\dot\\varphi_{j+1}","aliases":"dotφ_j+1 dot varphi","display":"\\dot\\varphi_{j+1}","kind":"use","occurrences":[3525,3534,3535]},{"id":"symbol-cd1a0c9fdd3cf81a","tex":"\\dot\\varphi_j^2","aliases":"dotφ_j^2 dot varphi","display":"\\dot\\varphi_j^2","kind":"use","occurrences":[3521,3525,3533,3546]},{"id":"symbol-302ba2bfdf019cf6","tex":"\\dot\\chi","aliases":"dotχ dot chi","display":"\\dot\\chi","kind":"use","occurrences":[2762,2764]},{"id":"symbol-0bcb61993d85cfe8","tex":"\\dot\\chi_j","aliases":"dotχ_j dot chi","display":"\\dot\\chi_j","kind":"use","occurrences":[2762,2764]},{"id":"symbol-56ba2edb3284c438","tex":"\\dot\\chi_{j+1}","aliases":"dotχ_j+1 dot chi","display":"\\dot\\chi_{j+1}","kind":"use","occurrences":[2764]},{"id":"symbol-47f08e6aab43dd03","tex":"\\dot\\psi_j","aliases":"dotψ_j dot psi","display":"\\dot\\psi_j","kind":"use","occurrences":[2726,2727,2730,2738,2753,2754,2755]},{"id":"symbol-6038007d6ae6dd28","tex":"\\dot\\psi_j^*","aliases":"dotψ_j^* dot psi","display":"\\dot\\psi_j^*","kind":"use","occurrences":[2726]},{"id":"symbol-bb79f280e36ef7a2","tex":"\\dot\\psi_r","aliases":"dotψ_r dot psi","display":"\\dot\\psi_r","kind":"use","occurrences":[819]},{"id":"symbol-8052e1ac0acaea2d","tex":"\\dot\\Omega","aliases":"dotΩ dot Omega","display":"\\dot\\Omega","kind":"use","occurrences":[3340]},{"id":"symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","occurrences":[3581,3930,4127]},{"id":"symbol-7015f1f6989ac898","tex":"\\mathbf E","aliases":"E mathbf","display":"\\mathbf E","kind":"use","occurrences":[1236,1240]},{"id":"symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","occurrences":[1925,1926,1930,1933,3158,3159,3160,3163,3379,3380,3381,3402,3403,3405,3623,3626,3631,3723,3724,3725,4059,4079,4080,4083,4087,4101,4103,4106]},{"id":"symbol-303b235f22c45f55","tex":"E'_{L,0}","aliases":"E'_L,0","display":"E'_{L,0}","kind":"use","occurrences":[1235]},{"id":"symbol-2099c82204eda1e6","tex":"E'_{L,j}","aliases":"E'_L,j","display":"E'_{L,j}","kind":"use","occurrences":[1232]},{"id":"symbol-67b08d219bc9d5a2","tex":"e^{(L_g+S_1/2)t}","aliases":"e^(L_g+S_1/2)t","display":"e^{(L_g+S_1/2)t}","kind":"use","occurrences":[3972]},{"id":"symbol-dbf30aa11f3ef3cc","tex":"e^{-(r-10)^2}","aliases":"e^-(r-10)^2","display":"e^{-(r-10)^2}","kind":"use","occurrences":[4131]},{"id":"symbol-d7344e7d866b8403","tex":"e^{-(r_i+r_j)t/2}","aliases":"e^-(r_i+r_j)t/2","display":"e^{-(r_i+r_j)t/2}","kind":"use","occurrences":[2168]},{"id":"symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","occurrences":[3416]},{"id":"symbol-fbb1629ff78a5896","tex":"e^{-2kt}","aliases":"e^-2kt","display":"e^{-2kt}","kind":"use","occurrences":[2171]},{"id":"symbol-a1b4272a33792a85","tex":"e^{-2P}","aliases":"e^-2P","display":"e^{-2P}","kind":"use","occurrences":[3261]},{"id":"symbol-e4256bd4d37cf206","tex":"e^{-2\\beta\\epsilon_m}","aliases":"e^-2βϵ_m beta epsilon","display":"e^{-2\\beta\\epsilon_m}","kind":"use","occurrences":[592]},{"id":"symbol-462eff40a0d76a62","tex":"e^{-2\\pi ijk/M}","aliases":"e^-2π ijk/M pi","display":"e^{-2\\pi ijk/M}","kind":"use","occurrences":[829]},{"id":"symbol-6332bad66e756810","tex":"e^{-2\\tau}","aliases":"e^-2τ tau","display":"e^{-2\\tau}","kind":"use","occurrences":[3620]},{"id":"symbol-414a8cb533bd9054","tex":"e^{-2\\phi}","aliases":"e^-2ϕ phi","display":"e^{-2\\phi}","kind":"use","occurrences":[3556,3569]},{"id":"symbol-6aa40be28a41f8ce","tex":"e^{-hA}","aliases":"e^-hA","display":"e^{-hA}","kind":"use","occurrences":[2845]},{"id":"symbol-bfb1cd35e1e6c75a","tex":"e^{-hB}","aliases":"e^-hB","display":"e^{-hB}","kind":"use","occurrences":[2845]},{"id":"symbol-46080bafb5da7953","tex":"e^{-i(s+t)E}","aliases":"e^-i(s+t)E","display":"e^{-i(s+t)E}","kind":"use","occurrences":[1651]},{"id":"symbol-515496daf359dcbc","tex":"e^{-i(t-s)A}","aliases":"e^-i(t-s)A","display":"e^{-i(t-s)A}","kind":"use","occurrences":[1880,1881,1884]},{"id":"symbol-b62117803ef27c0d","tex":"e^{-i(t-s)D}","aliases":"e^-i(t-s)D","display":"e^{-i(t-s)D}","kind":"use","occurrences":[926]},{"id":"symbol-2ea38b9782a5a3dd","tex":"e^{-i(t-s)H_\\lambda}","aliases":"e^-i(t-s)H_λ lambda","display":"e^{-i(t-s)H_\\lambda}","kind":"use","occurrences":[685]},{"id":"symbol-7e78dacd05eb2880","tex":"e^{-igtZ_{\\mathrm{sign}}}","aliases":"e^-igtZ_sign mathrm","display":"e^{-igtZ_{\\mathrm{sign}}}","kind":"use","occurrences":[656]},{"id":"symbol-426af39a89a86187","tex":"e^{-ih^2}","aliases":"e^-ih^2","display":"e^{-ih^2}","kind":"use","occurrences":[2800]},{"id":"symbol-ca72e42f27fac12f","tex":"e^{-ihA(x)}","aliases":"e^-ihA(x)","display":"e^{-ihA(x)}","kind":"use","occurrences":[2799]},{"id":"symbol-5168a10b25fab030","tex":"e^{-ihA(x+h/2)}","aliases":"e^-ihA(x+h/2)","display":"e^{-ihA(x+h/2)}","kind":"use","occurrences":[2790]},{"id":"symbol-06c04b308af64a05","tex":"e^{-ihA_j}","aliases":"e^-ihA_j","display":"e^{-ihA_j}","kind":"use","occurrences":[2723,2802]},{"id":"symbol-ba26c168a779953c","tex":"e^{-ihp_j(n)}","aliases":"e^-ihp_j(n)","display":"e^{-ihp_j(n)}","kind":"use","occurrences":[1580]},{"id":"symbol-938d1b8f1712512f","tex":"e^{-\\operatorname{Im}z\\,\\tau}","aliases":"e^-Imz τ operatorname tau","display":"e^{-\\operatorname{Im}z\\,\\tau}","kind":"use","occurrences":[932]},{"id":"symbol-a3fbb729a001bbbc","tex":"e^{-isB}","aliases":"e^-isB","display":"e^{-isB}","kind":"use","occurrences":[1873,1875,1880,1881,1884]},{"id":"symbol-a3cd2d38b3e555da","tex":"e^{-isH_0}","aliases":"e^-isH_0","display":"e^{-isH_0}","kind":"use","occurrences":[685,687]},{"id":"symbol-a01743648714134d","tex":"e^{-it}","aliases":"e^-it","display":"e^{-it}","kind":"use","occurrences":[1847,1850]},{"id":"symbol-4a353a46a2b6c1b4","tex":"e^{-it\\mathcal A H\\mathcal A^*}","aliases":"e^-it A H A^* mathcal mathcal","display":"e^{-it\\mathcal A H\\mathcal A^*}","kind":"use","occurrences":[814]},{"id":"symbol-bd6eab3f096ffab4","tex":"e^{-it_0\\epsilon_m}","aliases":"e^-it_0ϵ_m epsilon","display":"e^{-it_0\\epsilon_m}","kind":"use","occurrences":[531]},{"id":"symbol-9eaf8275847a72c8","tex":"e^{-itA}","aliases":"e^-itA","display":"e^{-itA}","kind":"use","occurrences":[1871,1882,1884]},{"id":"symbol-e39fb67a6a73c3c4","tex":"e^{-itB}","aliases":"e^-itB","display":"e^{-itB}","kind":"use","occurrences":[1871,1883,1884]},{"id":"symbol-ac8c969ea731f537","tex":"e^{-itD}","aliases":"e^-itD","display":"e^{-itD}","kind":"use","occurrences":[925,926,936]},{"id":"symbol-fae7499816e18e18","tex":"e^{-itE}","aliases":"e^-itE","display":"e^{-itE}","kind":"use","occurrences":[1612,1654]},{"id":"symbol-2637b6dd9fae991f","tex":"e^{-itH}","aliases":"e^-itH","display":"e^{-itH}","kind":"use","occurrences":[814,1603]},{"id":"symbol-71404fd53acc476a","tex":"e^{-itH_0}","aliases":"e^-itH_0","display":"e^{-itH_0}","kind":"use","occurrences":[684]},{"id":"symbol-33b5b515ddb5e759","tex":"e^{-itH_A}","aliases":"e^-itH_A","display":"e^{-itH_A}","kind":"use","occurrences":[623]},{"id":"symbol-46067ecf745d5b8c","tex":"e^{-\\mathrm{i}tH_c}","aliases":"e^-itH_c mathrm","display":"e^{-\\mathrm{i}tH_c}","kind":"use","occurrences":[249]},{"id":"symbol-7d3c46f1ac08755a","tex":"e^{-\\mathrm{i}tH_f}","aliases":"e^-itH_f mathrm","display":"e^{-\\mathrm{i}tH_f}","kind":"use","occurrences":[249,286]},{"id":"symbol-77fd83ae503c187e","tex":"e^{-itH_m^{\\mathrm{phys}}}","aliases":"e^-itH_m^phys mathrm","display":"e^{-itH_m^{\\mathrm{phys}}}","kind":"use","occurrences":[1355]},{"id":"symbol-dc773578bf2509d3","tex":"e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}","aliases":"e^-itH_p,h^Haar mathrm mathrm","display":"e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}","kind":"use","occurrences":[274]},{"id":"symbol-4ce53f9cf88dacbf","tex":"e^{-itH_\\epsilon}","aliases":"e^-itH_ϵ epsilon","display":"e^{-itH_\\epsilon}","kind":"use","occurrences":[528]},{"id":"symbol-6ee9845200fd38a4","tex":"e^{-itH_\\lambda}","aliases":"e^-itH_λ lambda","display":"e^{-itH_\\lambda}","kind":"use","occurrences":[677,685,694]},{"id":"symbol-2f444d0d93addeee","tex":"e^{-itH_\\infty}","aliases":"e^-itH_∞ infty","display":"e^{-itH_\\infty}","kind":"use","occurrences":[1284,1291,1300]},{"id":"symbol-f75d36d14a6c2f17","tex":"e^{-it\\widetilde H_M}","aliases":"e^-itwidetilde H_M widetilde","display":"e^{-it\\widetilde H_M}","kind":"use","occurrences":[558]},{"id":"symbol-c8ba533552f605b6","tex":"e^{-it\\epsilon}","aliases":"e^-itϵ epsilon","display":"e^{-it\\epsilon}","kind":"use","occurrences":[535]},{"id":"symbol-5ce269a7f5673f0b","tex":"e^{-it\\epsilon_m}","aliases":"e^-itϵ_m epsilon","display":"e^{-it\\epsilon_m}","kind":"use","occurrences":[528,531,542,553]},{"id":"symbol-a3585bee2603724b","tex":"e^{-\\mathrm{i}t\\epsilon_{p,0}}","aliases":"e^-itϵ_p,0 mathrm epsilon","display":"e^{-\\mathrm{i}t\\epsilon_{p,0}}","kind":"use","occurrences":[276]},{"id":"symbol-b9adfb35026d06fa","tex":"e^{-\\mathrm{i}t\\epsilon_{p,r}}","aliases":"e^-itϵ_p,r mathrm epsilon","display":"e^{-\\mathrm{i}t\\epsilon_{p,r}}","kind":"use","occurrences":[274,276]},{"id":"symbol-ce034acd93c2c8ef","tex":"e^{-it\\lambda}","aliases":"e^-itλ lambda","display":"e^{-it\\lambda}","kind":"use","occurrences":[1283,2449]},{"id":"symbol-9638a772c36bf7ae","tex":"e^{-it\\lambda_n}","aliases":"e^-itλ_n lambda","display":"e^{-it\\lambda_n}","kind":"use","occurrences":[2366,2377,2384]},{"id":"symbol-53b868ab17bc229d","tex":"e^{-iu}","aliases":"e^-iu","display":"e^{-iu}","kind":"use","occurrences":[2378]},{"id":"symbol-5a7fb061cea5b411","tex":"e^{-iuA}","aliases":"e^-iuA","display":"e^{-iuA}","kind":"use","occurrences":[1878,1885,1886]},{"id":"symbol-47ca6386cf2c63a4","tex":"e^{-iuK}","aliases":"e^-iuK","display":"e^{-iuK}","kind":"use","occurrences":[696]},{"id":"symbol-58d59de3254bae16","tex":"e^{-i\\alpha_q}","aliases":"e^-iα_q alpha","display":"e^{-i\\alpha_q}","kind":"use","occurrences":[854]},{"id":"symbol-f9083e6f281158cc","tex":"e^{-i\\theta}","aliases":"e^-iθ theta","display":"e^{-i\\theta}","kind":"use","occurrences":[841,844,2690]},{"id":"symbol-d91f717734c65b3e","tex":"e^{-i\\theta_j}","aliases":"e^-iθ_j theta","display":"e^{-i\\theta_j}","kind":"use","occurrences":[840]},{"id":"symbol-09975dc298fd8d03","tex":"e^{-i\\tau D}","aliases":"e^-iτ D tau","display":"e^{-i\\tau D}","kind":"use","occurrences":[925]},{"id":"symbol-032fc47c4d958027","tex":"e^{-i\\tau J_{1:h}}","aliases":"e^-iτ J_1:h tau","display":"e^{-i\\tau J_{1:h}}","kind":"use","occurrences":[1026]},{"id":"symbol-8c46d727c71945e9","tex":"e^{-kt}","aliases":"e^-kt","display":"e^{-kt}","kind":"use","occurrences":[2171]},{"id":"symbol-bd14305f0c1887f7","tex":"e^{-r^2/(2\\sigma^2)}","aliases":"e^-r^2/(2σ^2) sigma","display":"e^{-r^2/(2\\sigma^2)}","kind":"use","occurrences":[3604,3605]},{"id":"symbol-543348375bb29268","tex":"e^{-S/\\hbar}","aliases":"e^-S/ℏ hbar","display":"e^{-S/\\hbar}","kind":"use","occurrences":[3500]},{"id":"symbol-e7a07fecc096889f","tex":"e^{-t(N+1)}","aliases":"e^-t(N+1)","display":"e^{-t(N+1)}","kind":"use","occurrences":[2455]},{"id":"symbol-9524e9b54a42d871","tex":"e^{-tn}","aliases":"e^-tn","display":"e^{-tn}","kind":"use","occurrences":[2454]},{"id":"symbol-04c554b9071d41ff","tex":"e^{-|\\alpha|^2/2}","aliases":"e^-|α|^2/2 alpha","display":"e^{-|\\alpha|^2/2}","kind":"use","occurrences":[1329]},{"id":"symbol-ea459297b3497aa4","tex":"e^{-\\beta H}","aliases":"e^-β H beta","display":"e^{-\\beta H}","kind":"use","occurrences":[255]},{"id":"symbol-5b952d109b96b1f8","tex":"e^{-\\beta H_c}","aliases":"e^-β H_c beta","display":"e^{-\\beta H_c}","kind":"use","occurrences":[249]},{"id":"symbol-ec933d1e365cdeb7","tex":"e^{-\\beta H_f}","aliases":"e^-β H_f beta","display":"e^{-\\beta H_f}","kind":"use","occurrences":[249]},{"id":"symbol-a8526e6c4580f2c3","tex":"e^{-\\beta H_\\epsilon}","aliases":"e^-β H_ϵ beta epsilon","display":"e^{-\\beta H_\\epsilon}","kind":"use","occurrences":[591,593]},{"id":"symbol-a5580f2eb87c1778","tex":"e^{-\\beta\\epsilon_0}","aliases":"e^-βϵ_0 beta epsilon","display":"e^{-\\beta\\epsilon_0}","kind":"use","occurrences":[593]},{"id":"symbol-5961b144710322df","tex":"e^{-\\beta\\epsilon_m}","aliases":"e^-βϵ_m beta epsilon","display":"e^{-\\beta\\epsilon_m}","kind":"use","occurrences":[593]},{"id":"symbol-3f49e8cdefb83b41","tex":"e^{-\\mu}","aliases":"e^-μ mu","display":"e^{-\\mu}","kind":"use","occurrences":[1330,1335]},{"id":"symbol-53b84cd8e738559b","tex":"e^{-\\mu(1-\\cos(2\\pi/L))}","aliases":"e^-μ(1-cos(2π/L)) mu cos pi","display":"e^{-\\mu(1-\\cos(2\\pi/L))}","kind":"use","occurrences":[1339]},{"id":"symbol-2af989611d84b3ff","tex":"e^{-\\mu(1-\\cos(2\\pi\\ell/L))}","aliases":"e^-μ(1-cos(2πℓ/L)) mu cos pi ell","display":"e^{-\\mu(1-\\cos(2\\pi\\ell/L))}","kind":"use","occurrences":[1339]},{"id":"symbol-8e4608509fba303e","tex":"e^{-\\tau}","aliases":"e^-τ tau","display":"e^{-\\tau}","kind":"use","occurrences":[3260,3652,3653]},{"id":"symbol-15348357d90941e2","tex":"e^{-\\chi}","aliases":"e^-χ chi","display":"e^{-\\chi}","kind":"use","occurrences":[3566]},{"id":"symbol-99e4c97cf0f019b5","tex":"E^0","aliases":"E^0","display":"E^0","kind":"use","occurrences":[3370]},{"id":"symbol-1feddf7f1342e789","tex":"e^0","aliases":"e^0","display":"e^0","kind":"use","occurrences":[3290,3553]},{"id":"symbol-ec1ff447b7876d8d","tex":"e^1","aliases":"e^1","display":"e^1","kind":"use","occurrences":[3290,3553]},{"id":"symbol-26deaea0ab56be9e","tex":"e^{1/2}","aliases":"e^1/2","display":"e^{1/2}","kind":"use","occurrences":[3390,3391,3414,3417]},{"id":"symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","occurrences":[3291]},{"id":"symbol-2cc4e438f644083b","tex":"e^{2b\\,\\operatorname{area}(\\square)K}","aliases":"e^2b area(square)K operatorname square","display":"e^{2b\\,\\operatorname{area}(\\square)K}","kind":"use","occurrences":[3569]},{"id":"symbol-7dc913e427c2b1e7","tex":"e^{2P}","aliases":"e^2P","display":"e^{2P}","kind":"use","occurrences":[3261,3621,3623,3625,3627,3631,3637,3674,3678,3725]},{"id":"symbol-e9aa10317885fcb2","tex":"e^{2P_j}","aliases":"e^2P_j","display":"e^{2P_j}","kind":"use","occurrences":[3675]},{"id":"symbol-6ea4a26d6415bea6","tex":"e^{2|t|E_n}","aliases":"e^2|t|E_n","display":"e^{2|t|E_n}","kind":"use","occurrences":[1780]},{"id":"symbol-a7ea9b34c288dc20","tex":"e^{2|t||\\lambda|}","aliases":"e^2|t||λ| lambda","display":"e^{2|t||\\lambda|}","kind":"use","occurrences":[1296]},{"id":"symbol-23fccd81348570f0","tex":"e^{2\\pi i n\\cdot x}","aliases":"e^2π i n· x pi cdot","display":"e^{2\\pi i n\\cdot x}","kind":"use","occurrences":[1413,1522]},{"id":"symbol-ede1673d5c208439","tex":"e^{2\\pi i(k-n)\\cdot x}","aliases":"e^2π i(k-n)· x pi cdot","display":"e^{2\\pi i(k-n)\\cdot x}","kind":"use","occurrences":[1671]},{"id":"symbol-f81e7a1f8753f574","tex":"e^{2\\pi i(x_1+x_2+x_3)}","aliases":"e^2π i(x_1+x_2+x_3) pi","display":"e^{2\\pi i(x_1+x_2+x_3)}","kind":"use","occurrences":[1850,1855]},{"id":"symbol-11bafad14ed5d97c","tex":"e^{2\\pi i/L}","aliases":"e^2π i/L pi","display":"e^{2\\pi i/L}","kind":"use","occurrences":[1205]},{"id":"symbol-450272f382dba42d","tex":"e^{2\\pi ir}","aliases":"e^2π ir pi","display":"e^{2\\pi ir}","kind":"use","occurrences":[1678]},{"id":"symbol-212c2fd58b825ca3","tex":"e^{2\\pi irx}","aliases":"e^2π irx pi","display":"e^{2\\pi irx}","kind":"use","occurrences":[1673]},{"id":"symbol-014cd8dee73ba62b","tex":"e^{2\\pi i\\widehat N/L}","aliases":"e^2π iwidehat N/L pi widehat","display":"e^{2\\pi i\\widehat N/L}","kind":"use","occurrences":[1323]},{"id":"symbol-f430d97b17e0353a","tex":"e^{2\\pi ix_1}","aliases":"e^2π ix_1 pi","display":"e^{2\\pi ix_1}","kind":"use","occurrences":[1850,1855]},{"id":"symbol-0dedbe3b64a4ae5e","tex":"e^{2\\phi}","aliases":"e^2ϕ phi","display":"e^{2\\phi}","kind":"use","occurrences":[3553,3558,3570]},{"id":"symbol-22e244f5761b2581","tex":"e^3","aliases":"e^3","display":"e^3","kind":"use","occurrences":[3291]},{"id":"symbol-f79ae30615c4831b","tex":"\\mathcal E^a","aliases":"E^a mathcal","display":"\\mathcal E^a","kind":"use","occurrences":[3379,3380,3381,3402,3403,3405]},{"id":"symbol-b56e4649574b8fa0","tex":"e^a","aliases":"e^a","display":"e^a","kind":"use","occurrences":[1251,1252]},{"id":"symbol-ba0a0499ab6712e5","tex":"E^a_i","aliases":"E^a_i","display":"E^a_i","kind":"use","occurrences":[3232,3237,3238,3240,3243]},{"id":"symbol-bd4227990f2f867b","tex":"E^a_k","aliases":"E^a_k","display":"E^a_k","kind":"use","occurrences":[3237]},{"id":"symbol-42f98c4694f18001","tex":"\\mathcal E^b","aliases":"E^b mathcal","display":"\\mathcal E^b","kind":"use","occurrences":[3381]},{"id":"symbol-dd7eab8cdf3e9482","tex":"E^b_i","aliases":"E^b_i","display":"E^b_i","kind":"use","occurrences":[3237]},{"id":"symbol-66eb38162a08955e","tex":"E^b_j","aliases":"E^b_j","display":"E^b_j","kind":"use","occurrences":[3237,3238]},{"id":"symbol-4ac68a8f5d0d5665","tex":"E^{bi}","aliases":"E^bi","display":"E^{bi}","kind":"use","occurrences":[3243]},{"id":"symbol-2b1300a4ddb90178","tex":"E^c_k","aliases":"E^c_k","display":"E^c_k","kind":"use","occurrences":[3238]},{"id":"symbol-65082530d34bd1ec","tex":"e^{hA}","aliases":"e^hA","display":"e^{hA}","kind":"use","occurrences":[2845]},{"id":"symbol-cf13ff3dd80732f9","tex":"e^{hB}","aliases":"e^hB","display":"e^{hB}","kind":"use","occurrences":[2845]},{"id":"symbol-e0f8b8099b1a4d62","tex":"E^I","aliases":"E^I","display":"E^I","kind":"use","occurrences":[3337,3341]},{"id":"symbol-e9e2a2077329c507","tex":"E^i","aliases":"E^i","display":"E^i","kind":"use","occurrences":[3370]},{"id":"symbol-dfd4f967373eb983","tex":"e^I","aliases":"e^I","display":"e^I","kind":"use","occurrences":[3267,3272,3279,3286,3337]},{"id":"symbol-4bd6cf0e6c9e8055","tex":"e^{i p(n)\\cdot x}","aliases":"e^i p(n)· x cdot","display":"e^{i p(n)\\cdot x}","kind":"use","occurrences":[1522]},{"id":"symbol-6711f41ccb2a9da4","tex":"e^{i(s+t)E}","aliases":"e^i(s+t)E","display":"e^{i(s+t)E}","kind":"use","occurrences":[1651]},{"id":"symbol-3d7d478f333b07a6","tex":"e^{i(z-\\lambda)\\tau}","aliases":"e^i(z-λ)τ lambda tau","display":"e^{i(z-\\lambda)\\tau}","kind":"use","occurrences":[934]},{"id":"symbol-85b4fb052ac49250","tex":"e^{ia}","aliases":"e^ia","display":"e^{ia}","kind":"use","occurrences":[2792]},{"id":"symbol-f4ea3d6c305c710f","tex":"e^{ib}","aliases":"e^ib","display":"e^{ib}","kind":"use","occurrences":[2792]},{"id":"symbol-e94881ab2b7c428e","tex":"e^{ih^2F_{ij}}","aliases":"e^ih^2F_ij","display":"e^{ih^2F_{ij}}","kind":"use","occurrences":[3080]},{"id":"symbol-55f51293b683c65d","tex":"e^{ihp_j(n)}","aliases":"e^ihp_j(n)","display":"e^{ihp_j(n)}","kind":"use","occurrences":[1580]},{"id":"symbol-ed7844fbf82ff3f5","tex":"e^{ik\\cdot x}","aliases":"e^ik· x cdot","display":"e^{ik\\cdot x}","kind":"use","occurrences":[3220]},{"id":"symbol-24d92c2ffc4b376f","tex":"e^{ip\\alpha_q}","aliases":"e^ipα_q alpha","display":"e^{ip\\alpha_q}","kind":"use","occurrences":[855]},{"id":"symbol-7663ef411b37c4f9","tex":"e^{is}","aliases":"e^is","display":"e^{is}","kind":"use","occurrences":[3097]},{"id":"symbol-1d3442f53084db4f","tex":"e^{iS_C/\\hbar}","aliases":"e^iS_C/ℏ hbar","display":"e^{iS_C/\\hbar}","kind":"use","occurrences":[3505,3506]},{"id":"symbol-0d43ec957b56df25","tex":"e^{iS_F/\\hbar}","aliases":"e^iS_F/ℏ hbar","display":"e^{iS_F/\\hbar}","kind":"use","occurrences":[3505,3506]},{"id":"symbol-9fd148b730c54fcd","tex":"e^{it}","aliases":"e^it","display":"e^{it}","kind":"use","occurrences":[3081,3097]},{"id":"symbol-8499e0c2a6028581","tex":"e^{itD}","aliases":"e^itD","display":"e^{itD}","kind":"use","occurrences":[929]},{"id":"symbol-97c26b61dd48806a","tex":"e^{itE}","aliases":"e^itE","display":"e^{itE}","kind":"use","occurrences":[1612,1654]},{"id":"symbol-4ce7d415045642ef","tex":"e^{iuA}","aliases":"e^iuA","display":"e^{iuA}","kind":"use","occurrences":[1885,1886]},{"id":"symbol-a50b74d94a7f721f","tex":"e^{iz\\tau}","aliases":"e^izτ tau","display":"e^{iz\\tau}","kind":"use","occurrences":[928]},{"id":"symbol-d1901d33f29f2a30","tex":"e^{i\\alpha_qr}","aliases":"e^iα_qr alpha","display":"e^{i\\alpha_qr}","kind":"use","occurrences":[853]},{"id":"symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","occurrences":[855,2690,2697]},{"id":"symbol-add9683f9060c3e6","tex":"e^{i\\theta_f}","aliases":"e^iθ_f theta","display":"e^{i\\theta_f}","kind":"use","occurrences":[2694]},{"id":"symbol-f2fd45f05bfc81bb","tex":"e^{i\\theta_i(p,h)}","aliases":"e^iθ_i(p,h) theta","display":"e^{i\\theta_i(p,h)}","kind":"use","occurrences":[3070]},{"id":"symbol-01b838c630a22dad","tex":"e^{i\\chi_j}","aliases":"e^iχ_j chi","display":"e^{i\\chi_j}","kind":"use","occurrences":[2758]},{"id":"symbol-c115057b598e60d1","tex":"E^J","aliases":"E^J","display":"E^J","kind":"use","occurrences":[3341,3345]},{"id":"symbol-5d361bdf2478b59b","tex":"E^j","aliases":"E^j","display":"E^j","kind":"use","occurrences":[3374]},{"id":"symbol-c77a77855f0f7f8b","tex":"e^J","aliases":"e^J","display":"e^J","kind":"use","occurrences":[3272,3279,3286]},{"id":"symbol-0c9eb1acc19f1994","tex":"E^K","aliases":"E^K","display":"E^K","kind":"use","occurrences":[3341,3345]},{"id":"symbol-b607a74a7267e802","tex":"E^k","aliases":"E^k","display":"E^k","kind":"use","occurrences":[1629,3374]},{"id":"symbol-420daa57bb2be484","tex":"e^K","aliases":"e^K","display":"e^K","kind":"use","occurrences":[3272,3397]},{"id":"symbol-52ee39a17082ee58","tex":"E^L","aliases":"E^L","display":"E^L","kind":"use","occurrences":[3341,3345]},{"id":"symbol-e0cff6c1252f56cd","tex":"e^L","aliases":"e^L","display":"e^L","kind":"use","occurrences":[3272,3397]},{"id":"symbol-bb582ca6960394af","tex":"e^{LT}","aliases":"e^LT","display":"e^{LT}","kind":"use","occurrences":[3715]},{"id":"symbol-ed38d997f7481198","tex":"e^R","aliases":"e^R","display":"e^R","kind":"use","occurrences":[1772]},{"id":"symbol-4b68cf7b255e2e9f","tex":"e^{R\\|B\\|}","aliases":"e^R B","display":"e^{R\\|B\\|}","kind":"use","occurrences":[1874]},{"id":"symbol-3dd973573165efc7","tex":"e^{tA_0}","aliases":"e^tA_0","display":"e^{tA_0}","kind":"use","occurrences":[1249]},{"id":"symbol-eb1d3777cb0c1a31","tex":"e^X","aliases":"e^X","display":"e^X","kind":"use","occurrences":[1772]},{"id":"symbol-1df81a064aee7511","tex":"e^Y","aliases":"e^Y","display":"e^Y","kind":"use","occurrences":[1772]},{"id":"symbol-8611063e24d14e09","tex":"e^z","aliases":"e^z","display":"e^z","kind":"use","occurrences":[1215]},{"id":"symbol-46b65184169dd9a1","tex":"e^{zA}","aliases":"e^zA","display":"e^{zA}","kind":"use","occurrences":[1795]},{"id":"symbol-cecf606487256a5e","tex":"e^{zA^{\\mathcal H}}","aliases":"e^zA^ H mathcal","display":"e^{zA^{\\mathcal H}}","kind":"use","occurrences":[1795]},{"id":"symbol-4eac7219693d8601","tex":"e^{zC_L}","aliases":"e^zC_L","display":"e^{zC_L}","kind":"use","occurrences":[1240]},{"id":"symbol-780a43111d19c036","tex":"e^{|t|E_n}","aliases":"e^|t|E_n","display":"e^{|t|E_n}","kind":"use","occurrences":[1778]},{"id":"symbol-cce7c34a27a2a30a","tex":"e^{|t||\\lambda|}","aliases":"e^|t||λ| lambda","display":"e^{|t||\\lambda|}","kind":"use","occurrences":[1297]},{"id":"symbol-c99f921ba3d45581","tex":"e^{\\pm itE_{h,n}}","aliases":"e^± itE_h,n pm","display":"e^{\\pm itE_{h,n}}","kind":"use","occurrences":[1755]},{"id":"symbol-d9d6541bc9c84d0f","tex":"e^{\\pm itE_n}","aliases":"e^± itE_n pm","display":"e^{\\pm itE_n}","kind":"use","occurrences":[1755]},{"id":"symbol-e19ad9bc486a5d76","tex":"e^{\\pm its}","aliases":"e^± its pm","display":"e^{\\pm its}","kind":"use","occurrences":[1756]},{"id":"symbol-356289f0a237ed70","tex":"e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}","aliases":"e^μ(Reω_L^ℓ-1) mu operatorname omega ell","display":"e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}","kind":"use","occurrences":[1342]},{"id":"symbol-cf514f58b73ee1a0","tex":"e^{\\mu(\\omega_L^\\ell-1)}","aliases":"e^μ(ω_L^ℓ-1) mu omega ell","display":"e^{\\mu(\\omega_L^\\ell-1)}","kind":"use","occurrences":[1330]},{"id":"symbol-b70233ddf00e967e","tex":"e^\\phi","aliases":"e^ϕ phi","display":"e^\\phi","kind":"use","occurrences":[3553]},{"id":"symbol-3980ccd55c0d7847","tex":"e^{\\chi}","aliases":"e^χ chi","display":"e^{\\chi}","kind":"use","occurrences":[3565]},{"id":"symbol-0073819703c9eac7","tex":"e^{\\chi K}","aliases":"e^χ K chi","display":"e^{\\chi K}","kind":"use","occurrences":[3563]},{"id":"symbol-28c070e65388214e","tex":"e^{\\chi(t,x)K}","aliases":"e^χ(t,x)K chi","display":"e^{\\chi(t,x)K}","kind":"use","occurrences":[3572]},{"id":"symbol-6956e1ff41ab0d60","tex":"e^{\\omega_L^\\ell tA_0}","aliases":"e^ω_L^ℓ tA_0 omega ell","display":"e^{\\omega_L^\\ell tA_0}","kind":"use","occurrences":[1249]},{"id":"symbol-5e6baee59f187d10","tex":"e^{\\omega_L^\\ell z}","aliases":"e^ω_L^ℓ z omega ell","display":"e^{\\omega_L^\\ell z}","kind":"use","occurrences":[1243]},{"id":"symbol-a48ab850207c4126","tex":"\\mathsf E_{\\mathbf h}","aliases":"E_ h mathsf mathbf","display":"\\mathsf E_{\\mathbf h}","kind":"use","occurrences":[606,607,612,622,623,634]},{"id":"symbol-db8e0e8ced5b248a","tex":"\\mathsf E_{\\mathbf k}","aliases":"E_ k mathsf mathbf","display":"\\mathsf E_{\\mathbf k}","kind":"use","occurrences":[607]},{"id":"symbol-d8be2ba7e790af78","tex":"E_+","aliases":"E_+","display":"E_+","kind":"use","occurrences":[112,113]},{"id":"symbol-ca57caa14f72ba3b","tex":"E_-","aliases":"E_-","display":"E_-","kind":"use","occurrences":[112,113]},{"id":"symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","occurrences":[149]},{"id":"symbol-f912500efb3ee692","tex":"\\mathsf E_0","aliases":"E_0 mathsf","display":"\\mathsf E_0","kind":"use","occurrences":[447,460]},{"id":"symbol-3536644415c7e956","tex":"E_0^{(h)}","aliases":"E_0^(h)","display":"E_0^{(h)}","kind":"use","occurrences":[134]},{"id":"symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","occurrences":[2091,2144,2149,3814]},{"id":"symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","occurrences":[3815]},{"id":"symbol-75efec2c359421e6","tex":"E_{2,0}","aliases":"E_2,0","display":"E_{2,0}","kind":"use","occurrences":[1261]},{"id":"symbol-fdddb5e6146d680d","tex":"E_{2,1}","aliases":"E_2,1","display":"E_{2,1}","kind":"use","occurrences":[1262]},{"id":"symbol-e9aaef4e6308432c","tex":"e_a","aliases":"e_a","display":"e_a","kind":"use","occurrences":[383,384,385,388,392,2151,2155,2486,2487,2488,2491,2492,3370,3371,3399,3415,3803,3805,3807]},{"id":"symbol-ce80e19252178918","tex":"e_{a+d p^m}","aliases":"e_a+d p^m","display":"e_{a+d p^m}","kind":"use","occurrences":[392]},{"id":"symbol-274edfe3f1c1c948","tex":"e_a^2","aliases":"e_a^2","display":"e_a^2","kind":"use","occurrences":[2487,3805]},{"id":"symbol-b03eb1832ae32d9a","tex":"E_a^I","aliases":"E_a^I","display":"E_a^I","kind":"use","occurrences":[3359,3361,3363]},{"id":"symbol-b207391fd280ff7e","tex":"E_a^J","aliases":"E_a^J","display":"E_a^J","kind":"use","occurrences":[3360]},{"id":"symbol-fc6e8ae822395078","tex":"E_A^{\\rm con}","aliases":"E_A^rm con rm","display":"E_A^{\\rm con}","kind":"use","occurrences":[3773]},{"id":"symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","occurrences":[385,2155,3371]},{"id":"symbol-eb1e556f8b1d6a8e","tex":"E_b^J","aliases":"E_b^J","display":"E_b^J","kind":"use","occurrences":[3359]},{"id":"symbol-a8691c09c2b87496","tex":"E_b^K","aliases":"E_b^K","display":"E_b^K","kind":"use","occurrences":[3360]},{"id":"symbol-6500ef20b6c4932a","tex":"e_c","aliases":"e_c","display":"e_c","kind":"use","occurrences":[3371]},{"id":"symbol-3174fd0f13b3655d","tex":"E_c^L","aliases":"E_c^L","display":"E_c^L","kind":"use","occurrences":[3360]},{"id":"symbol-dfb98ad8aedbb4dd","tex":"E_h","aliases":"E_h","display":"E_h","kind":"use","occurrences":[3527]},{"id":"symbol-e8d4f703100b5fef","tex":"\\mathcal E_h","aliases":"E_h mathcal","display":"\\mathcal E_h","kind":"use","occurrences":[1926,1930,1933]},{"id":"symbol-567fce9b1e56d505","tex":"E_{h,n}","aliases":"E_h,n","display":"E_{h,n}","kind":"use","occurrences":[1592,1641,1642,1738,1741,1749,1752,1755,1773,1774,1828]},{"id":"symbol-63ddef0ed4a25d17","tex":"E_{h,n}^2","aliases":"E_h,n^2","display":"E_{h,n}^2","kind":"use","occurrences":[1592,1743]},{"id":"symbol-5479790a579a7cad","tex":"E_h^{(h)}","aliases":"E_h^(h)","display":"E_h^{(h)}","kind":"use","occurrences":[133]},{"id":"symbol-81545aa21397b4c1","tex":"E_h^{(h+1)}","aliases":"E_h^(h+1)","display":"E_h^{(h+1)}","kind":"use","occurrences":[189]},{"id":"symbol-09f94d8cb8d59698","tex":"e_h^2","aliases":"e_h^2","display":"e_h^2","kind":"use","occurrences":[3977]},{"id":"symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","occurrences":[3070,3074,3075,3076,3089,3091]},{"id":"symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","occurrences":[2722,2726,2728,2730,2734,2751,2756,2758,2765]},{"id":"symbol-3296004731e3487c","tex":"E_{j-1}","aliases":"E_j-1","display":"E_{j-1}","kind":"use","occurrences":[2734]},{"id":"symbol-204f9ab8a5b56738","tex":"E_j^2","aliases":"E_j^2","display":"E_j^2","kind":"use","occurrences":[2725]},{"id":"symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","occurrences":[2137,2141]},{"id":"symbol-fcbad8d16f70335c","tex":"\\mathbf E_L","aliases":"E_L mathbf","display":"\\mathbf E_L","kind":"use","occurrences":[1236,1240]},{"id":"symbol-568316c37cf711b8","tex":"\\mathbf E_L'","aliases":"E_L' mathbf","display":"\\mathbf E_L'","kind":"use","occurrences":[1240]},{"id":"symbol-b45e7eb6ee17b6f3","tex":"E_{L,0}","aliases":"E_L,0","display":"E_{L,0}","kind":"use","occurrences":[1236]},{"id":"symbol-0f77f254fddb0f70","tex":"E_{L,j-1}","aliases":"E_L,j-1","display":"E_{L,j-1}","kind":"use","occurrences":[1232]},{"id":"symbol-b6b4a7ea5864023d","tex":"E_{L,L-1}","aliases":"E_L,L-1","display":"E_{L,L-1}","kind":"use","occurrences":[1235,1236]},{"id":"symbol-c7c5122c570797bd","tex":"\\mathsf E_M","aliases":"E_M mathsf","display":"\\mathsf E_M","kind":"use","occurrences":[453,479,481,482,487,488,509,510,511,542,547,556,558,560,568,577]},{"id":"symbol-0972d4ba92947985","tex":"\\mathsf E_{m-1}","aliases":"E_m-1 mathsf","display":"\\mathsf E_{m-1}","kind":"use","occurrences":[448,455,456,458,783]},{"id":"symbol-bc1cb9d9d07da221","tex":"E_m^{(h)}","aliases":"E_m^(h)","display":"E_m^{(h)}","kind":"use","occurrences":[152,153]},{"id":"symbol-6fe2642a9fd5fc0e","tex":"\\mathsf E_m^*","aliases":"E_m^* mathsf","display":"\\mathsf E_m^*","kind":"use","occurrences":[414]},{"id":"symbol-4d74754653932380","tex":"\\mathsf E_m^2","aliases":"E_m^2 mathsf","display":"\\mathsf E_m^2","kind":"use","occurrences":[414]},{"id":"symbol-5ee8d3321f36c338","tex":"\\mathsf E_{\\min(\\mathbf h,\\mathbf k)}","aliases":"E_min( h, k) mathsf min mathbf mathbf","display":"\\mathsf E_{\\min(\\mathbf h,\\mathbf k)}","kind":"use","occurrences":[607]},{"id":"symbol-30864bbcd81fdb76","tex":"\\mathsf E_{\\min(m,n)}","aliases":"E_min(m,n) mathsf min","display":"\\mathsf E_{\\min(m,n)}","kind":"use","occurrences":[414]},{"id":"symbol-253619f7a5d93163","tex":"E_{\\min(r,s)}^{(h)}","aliases":"E_min(r,s)^(h) min","display":"E_{\\min(r,s)}^{(h)}","kind":"use","occurrences":[133]},{"id":"symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","occurrences":[1591,1700,1738,1741,1747,1749,1750,1754,1755,1773,1774,1777,1778,1780,1827]},{"id":"symbol-3d0abe95633dd7c9","tex":"\\mathsf E_n","aliases":"E_n mathsf","display":"\\mathsf E_n","kind":"use","occurrences":[414,455,460]},{"id":"symbol-bade16369687b04f","tex":"e_N","aliases":"e_N","display":"e_N","kind":"use","occurrences":[2301,2304]},{"id":"symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","occurrences":[1413,1417,1522,1530,1566,1567,1568,1579,1580,1665,1684,1706,1707,2300,2311,2316,2329,2392,3712,3713]},{"id":"symbol-bd332fc3db09af0f","tex":"e_{N+1}","aliases":"e_N+1","display":"e_{N+1}","kind":"use","occurrences":[2444]},{"id":"symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","occurrences":[3713]},{"id":"symbol-7bfa75c99a6ca248","tex":"\\mathsf E_{n-1}","aliases":"E_n-1 mathsf","display":"\\mathsf E_{n-1}","kind":"use","occurrences":[455]},{"id":"symbol-5d6f034bf6418f2e","tex":"E_n^2","aliases":"E_n^2","display":"E_n^2","kind":"use","occurrences":[1591,1743,1748]},{"id":"symbol-a48064d29c441e34","tex":"e_{n\\bmod L}","aliases":"e_nbmod L bmod","display":"e_{n\\bmod L}","kind":"use","occurrences":[1241]},{"id":"symbol-a06214110d9137a5","tex":"\\mathcal E_P","aliases":"E_P mathcal","display":"\\mathcal E_P","kind":"use","occurrences":[3623,3626,3631]},{"id":"symbol-cf5bac72f2b1a2ac","tex":"\\mathcal E_{P,h}","aliases":"E_P,h mathcal","display":"\\mathcal E_{P,h}","kind":"use","occurrences":[3723,3725]},{"id":"symbol-b6b70465057ce11c","tex":"\\mathsf E_{p,h_p}","aliases":"E_p,h_p mathsf","display":"\\mathsf E_{p,h_p}","kind":"use","occurrences":[606]},{"id":"symbol-98f6149adeb54f57","tex":"E_{pL,j+aL}","aliases":"E_pL,j+aL","display":"E_{pL,j+aL}","kind":"use","occurrences":[1318]},{"id":"symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","occurrences":[847]},{"id":"symbol-f7f8687050d24b46","tex":"\\mathcal E_Q","aliases":"E_Q mathcal","display":"\\mathcal E_Q","kind":"use","occurrences":[3623,3626,3631]},{"id":"symbol-ec5435df861a0c30","tex":"\\mathcal E_{Q,h}","aliases":"E_Q,h mathcal","display":"\\mathcal E_{Q,h}","kind":"use","occurrences":[3724,3725]},{"id":"symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","occurrences":[140,141,143,145,147,781,1843,1844,1845,1856]},{"id":"symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","occurrences":[841,959,966]},{"id":"symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"use","occurrences":[841]},{"id":"symbol-47afbcdfac96063a","tex":"E_{r-1}","aliases":"E_r-1","display":"E_{r-1}","kind":"use","occurrences":[143,145,147]},{"id":"symbol-aa3a317d7ce08707","tex":"E_{r-1}^{(h)}","aliases":"E_r-1^(h)","display":"E_{r-1}^{(h)}","kind":"use","occurrences":[134]},{"id":"symbol-a7d70bc7f1f74924","tex":"E_r^{(h)}","aliases":"E_r^(h)","display":"E_r^{(h)}","kind":"use","occurrences":[129,130,133,134,190,272,275]},{"id":"symbol-95f29086cb5fde1d","tex":"E_r^{(h+1)}","aliases":"E_r^(h+1)","display":"E_r^{(h+1)}","kind":"use","occurrences":[190]},{"id":"symbol-9a38932300c921be","tex":"E_s","aliases":"E_s","display":"E_s","kind":"use","occurrences":[140,141,145]},{"id":"symbol-cfc2ed00b0c304ae","tex":"E_{s-1}","aliases":"E_s-1","display":"E_{s-1}","kind":"use","occurrences":[145]},{"id":"symbol-6e7970e33de410d2","tex":"E_s^{(h)}","aliases":"E_s^(h)","display":"E_s^{(h)}","kind":"use","occurrences":[133]},{"id":"symbol-6cb49e563b7d9a67","tex":"e_y","aliases":"e_y","display":"e_y","kind":"use","occurrences":[3095]},{"id":"symbol-23943c7271363293","tex":"E_\\pm","aliases":"E_± pm","display":"E_\\pm","kind":"use","occurrences":[110,111,119]},{"id":"symbol-eefa259b77b7a2fe","tex":"E_\\pm^2","aliases":"E_±^2 pm","display":"E_\\pm^2","kind":"use","occurrences":[111]},{"id":"symbol-74e2d51d64f7fad4","tex":"E_\\lambda","aliases":"E_λ lambda","display":"E_\\lambda","kind":"use","occurrences":[897,898,899]},{"id":"symbol-9bd68f1d61a4d151","tex":"e_\\perp^I","aliases":"e_⊥^I perp","display":"e_\\perp^I","kind":"use","occurrences":[3337]},{"id":"symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","occurrences":[2079,2081,2089,2095,3507]},{"id":"symbol-a66b2c302d0c8a7d","tex":"\\mathrm{emb}","aliases":"emb mathrm","display":"\\mathrm{emb}","kind":"use","occurrences":[542,547,552,558,563]},{"id":"symbol-0bff408d7104cbba","tex":"\\operatorname{erf}","aliases":"erf operatorname","display":"\\operatorname{erf}","kind":"use","occurrences":[3605]},{"id":"symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","occurrences":[1532,1685,1693,1697,1931,2516,2517,3339,3340,3345,3352,3381]},{"id":"symbol-28240dde4f98a443","tex":"\\mathrm f","aliases":"f mathrm","display":"\\mathrm f","kind":"use","occurrences":[2053]},{"id":"symbol-aaec5c835f65f308","tex":"\\mathsf F","aliases":"F mathsf","display":"\\mathsf F","kind":"use","occurrences":[180,447,448,453,461,470,476,479,486,494,499,526,528,531,532,542,552,553,556,564,592,611,615,617,622,627,783]},{"id":"symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","occurrences":[1881]},{"id":"symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","occurrences":[2184,3821,3823,3927]},{"id":"symbol-9337f69b3f162423","tex":"f''","aliases":"f''","display":"f''","kind":"use","occurrences":[3823]},{"id":"symbol-058246411cc5b595","tex":"f'''","aliases":"f'''","display":"f'''","kind":"use","occurrences":[3821]},{"id":"symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","occurrences":[2044]},{"id":"symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","occurrences":[3897]},{"id":"symbol-8c4caf5c79a35c9c","tex":"f^2","aliases":"f^2","display":"f^2","kind":"use","occurrences":[150,187]},{"id":"symbol-c9a3c3f9d72a35e1","tex":"F^{IJ}","aliases":"F^IJ","display":"F^{IJ}","kind":"use","occurrences":[3294]},{"id":"symbol-b9284f7a50c1ed02","tex":"f_{\\mathfrak H_p^s}","aliases":"f_ H_p^s mathfrak","display":"f_{\\mathfrak H_p^s}","kind":"use","occurrences":[550]},{"id":"symbol-71bf007f614b8921","tex":"\\mathsf F_{\\mathbf r}","aliases":"F_ r mathsf mathbf","display":"\\mathsf F_{\\mathbf r}","kind":"use","occurrences":[611,615,617,622,627]},{"id":"symbol-0e24ae7f63f35dfa","tex":"f_{\\mathbf r}","aliases":"f_ r mathbf","display":"f_{\\mathbf r}","kind":"use","occurrences":[630,631]},{"id":"symbol-dcde29ad44d73958","tex":"F_+","aliases":"F_+","display":"F_+","kind":"use","occurrences":[2229,2235,2281,2282]},{"id":"symbol-1cef4fb299762615","tex":"F_-","aliases":"F_-","display":"F_-","kind":"use","occurrences":[2229,2235,2282]},{"id":"symbol-b910d55bebfcc906","tex":"\\mathsf F_0","aliases":"F_0 mathsf","display":"\\mathsf F_0","kind":"use","occurrences":[447]},{"id":"symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","occurrences":[1286]},{"id":"symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","occurrences":[1263,3039,3041,3043]},{"id":"symbol-3432ebe25e9aabbb","tex":"f_{1/12}","aliases":"f_1/12","display":"f_{1/12}","kind":"use","occurrences":[1921]},{"id":"symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","occurrences":[513,527,1286]},{"id":"symbol-eea600dc4f325e8a","tex":"F_{2^{h_2}}","aliases":"F_2^h_2","display":"F_{2^{h_2}}","kind":"use","occurrences":[78]},{"id":"symbol-6e38e6f593dba41e","tex":"F_{2L-1}","aliases":"F_2L-1","display":"F_{2L-1}","kind":"use","occurrences":[1267]},{"id":"symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"use","occurrences":[1530,1532,1691,1855]},{"id":"symbol-1438a4d487c7b22d","tex":"f_{a+1}","aliases":"f_a+1","display":"f_{a+1}","kind":"use","occurrences":[3812]},{"id":"symbol-cacee2cb9dea5558","tex":"f_{a-1}","aliases":"f_a-1","display":"f_{a-1}","kind":"use","occurrences":[3812]},{"id":"symbol-6f98938f2b1b013e","tex":"F_a^{1/2}","aliases":"F_a^1/2","display":"F_a^{1/2}","kind":"use","occurrences":[2266]},{"id":"symbol-0fb521484b5b5711","tex":"F_a^2","aliases":"F_a^2","display":"F_a^2","kind":"use","occurrences":[2226,2227,2276,2277]},{"id":"symbol-dd0b7a11832442ee","tex":"F_{ab}^i","aliases":"F_ab^i","display":"F_{ab}^i","kind":"use","occurrences":[3237]},{"id":"symbol-08d88035e5bf8864","tex":"F_{ab}^k","aliases":"F_ab^k","display":"F_{ab}^k","kind":"use","occurrences":[3237]},{"id":"symbol-36f0dfbeaf4bfcdb","tex":"F_{a\\leftarrow b}","aliases":"F_a← b leftarrow","display":"F_{a\\leftarrow b}","kind":"use","occurrences":[2158]},{"id":"symbol-54b44dec26d2ba54","tex":"F_b","aliases":"F_b","display":"F_b","kind":"use","occurrences":[83,2237,2238,2240,2242,2243,2246,2280]},{"id":"symbol-9e7bf13ec229e895","tex":"\\mathcal F_{bc}","aliases":"F_bc mathcal","display":"\\mathcal F_{bc}","kind":"use","occurrences":[3381]},{"id":"symbol-fb505b26292653fe","tex":"F_{b\\leftarrow a}","aliases":"F_b← a leftarrow","display":"F_{b\\leftarrow a}","kind":"use","occurrences":[2158,2199,2202]},{"id":"symbol-e982a068c0aac4ff","tex":"F_h","aliases":"F_h","display":"F_h","kind":"use","occurrences":[2851,2852,2855,2864]},{"id":"symbol-1551bc759aa860fc","tex":"f_h","aliases":"f_h","display":"f_h","kind":"use","occurrences":[1684,1691,1694,1764,1765,1766,1780,1850,1859,1866,1900,1902,1922,3672,3682,3683]},{"id":"symbol-fc9d2a85cc280a7e","tex":"f_{h,a}","aliases":"f_h,a","display":"f_{h,a}","kind":"use","occurrences":[1696]},{"id":"symbol-4568892b4cd9f787","tex":"f_{h,b}","aliases":"f_h,b","display":"f_{h,b}","kind":"use","occurrences":[1696]},{"id":"symbol-e8a91baf437e5b29","tex":"F_{h/2}","aliases":"F_h/2","display":"F_{h/2}","kind":"use","occurrences":[2852,2864]},{"id":"symbol-8a63a9f5145880e2","tex":"F_h^{[1]}","aliases":"F_h^[1]","display":"F_h^{[1]}","kind":"use","occurrences":[2852,2856]},{"id":"symbol-74eec0b63ac7044d","tex":"F_{IJ}","aliases":"F_IJ","display":"F_{IJ}","kind":"use","occurrences":[3294]},{"id":"symbol-09888c9e8426e6e3","tex":"F_{ij}","aliases":"F_ij","display":"F_{ij}","kind":"use","occurrences":[3078,3079,3080,3099]},{"id":"symbol-967ebb9573a3e321","tex":"\\mathcal F_{ij}","aliases":"F_ij mathcal","display":"\\mathcal F_{ij}","kind":"use","occurrences":[2516,2517]},{"id":"symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","occurrences":[3036,3038,3042]},{"id":"symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","occurrences":[1280,1281,1282,1283,1284,1297,1300,1301,2497,2499,3228,3230,3668]},{"id":"symbol-8576ad4cc4bc1341","tex":"F_{j+1}","aliases":"F_j+1","display":"F_{j+1}","kind":"use","occurrences":[3041]},{"id":"symbol-0fe97be4e95beb8d","tex":"f_{j+1}","aliases":"f_j+1","display":"f_{j+1}","kind":"use","occurrences":[2497,3668]},{"id":"symbol-44073352c09346c9","tex":"F_{j+1/2}","aliases":"F_j+1/2","display":"F_{j+1/2}","kind":"use","occurrences":[3525,3526]},{"id":"symbol-d54d63626059c12d","tex":"f_{j-1}","aliases":"f_j-1","display":"f_{j-1}","kind":"use","occurrences":[3668]},{"id":"symbol-d63682be513a74f9","tex":"F_{j-1/2}","aliases":"F_j-1/2","display":"F_{j-1/2}","kind":"use","occurrences":[3526]},{"id":"symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"use","occurrences":[829]},{"id":"symbol-3b20c59729d1aa1f","tex":"\\mathsf F_m","aliases":"F_m mathsf","display":"\\mathsf F_m","kind":"use","occurrences":[448,453,461,470,476,479,494,499,526,528,531,532,542,552,553,556,564,592,783]},{"id":"symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","occurrences":[499,3126]},{"id":"symbol-8b7438119014ae74","tex":"\\mathsf F_{M+1}","aliases":"F_M+1 mathsf","display":"\\mathsf F_{M+1}","kind":"use","occurrences":[486]},{"id":"symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","occurrences":[83,3041,3043]},{"id":"symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","occurrences":[77]},{"id":"symbol-e213f3328b138478","tex":"\\mathsf F_n","aliases":"F_n mathsf","display":"\\mathsf F_n","kind":"use","occurrences":[453]},{"id":"symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","occurrences":[3120,3121,3122,3124,3126,3129]},{"id":"symbol-af40e62da957c071","tex":"f_{n+1}","aliases":"f_n+1","display":"f_{n+1}","kind":"use","occurrences":[3121]},{"id":"symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"use","occurrences":[215,216]},{"id":"symbol-3662907c3ec44b16","tex":"\\mathsf F_{p,r_p}","aliases":"F_p,r_p mathsf","display":"\\mathsf F_{p,r_p}","kind":"use","occurrences":[611]},{"id":"symbol-c05872ad0a26184c","tex":"F_{p-1}","aliases":"F_p-1","display":"F_{p-1}","kind":"use","occurrences":[78]},{"id":"symbol-fb50aad1fbf28b7f","tex":"F_{p^{h_p}}","aliases":"F_p^h_p","display":"F_{p^{h_p}}","kind":"use","occurrences":[78]},{"id":"symbol-7b7f016ef3a20773","tex":"F_{p_a}","aliases":"F_p_a","display":"F_{p_a}","kind":"use","occurrences":[3836]},{"id":"symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","occurrences":[566,567,568,569,571]},{"id":"symbol-b2fb504bd6f0611c","tex":"F_{q_a}","aliases":"F_q_a","display":"F_{q_a}","kind":"use","occurrences":[3836]},{"id":"symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","occurrences":[3871,3873,3874,3875,3878,3884,3896,3897,3905,3914,3915]},{"id":"symbol-92f4bda95e0af13d","tex":"\\mathsf F_r","aliases":"F_r mathsf","display":"\\mathsf F_r","kind":"use","occurrences":[180]},{"id":"symbol-142a68e175e6731e","tex":"F_{r-1/2}","aliases":"F_r-1/2","display":"F_{r-1/2}","kind":"use","occurrences":[3541]},{"id":"symbol-a46e3f3e2645bd3d","tex":"\\mathcal F_S","aliases":"F_S mathcal","display":"\\mathcal F_S","kind":"use","occurrences":[1532,1685,1693,1697,1931]},{"id":"symbol-2ef3bbb91e99e25c","tex":"F_{s+1/2}","aliases":"F_s+1/2","display":"F_{s+1/2}","kind":"use","occurrences":[3541]},{"id":"symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","occurrences":[3873,3874,3887,3896]},{"id":"symbol-392398bb2bd705e1","tex":"F_{tr}","aliases":"F_tr","display":"F_{tr}","kind":"use","occurrences":[3891,3892,3896]},{"id":"symbol-0ddd8046eaa27be9","tex":"F_{U_M}","aliases":"F_U_M","display":"F_{U_M}","kind":"use","occurrences":[78]},{"id":"symbol-308fda5bda779517","tex":"f_x","aliases":"f_x","display":"f_x","kind":"use","occurrences":[2780,2781,2782,2784,2785]},{"id":"symbol-df08b9f9771e4e17","tex":"f_x'","aliases":"f_x'","display":"f_x'","kind":"use","occurrences":[2783,2785]},{"id":"symbol-faca7756d6387453","tex":"f_x'''","aliases":"f_x'''","display":"f_x'''","kind":"use","occurrences":[2789]},{"id":"symbol-7a3e381615a63515","tex":"F_{xy}","aliases":"F_xy","display":"F_{xy}","kind":"use","occurrences":[3082,3110]},{"id":"symbol-c347f30d5a08d9fc","tex":"F_{xz}","aliases":"F_xz","display":"F_{xz}","kind":"use","occurrences":[3082]},{"id":"symbol-4b8a2189d0ac907c","tex":"F_{yz}","aliases":"F_yz","display":"F_{yz}","kind":"use","occurrences":[3082]},{"id":"symbol-3aeda5cb74c7b3c2","tex":"F_{\\mu\\nu}","aliases":"F_μν mu nu","display":"F_{\\mu\\nu}","kind":"use","occurrences":[2698,2700,2706,2707]},{"id":"symbol-cf0883ad1e961a59","tex":"F_\\infty","aliases":"F_∞ infty","display":"F_\\infty","kind":"use","occurrences":[634]},{"id":"symbol-9a7b27c4eeefb975","tex":"f_\\infty","aliases":"f_∞ infty","display":"f_\\infty","kind":"use","occurrences":[3123,3124]},{"id":"symbol-ed487b1e5f57fec8","tex":"\\mathrm{fine}","aliases":"fine mathrm","display":"\\mathrm{fine}","kind":"use","occurrences":[2998]},{"id":"symbol-ff9bce76b90458b5","tex":"\\mathrm{fld}","aliases":"fld mathrm","display":"\\mathrm{fld}","kind":"use","occurrences":[3300]},{"id":"symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","occurrences":[715,2734,2736,2748,3237,3239,3346,3347,3405,3735,3736,3737,3740,3742,3748,3750,3764,3776,3850,3851,3856,3858,3859]},{"id":"symbol-b7b55fe92541ca0f","tex":"\\mathrm G","aliases":"G mathrm","display":"\\mathrm G","kind":"use","occurrences":[3278,3279,3344,3345]},{"id":"symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","occurrences":[3823,3971,3976]},{"id":"symbol-eb695a8d9657252e","tex":"g''","aliases":"g''","display":"g''","kind":"use","occurrences":[3823]},{"id":"symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","occurrences":[3310,3313,3315,3697]},{"id":"symbol-1dd057a76e8bf9c5","tex":"G^n","aliases":"G^n","display":"G^n","kind":"use","occurrences":[2647]},{"id":"symbol-8247fba132aaeb48","tex":"G^{n+1}","aliases":"G^n+1","display":"G^{n+1}","kind":"use","occurrences":[2647]},{"id":"symbol-ac1e4b86b2063762","tex":"G^T","aliases":"G^T","display":"G^T","kind":"use","occurrences":[3428,3429]},{"id":"symbol-531a7fd60afe23fd","tex":"G_{\\mathrm L}","aliases":"G_ L mathrm","display":"G_{\\mathrm L}","kind":"use","occurrences":[3348,3349,3353]},{"id":"symbol-57321f6d631ce0e0","tex":"G_{\\mathrm U}","aliases":"G_ U mathrm","display":"G_{\\mathrm U}","kind":"use","occurrences":[3348,3349,3353]},{"id":"symbol-f28a064eaf92ff39","tex":"\\mathcal G_{\\mathrm U}","aliases":"G_ U mathcal mathrm","display":"\\mathcal G_{\\mathrm U}","kind":"use","occurrences":[3346,3347]},{"id":"symbol-30418dea0b4dc62f","tex":"\\mathcal G_{\\mathrm U}^{\\mathrm{red}}","aliases":"G_ U^red mathcal mathrm mathrm","display":"\\mathcal G_{\\mathrm U}^{\\mathrm{red}}","kind":"use","occurrences":[3405]},{"id":"symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","occurrences":[1003]},{"id":"symbol-a43ad99289f696af","tex":"G_g","aliases":"G_g","display":"G_g","kind":"use","occurrences":[2671]},{"id":"symbol-624b8e0e8ca99130","tex":"G_h","aliases":"G_h","display":"G_h","kind":"use","occurrences":[3679,3682]},{"id":"symbol-6ba4cd8d81889283","tex":"g_h","aliases":"g_h","display":"g_h","kind":"use","occurrences":[1000]},{"id":"symbol-d5c5a1afa19b22e1","tex":"\\mathcal G_i","aliases":"G_i mathcal","display":"\\mathcal G_i","kind":"use","occurrences":[3237,3239]},{"id":"symbol-967e5345c58b31c4","tex":"\\mathcal G_{IJ}","aliases":"G_IJ mathcal","display":"\\mathcal G_{IJ}","kind":"use","occurrences":[3346,3347]},{"id":"symbol-4e1d92ceb75803b7","tex":"\\mathcal G_{ij}","aliases":"G_ij mathcal","display":"\\mathcal G_{ij}","kind":"use","occurrences":[3735,3736,3737,3748,3776]},{"id":"symbol-9ef43ece144c5d2a","tex":"\\mathcal G_j","aliases":"G_j mathcal","display":"\\mathcal G_j","kind":"use","occurrences":[2734,2736,2748]},{"id":"symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","occurrences":[2499,2885]},{"id":"symbol-b371104368ef3d3d","tex":"g_{j+1}^{-1}","aliases":"g_j+1^-1","display":"g_{j+1}^{-1}","kind":"use","occurrences":[2885]},{"id":"symbol-89aa350d075d9b5d","tex":"g_{j+1}^n","aliases":"g_j+1^n","display":"g_{j+1}^n","kind":"use","occurrences":[2644]},{"id":"symbol-5c7ba05d541dd915","tex":"g_j^n","aliases":"g_j^n","display":"g_j^n","kind":"use","occurrences":[2643,2644,2647,2656]},{"id":"symbol-65205833b6a4bc33","tex":"g_j^{n+1}","aliases":"g_j^n+1","display":"g_j^{n+1}","kind":"use","occurrences":[2644]},{"id":"symbol-1a4a7f45555ebe7e","tex":"g_m","aliases":"g_m","display":"g_m","kind":"use","occurrences":[503,505]},{"id":"symbol-a3d27b1546a14b2e","tex":"G_{p,h}","aliases":"G_p,h","display":"G_{p,h}","kind":"use","occurrences":[123]},{"id":"symbol-bbd4724f38071f66","tex":"G_{p_a}","aliases":"G_p_a","display":"G_{p_a}","kind":"use","occurrences":[3836]},{"id":"symbol-7eadc3987326373a","tex":"g_q","aliases":"g_q","display":"g_q","kind":"use","occurrences":[1307]},{"id":"symbol-bf4f844ef337a256","tex":"G_{q_a}","aliases":"G_q_a","display":"G_{q_a}","kind":"use","occurrences":[3836]},{"id":"symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"use","occurrences":[999,1000,1007,1008]},{"id":"symbol-c9668d016de81fd0","tex":"g_{r+1}","aliases":"g_r+1","display":"g_{r+1}","kind":"use","occurrences":[1000]},{"id":"symbol-169c77a4f93260fb","tex":"g_U","aliases":"g_U","display":"g_U","kind":"use","occurrences":[3678,3679,3680,3717]},{"id":"symbol-08d86714390ee3fe","tex":"g_V","aliases":"g_V","display":"g_V","kind":"use","occurrences":[3678,3679,3680,3717]},{"id":"symbol-310a48051f387508","tex":"g_x","aliases":"g_x","display":"g_x","kind":"use","occurrences":[2607,2608,2609,2621,2622,2623,2682,2687,2805,2808,2813]},{"id":"symbol-7b7b7cb3a0e5cda8","tex":"g_{x+\\widehat\\mu}","aliases":"g_x+widehatμ widehat mu","display":"g_{x+\\widehat\\mu}","kind":"use","occurrences":[2612]},{"id":"symbol-c1bd1851814a55bc","tex":"g_{x+\\widehat\\mu}^{-1}","aliases":"g_x+widehatμ^-1 widehat mu","display":"g_{x+\\widehat\\mu}^{-1}","kind":"use","occurrences":[2609]},{"id":"symbol-badd526519f7d4c1","tex":"g_{x-\\widehat\\mu}","aliases":"g_x-widehatμ widehat mu","display":"g_{x-\\widehat\\mu}","kind":"use","occurrences":[2622]},{"id":"symbol-5feaf61feae29f5f","tex":"g_x^*","aliases":"g_x^*","display":"g_x^*","kind":"use","occurrences":[2807,2813]},{"id":"symbol-c4b741b3ca6387a1","tex":"g_x^{-1}","aliases":"g_x^-1","display":"g_x^{-1}","kind":"use","occurrences":[2611,2612,2682,2875]},{"id":"symbol-889e84d345b27e17","tex":"g_y","aliases":"g_y","display":"g_y","kind":"use","occurrences":[2611,2875]},{"id":"symbol-e34f9d3fec41ee83","tex":"g_{\\mathrm{YM}}","aliases":"g_YM mathrm","display":"g_{\\mathrm{YM}}","kind":"use","occurrences":[2686]},{"id":"symbol-c445feaff50474e5","tex":"g_{\\mathrm{YM}}^2","aliases":"g_YM^2 mathrm","display":"g_{\\mathrm{YM}}^2","kind":"use","occurrences":[2686]},{"id":"symbol-876a72a7054676bb","tex":"G_\\beta","aliases":"G_β beta","display":"G_\\beta","kind":"use","occurrences":[2250]},{"id":"symbol-f907805278b68599","tex":"\\operatorname{gh}","aliases":"gh operatorname","display":"\\operatorname{gh}","kind":"use","occurrences":[3318]},{"id":"symbol-1725e2bf2fce3bf7","tex":"\\mathrm{graph}","aliases":"graph mathrm","display":"\\mathrm{graph}","kind":"use","occurrences":[1466,1493]},{"id":"symbol-b75cc7bdfa6f5f94","tex":"\\mathbf h","aliases":"h mathbf","display":"\\mathbf h","kind":"use","occurrences":[605,606,607,612,620,622,623,634]},{"id":"symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","occurrences":[123,150,225,1787,1788,1793,1795,1802,3237,3239,3261,3264,3381,3850,3852,3856,3858,3886,4044]},{"id":"symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"use","occurrences":[476,480,481,549,550]},{"id":"symbol-823170eb09d88517","tex":"\\mathsf H","aliases":"H mathsf","display":"\\mathsf H","kind":"use","occurrences":[841,843,846]},{"id":"symbol-b8bdb1c02444ca91","tex":"H'","aliases":"H'","display":"H'","kind":"use","occurrences":[780]},{"id":"symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","occurrences":[809,2030,2109]},{"id":"symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","occurrences":[2775]},{"id":"symbol-fa0621c04438e85a","tex":"h^{-2}","aliases":"h^-2","display":"h^{-2}","kind":"use","occurrences":[2626,2851,2866]},{"id":"symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","occurrences":[1478]},{"id":"symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","occurrences":[3185]},{"id":"symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","occurrences":[1395,1398,1411,1705,1707,1721,1723,1729,1731,1733,1738,1742,1743,1749,1761,1764,1778,1780,1899,1900,1902,1905,1933,1935,2595,2596,2617,2778,2796,2800,2852,2858,2860,2862,2865,3077,3078,3080,3082,3096,3102,3107,3114,3522,3550,3593,3668,3777,3786,3789,3821,3823,3827,3828]},{"id":"symbol-9217de3017a2652b","tex":"h^{2.084394965}","aliases":"h^2.084394965","display":"h^{2.084394965}","kind":"use","occurrences":[4149]},{"id":"symbol-92940262b09349da","tex":"h^{2.173853282}","aliases":"h^2.173853282","display":"h^{2.173853282}","kind":"use","occurrences":[4149]},{"id":"symbol-fc3c9dc938c10528","tex":"h^3","aliases":"h^3","display":"h^3","kind":"use","occurrences":[2595,2596,2785,2791,2794,2800,2858,3773]},{"id":"symbol-fd80ec0462f00d62","tex":"h^4","aliases":"h^4","display":"h^4","kind":"use","occurrences":[1742,1743,1746,1749,2858,2861,3550,3789,3821,3823]},{"id":"symbol-425652a73ca954b2","tex":"H^k","aliases":"H^k","display":"H^k","kind":"use","occurrences":[1629]},{"id":"symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","occurrences":[815]},{"id":"symbol-858ebfe3a6d1e027","tex":"h^r","aliases":"h^r","display":"h^r","kind":"use","occurrences":[2567,2576,3214,3216]},{"id":"symbol-1fe35faf74f825f6","tex":"H^T","aliases":"H^T","display":"H^T","kind":"use","occurrences":[3439,3460]},{"id":"symbol-7e488733e11e45b8","tex":"H_{\\mathrm m}","aliases":"H_ m mathrm","display":"H_{\\mathrm m}","kind":"use","occurrences":[2725,2753,2756]},{"id":"symbol-16ef1e898cbf3ee0","tex":"\\mathcal H_{\\mathrm M}","aliases":"H_ M mathcal mathrm","display":"\\mathcal H_{\\mathrm M}","kind":"use","occurrences":[3381]},{"id":"symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","occurrences":[669,670,674,683,684,685,686,687,689,694,698,706]},{"id":"symbol-49ecf53502f364f0","tex":"H_0^*","aliases":"H_0^*","display":"H_0^*","kind":"use","occurrences":[669]},{"id":"symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","occurrences":[72,74,76,78,663]},{"id":"symbol-bea5d0e7eb485d1c","tex":"H_A","aliases":"H_A","display":"H_A","kind":"use","occurrences":[617,618,619,622,623,629,632]},{"id":"symbol-e337b6fe26be0666","tex":"H_A^{(\\mathbf h)}","aliases":"H_A^( h) mathbf","display":"H_A^{(\\mathbf h)}","kind":"use","occurrences":[620]},{"id":"symbol-c54ddda591ca604e","tex":"H_A^*","aliases":"H_A^*","display":"H_A^*","kind":"use","occurrences":[625,629]},{"id":"symbol-5294eca4b0cb55fe","tex":"H_{AA}","aliases":"H_AA","display":"H_{AA}","kind":"use","occurrences":[2008]},{"id":"symbol-c9c78162ab0afe1d","tex":"H_{AB}","aliases":"H_AB","display":"H_{AB}","kind":"use","occurrences":[2008]},{"id":"symbol-587cf05864943581","tex":"H_{BA}","aliases":"H_BA","display":"H_{BA}","kind":"use","occurrences":[2008]},{"id":"symbol-8b69ca3ee83e3f48","tex":"H_{ba}","aliases":"H_ba","display":"H_{ba}","kind":"use","occurrences":[2121]},{"id":"symbol-7cee56da1c8c7844","tex":"H_{BA}^*","aliases":"H_BA^*","display":"H_{BA}^*","kind":"use","occurrences":[2008]},{"id":"symbol-cca8f5fa2a64839a","tex":"H_{BB}","aliases":"H_BB","display":"H_{BB}","kind":"use","occurrences":[2008]},{"id":"symbol-b8541d1f64196274","tex":"H_c","aliases":"H_c","display":"H_c","kind":"use","occurrences":[228,229,231,235,240,242,245,249,252,670,677]},{"id":"symbol-2b355180d8491077","tex":"\\mathcal H_c","aliases":"H_c mathcal","display":"\\mathcal H_c","kind":"use","occurrences":[225]},{"id":"symbol-559e90a1829db889","tex":"H_D","aliases":"H_D","display":"H_D","kind":"use","occurrences":[1010]},{"id":"symbol-0028a7a569a1dcc9","tex":"H_E","aliases":"H_E","display":"H_E","kind":"use","occurrences":[2725]},{"id":"symbol-7b786d951bfb29bf","tex":"H_{\\mathrm{eff}}","aliases":"H_eff mathrm","display":"H_{\\mathrm{eff}}","kind":"use","occurrences":[2089]},{"id":"symbol-399afbaa0e9a8610","tex":"H_f","aliases":"H_f","display":"H_f","kind":"use","occurrences":[228,229,231,235,240,241,242,243,245,248,249,253,283,286,287,290,294,295]},{"id":"symbol-4bc955a2f0ca796d","tex":"\\mathcal H_f","aliases":"H_f mathcal","display":"\\mathcal H_f","kind":"use","occurrences":[225]},{"id":"symbol-c2c11852c7f707dc","tex":"H_G","aliases":"H_G","display":"H_G","kind":"use","occurrences":[3627,3628,3631,3637,3643,3647]},{"id":"symbol-102402835dbe0bc2","tex":"H_{G,h}","aliases":"H_G,h","display":"H_{G,h}","kind":"use","occurrences":[3721,3725]},{"id":"symbol-820491df7b55e699","tex":"H_h","aliases":"H_h","display":"H_h","kind":"use","occurrences":[2725,2726,2727,2728,2729]},{"id":"symbol-027276bb71e82248","tex":"H_{h,n}^2","aliases":"H_h,n^2","display":"H_{h,n}^2","kind":"use","occurrences":[1592]},{"id":"symbol-46e2d2e905a9d0ff","tex":"h_{ij}","aliases":"h_ij","display":"h_{ij}","kind":"use","occurrences":[3730,3735,3737,3770,3775,3776]},{"id":"symbol-b90556c898d304c9","tex":"h_{ik}","aliases":"h_ik","display":"h_{ik}","kind":"use","occurrences":[3735,3737]},{"id":"symbol-60f212957626b8f2","tex":"H_{\\mathrm{int}}","aliases":"H_int mathrm","display":"H_{\\mathrm{int}}","kind":"use","occurrences":[307,310,311,315,349,655]},{"id":"symbol-e6e1d7ad2fc5bdfa","tex":"h_{ji}","aliases":"h_ji","display":"h_{ji}","kind":"use","occurrences":[3730]},{"id":"symbol-957daf505ba79710","tex":"h_{jk}","aliases":"h_jk","display":"h_{jk}","kind":"use","occurrences":[3735,3737]},{"id":"symbol-95273a360d135670","tex":"h_{kk}","aliases":"h_kk","display":"h_{kk}","kind":"use","occurrences":[3733]},{"id":"symbol-df3ba0db246bbd1a","tex":"h_{kl}","aliases":"h_kl","display":"h_{kl}","kind":"use","occurrences":[3737]},{"id":"symbol-0239d5e810818427","tex":"H_m^{\\mathrm{phys}}","aliases":"H_m^phys mathrm","display":"H_m^{\\mathrm{phys}}","kind":"use","occurrences":[1355,1496,1498,1499,1502]},{"id":"symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","occurrences":[3117]},{"id":"symbol-0ced436a17ff0e3c","tex":"H_n^*","aliases":"H_n^*","display":"H_n^*","kind":"use","occurrences":[1467,1469]},{"id":"symbol-921fd2ec6756cc1a","tex":"H_n^2","aliases":"H_n^2","display":"H_n^2","kind":"use","occurrences":[1468,1591]},{"id":"symbol-0614636d0bfeab26","tex":"\\mathsf H_p","aliases":"H_p mathsf","display":"\\mathsf H_p","kind":"use","occurrences":[841,843,846]},{"id":"symbol-d54b1a6be7c911b2","tex":"h_p","aliases":"h_p","display":"h_p","kind":"use","occurrences":[72,74,76,78,604,605,606,607]},{"id":"symbol-263fca6973f674db","tex":"H_{p,h}","aliases":"H_p,h","display":"H_{p,h}","kind":"use","occurrences":[270]},{"id":"symbol-5d63a92771aeec1f","tex":"\\mathcal H_{p,h}","aliases":"H_p,h mathcal","display":"\\mathcal H_{p,h}","kind":"use","occurrences":[123,150]},{"id":"symbol-44144103f9354925","tex":"H_{p,h+1}^{\\mathrm{Haar}}","aliases":"H_p,h+1^Haar mathrm","display":"H_{p,h+1}^{\\mathrm{Haar}}","kind":"use","occurrences":[260]},{"id":"symbol-8032d1774cbfdf13","tex":"H_{p,h}^{\\mathrm{Haar}}","aliases":"H_p,h^Haar mathrm","display":"H_{p,h}^{\\mathrm{Haar}}","kind":"use","occurrences":[257,260,262,270,274]},{"id":"symbol-3a2ec2244c17754a","tex":"\\mathfrak H_p^s","aliases":"H_p^s mathfrak","display":"\\mathfrak H_p^s","kind":"use","occurrences":[476,480,481,549,550]},{"id":"symbol-d83201f3b36e1284","tex":"H_Q","aliases":"H_Q","display":"H_Q","kind":"use","occurrences":[231,232,294,295]},{"id":"symbol-a06593e7c86b3be5","tex":"H_q","aliases":"H_q","display":"H_q","kind":"use","occurrences":[2191]},{"id":"symbol-e3b81d1ac3c985ef","tex":"H_{\\rm eff}^{(2)}","aliases":"H_rm eff^(2) rm","display":"H_{\\rm eff}^{(2)}","kind":"use","occurrences":[1083]},{"id":"symbol-b60b9efbddaafac9","tex":"H_{\\rm eff}^{(p)}","aliases":"H_rm eff^(p) rm","display":"H_{\\rm eff}^{(p)}","kind":"use","occurrences":[1063,1064]},{"id":"symbol-da0dbd41a50083a7","tex":"H_{\\rm full}","aliases":"H_rm full rm","display":"H_{\\rm full}","kind":"use","occurrences":[1015]},{"id":"symbol-a3001aa6263f9409","tex":"\\mathcal H_{\\rm matt}","aliases":"H_rm matt mathcal rm","display":"\\mathcal H_{\\rm matt}","kind":"use","occurrences":[4044]},{"id":"symbol-3d87103c514a144c","tex":"h_{\\rm tr}","aliases":"h_rm tr rm","display":"h_{\\rm tr}","kind":"use","occurrences":[3733,3735,3737,3775]},{"id":"symbol-b6bc647753035f9f","tex":"H_{\\rm TT}","aliases":"H_rm TT rm","display":"H_{\\rm TT}","kind":"use","occurrences":[3777]},{"id":"symbol-00e5cb4d6d94dccd","tex":"H_{rs}","aliases":"H_rs","display":"H_{rs}","kind":"use","occurrences":[819,822,824]},{"id":"symbol-668caf8df5eeb048","tex":"H_{rs}^*","aliases":"H_rs^*","display":"H_{rs}^*","kind":"use","occurrences":[810,820]},{"id":"symbol-f2b2331953e04c5c","tex":"H_{sr}","aliases":"H_sr","display":"H_{sr}","kind":"use","occurrences":[810,820,825]},{"id":"symbol-6ea1daba7a0d5088","tex":"\\mathcal H_t","aliases":"H_t mathcal","display":"\\mathcal H_t","kind":"use","occurrences":[3261,3264]},{"id":"symbol-95f7cb457015e1aa","tex":"H_\\epsilon","aliases":"H_ϵ epsilon","display":"H_\\epsilon","kind":"use","occurrences":[494,497,506,507,508,511,513,518,524,526,527,528,534,536,539,551,552,554,556,560,562,569,571,577,578,589,591,593]},{"id":"symbol-49ad7821cf07d564","tex":"H_\\epsilon^{(M)}","aliases":"H_ϵ^(M) epsilon","display":"H_\\epsilon^{(M)}","kind":"use","occurrences":[538]},{"id":"symbol-15016e5fa808b1d5","tex":"H_\\epsilon^*","aliases":"H_ϵ^* epsilon","display":"H_\\epsilon^*","kind":"use","occurrences":[500,501]},{"id":"symbol-18adc48f226810ce","tex":"H_\\lambda","aliases":"H_λ lambda","display":"H_\\lambda","kind":"use","occurrences":[674,677,678,679,681,685,690,691,694,698,706,2306,2307,2308,2315,2317,2321,2324,2374,2375,2376,2388,2389,2390,2397,2398,2403,2407,2447,2448,2459,2460,2461]},{"id":"symbol-4f4f51cfd6f3785b","tex":"H_\\lambda^*","aliases":"H_λ^* lambda","display":"H_\\lambda^*","kind":"use","occurrences":[2309,2310]},{"id":"symbol-e1ec9ac41ad6dd0e","tex":"H_\\lambda^{1/2}","aliases":"H_λ^1/2 lambda","display":"H_\\lambda^{1/2}","kind":"use","occurrences":[2320,2324]},{"id":"symbol-743400708b98042d","tex":"H_\\infty","aliases":"H_∞ infty","display":"H_\\infty","kind":"use","occurrences":[1277,1281,1282,1284,1287,1291,1294,1295,1300,1301]},{"id":"symbol-7fa5c4539bb70667","tex":"\\mathrm{Haar}","aliases":"Haar mathrm","display":"\\mathrm{Haar}","kind":"use","occurrences":[257,260,262,270,274,397]},{"id":"symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","occurrences":[1764]},{"id":"symbol-04e8653f90df6217","tex":"\\HH_{\\mathcal P}^{\\infty}","aliases":"HH_ P^∞ HH mathcal infty","display":null,"kind":"use","occurrences":[603,616]},{"id":"symbol-7f4a69bec175bb06","tex":"\\HH_A","aliases":"HH_A HH","display":null,"kind":"use","occurrences":[2127]},{"id":"symbol-e68521b49f17a05c","tex":"\\HH_B","aliases":"HH_B HH","display":null,"kind":"use","occurrences":[2128,2138]},{"id":"symbol-02afd49e2184a454","tex":"\\HH_D","aliases":"HH_D HH","display":null,"kind":"use","occurrences":[864,866]},{"id":"symbol-169da2322bee3862","tex":"\\HH_L","aliases":"HH_L HH","display":null,"kind":"use","occurrences":[864,866,868,893]},{"id":"symbol-e1e857460125b3b5","tex":"\\HH_{p,m}","aliases":"HH_p,m HH","display":null,"kind":"use","occurrences":[382,439]},{"id":"symbol-540b985016256f81","tex":"\\HH_S","aliases":"HH_S HH","display":null,"kind":"use","occurrences":[3019]},{"id":"symbol-26ff68a22dbb0da1","tex":"\\HH_x","aliases":"HH_x HH","display":null,"kind":"use","occurrences":[2606,2612]},{"id":"symbol-b70e977f58fbef0b","tex":"\\HH_{x+\\widehat\\mu}","aliases":"HH_x+widehatμ HH widehat mu","display":null,"kind":"use","occurrences":[2606,2612]},{"id":"symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","occurrences":[2324]},{"id":"symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"use","occurrences":[3725,3726]},{"id":"symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","occurrences":[1012,1013,1014,1027,1545,1553]},{"id":"symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","occurrences":[1468,1548,1557,1588,1591,1592,1597,1598,1606,1610,1612,1615,1620,1621,1622,1623,1624,1628,1635,1645,1646,1652,1657,1725,1907]},{"id":"symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","occurrences":[2913,2920,2931,2935,2936,2940,3006,3008,3009,3032]},{"id":"symbol-d971406bf59d1ecc","tex":"\\mathcal I_h","aliases":"I_h mathcal","display":"\\mathcal I_h","kind":"use","occurrences":[3725,3726]},{"id":"symbol-1838bc7ba65331a7","tex":"I_{\\KK}","aliases":"I_KK KK","display":null,"kind":"use","occurrences":[2273]},{"id":"symbol-136567efa8d010b5","tex":"I_{\\KK_{\\mathrm{tor}}}","aliases":"I_KK_tor KK mathrm","display":null,"kind":"use","occurrences":[604,606,611]},{"id":"symbol-3685c5d2d3894445","tex":"I_{K\\to L}","aliases":"I_K→ L to","display":"I_{K\\to L}","kind":"use","occurrences":[3251,3475,3478]},{"id":"symbol-acc489cb07feb1b4","tex":"I_{K\\to M}","aliases":"I_K→ M to","display":"I_{K\\to M}","kind":"use","occurrences":[3251]},{"id":"symbol-03e4b06b60f9a7dc","tex":"I_{L\\to M}","aliases":"I_L→ M to","display":"I_{L\\to M}","kind":"use","occurrences":[3251]},{"id":"symbol-a3ecf779d0b53571","tex":"I_M","aliases":"I_M","display":"I_M","kind":"use","occurrences":[798,812,1062]},{"id":"symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","occurrences":[215,218]},{"id":"symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","occurrences":[799]},{"id":"symbol-c7b33cae672307e6","tex":"I_p","aliases":"I_p","display":"I_p","kind":"use","occurrences":[202,218,841,844]},{"id":"symbol-89e0ceb2ea34de35","tex":"I_{\\mathrm{phase},\\mathbf r}","aliases":"I_phase, r mathrm mathbf","display":"I_{\\mathrm{phase},\\mathbf r}","kind":"use","occurrences":[635]},{"id":"symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","occurrences":[2993,2998]},{"id":"symbol-fad10e84b7053877","tex":"I_\\delta","aliases":"I_δ delta","display":"I_\\delta","kind":"use","occurrences":[3188,3189,3190,3192,3194,3195,3197,3202]},{"id":"symbol-f2a427dbf773b62b","tex":"I_\\ell","aliases":"I_ℓ ell","display":"I_\\ell","kind":"use","occurrences":[2974]},{"id":"symbol-b1185bc920eea10a","tex":"\\id_A","aliases":"id_A id","display":null,"kind":"use","occurrences":[2133,2146]},{"id":"symbol-b6db13550ab9effd","tex":"\\id_B","aliases":"id_B id","display":null,"kind":"use","occurrences":[2133,2136,2146]},{"id":"symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","occurrences":[560,822,904,913,914,927,932,2032,2033,2037,2110,2115,2121,2336,2338,2353,2725,2732,2810]},{"id":"symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"use","occurrences":[82,83]},{"id":"symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"use","occurrences":[307,310,311,315,349,655,3349,3350]},{"id":"symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","occurrences":[822,4072,4079,4087,4092,4101]},{"id":"symbol-41076cb14175109c","tex":"J^{\\mathrm c}","aliases":"J^ c mathrm","display":"J^{\\mathrm c}","kind":"use","occurrences":[2053,2066]},{"id":"symbol-eda0982baceee7a4","tex":"J^{\\mathrm c}_{B\\leftarrow A}","aliases":"J^ c_B← A mathrm leftarrow","display":"J^{\\mathrm c}_{B\\leftarrow A}","kind":"use","occurrences":[2062]},{"id":"symbol-6d08297f6303506d","tex":"J^{(\\rho)}_{a\\leftarrow b}","aliases":"J^(ρ)_a← b rho leftarrow","display":"J^{(\\rho)}_{a\\leftarrow b}","kind":"use","occurrences":[2112]},{"id":"symbol-e17cbcbf5402efc6","tex":"J^{(\\rho)}_{b\\leftarrow a}","aliases":"J^(ρ)_b← a rho leftarrow","display":"J^{(\\rho)}_{b\\leftarrow a}","kind":"use","occurrences":[2110,2112]},{"id":"symbol-35b3ffd4dcaf66dc","tex":"J^*","aliases":"J^*","display":"J^*","kind":"use","occurrences":[770]},{"id":"symbol-cd5b3dbd78a34d9d","tex":"J^F","aliases":"J^F","display":"J^F","kind":"use","occurrences":[2252]},{"id":"symbol-5e0b1d825c322ca4","tex":"J^F_{ba}","aliases":"J^F_ba","display":"J^F_{ba}","kind":"use","occurrences":[2238,2239,2241,2246,2251,2280]},{"id":"symbol-34af18168642b919","tex":"J^{F}_{ba}","aliases":"J^F_ba","display":"J^{F}_{ba}","kind":"use","occurrences":[2237]},{"id":"symbol-ae0b0d392d2d752f","tex":"J^{\\rm c}_{\\beta\\alpha}","aliases":"J^rm c_βα rm beta alpha","display":"J^{\\rm c}_{\\beta\\alpha}","kind":"use","occurrences":[2251,2353]},{"id":"symbol-e9d5a89b8a95b615","tex":"J_{\\mathrm D}^{\\Delta}","aliases":"J_ D^Δ mathrm Delta","display":"J_{\\mathrm D}^{\\Delta}","kind":"use","occurrences":[3429,3430,3432,3434,3438,3439,3459]},{"id":"symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","occurrences":[1265,3650,3651,3652,3654,3659]},{"id":"symbol-67957011b9286dc7","tex":"J_0'","aliases":"J_0'","display":"J_0'","kind":"use","occurrences":[3651,3654]},{"id":"symbol-351696876f533574","tex":"J_0''","aliases":"J_0''","display":"J_0''","kind":"use","occurrences":[3651,3654]},{"id":"symbol-1048fbee73eb4c54","tex":"J_{1:h}","aliases":"J_1:h","display":"J_{1:h}","kind":"use","occurrences":[1012,1014,1026]},{"id":"symbol-2ac431dc5751a954","tex":"J_{1\\leftarrow4}","aliases":"J_1←4 leftarrow","display":"J_{1\\leftarrow4}","kind":"use","occurrences":[2056]},{"id":"symbol-02186f046775278e","tex":"J_{2\\leftarrow1}","aliases":"J_2←1 leftarrow","display":"J_{2\\leftarrow1}","kind":"use","occurrences":[2037,2056,2098,2125]},{"id":"symbol-86b61ad61a12e40d","tex":"J_{3\\leftarrow2}","aliases":"J_3←2 leftarrow","display":"J_{3\\leftarrow2}","kind":"use","occurrences":[2056]},{"id":"symbol-2e4348540a1ad514","tex":"J_{4\\leftarrow3}","aliases":"J_4←3 leftarrow","display":"J_{4\\leftarrow3}","kind":"use","occurrences":[2056]},{"id":"symbol-9448a796bda79d54","tex":"J_5^0","aliases":"J_5^0","display":"J_5^0","kind":"use","occurrences":[3399]},{"id":"symbol-07efa069ab951ea9","tex":"J_5^I","aliases":"J_5^I","display":"J_5^I","kind":"use","occurrences":[3396,3401]},{"id":"symbol-aa1c6e5f9fe8f17e","tex":"J_5^J","aliases":"J_5^J","display":"J_5^J","kind":"use","occurrences":[3397]},{"id":"symbol-7fbcde5169efaf2d","tex":"J_5^K","aliases":"J_5^K","display":"J_5^K","kind":"use","occurrences":[3397]},{"id":"symbol-a7ed374167b55075","tex":"J_{5I}","aliases":"J_5I","display":"J_{5I}","kind":"use","occurrences":[3401]},{"id":"symbol-4740cf6406194851","tex":"J_{a-1}","aliases":"J_a-1","display":"J_{a-1}","kind":"use","occurrences":[2750]},{"id":"symbol-e268ef5e9ac6d864","tex":"J_{ab}","aliases":"J_ab","display":"J_{ab}","kind":"use","occurrences":[2337]},{"id":"symbol-8a5d3aa52b0f5bb8","tex":"J_{a\\leftarrow a}","aliases":"J_a← a leftarrow","display":"J_{a\\leftarrow a}","kind":"use","occurrences":[2019]},{"id":"symbol-777ddad06f846983","tex":"J_{a\\leftarrow b}","aliases":"J_a← b leftarrow","display":"J_{a\\leftarrow b}","kind":"use","occurrences":[2019]},{"id":"symbol-03acd6e46a6b3e27","tex":"J_{A\\leftarrow B}^{\\mathrm c}","aliases":"J_A← B^ c leftarrow mathrm","display":"J_{A\\leftarrow B}^{\\mathrm c}","kind":"use","occurrences":[2050]},{"id":"symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","occurrences":[2750]},{"id":"symbol-b1866826ed9899dc","tex":"J_{ba}","aliases":"J_ba","display":"J_{ba}","kind":"use","occurrences":[2213,2336,2337,2345,2346,2347,2353]},{"id":"symbol-8c11d396019c9a67","tex":"J_{B\\leftarrow A}^{\\mathrm c}","aliases":"J_B← A^ c leftarrow mathrm","display":"J_{B\\leftarrow A}^{\\mathrm c}","kind":"use","occurrences":[2048,2049,2050]},{"id":"symbol-6e30319bc4adb843","tex":"J_h","aliases":"J_h","display":"J_h","kind":"use","occurrences":[961,962]},{"id":"symbol-2669ee857bf534f7","tex":"J_j","aliases":"J_j","display":"J_j","kind":"use","occurrences":[2730,2733,2735,2740,2746,2751,2752,2756]},{"id":"symbol-d7945cdcf60e1077","tex":"J_{j+1/2}^n","aliases":"J_j+1/2^n","display":"J_{j+1/2}^n","kind":"use","occurrences":[2649,2650]},{"id":"symbol-14e9327ada8965fb","tex":"J_{j-1}","aliases":"J_j-1","display":"J_{j-1}","kind":"use","occurrences":[2735,2744,2746]},{"id":"symbol-e866d0c3b0e9c007","tex":"J_{j-1/2}^n","aliases":"J_j-1/2^n","display":"J_{j-1/2}^n","kind":"use","occurrences":[2650]},{"id":"symbol-b78dc9dea4a072f6","tex":"J_k","aliases":"J_k","display":"J_k","kind":"use","occurrences":[1668,1671]},{"id":"symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","occurrences":[1664,1665,1668,1669,1670,1671]},{"id":"symbol-fe88e020806e1739","tex":"J_r","aliases":"J_r","display":"J_r","kind":"use","occurrences":[796,797,799,806,807]},{"id":"symbol-e572acf19e0e35d0","tex":"J_{r:h}","aliases":"J_r:h","display":"J_{r:h}","kind":"use","occurrences":[998,999]},{"id":"symbol-3a4b59b7e16443be","tex":"J_r^*","aliases":"J_r^*","display":"J_r^*","kind":"use","occurrences":[798,799,805,807,812]},{"id":"symbol-dacf8aa7e30755b8","tex":"\\mathcal J_{r\\leftarrow s}","aliases":"J_r← s mathcal leftarrow","display":"\\mathcal J_{r\\leftarrow s}","kind":"use","occurrences":[822]},{"id":"symbol-b7b5625b6d739181","tex":"J_s","aliases":"J_s","display":"J_s","kind":"use","occurrences":[798,805]},{"id":"symbol-6e7616576e5283b5","tex":"J_s^*","aliases":"J_s^*","display":"J_s^*","kind":"use","occurrences":[806,812]},{"id":"symbol-faef0668e8df6422","tex":"\\mathcal J_{s\\leftarrow r}","aliases":"J_s← r mathcal leftarrow","display":"\\mathcal J_{s\\leftarrow r}","kind":"use","occurrences":[822]},{"id":"symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","occurrences":[812]},{"id":"symbol-f642708102f36295","tex":"j_{x,\\mu}","aliases":"j_x,μ mu","display":"j_{x,\\mu}","kind":"use","occurrences":[2810]},{"id":"symbol-be10909ba1926f76","tex":"J_\\Sigma","aliases":"J_Σ Sigma","display":"J_\\Sigma","kind":"use","occurrences":[3353,3356]},{"id":"symbol-39da9152b74aaf7c","tex":"J_{\\partial_+A}^n","aliases":"J_∂_+A^n partial","display":"J_{\\partial_+A}^n","kind":"use","occurrences":[2652]},{"id":"symbol-175fcc560bff2509","tex":"J_{\\partial_-A}^n","aliases":"J_∂_-A^n partial","display":"J_{\\partial_-A}^n","kind":"use","occurrences":[2652]},{"id":"symbol-655e6df9493408f0","tex":"\\mathbf k","aliases":"k mathbf","display":"\\mathbf k","kind":"use","occurrences":[607,3787,3789,3790]},{"id":"symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","occurrences":[1167,1170,1172,1183,1185,1186,1531,1532,1533,1931,3917,3924,3925]},{"id":"symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","occurrences":[16,20,21,24,54,56,77,82,86,87,88,89,90,94,197,280,281,501,503,504,625,627,628,797,801,802,829,832,835,836,838,839,858,859,860,1010,1011,1015,1016,1019,1114,1115,1154,1164,1180,1210,1221,1253,1254,1326,1479,1480,1481,1483,1485,1549,1556,1595,1596,1603,1604,1629,1630,1666,1668,1671,1680,1681,1768,1769,1771,1772,1798,1799,1802,1874,1887,1888,1889,2137,2139,2141,2169,2171,2567,2575,2581,3124,3126,3220,3237,3238,3243,3371,3373,3374,3733,3734,3735,3737,3740,3762,3771,3777,3778,3779,3781,3786,3789]},{"id":"symbol-1406e790403afdc4","tex":"k^{-2s}","aliases":"k^-2s","display":"k^{-2s}","kind":"use","occurrences":[1483]},{"id":"symbol-e47a0b4ba7b7781f","tex":"k^{-s}","aliases":"k^-s","display":"k^{-s}","kind":"use","occurrences":[1481]},{"id":"symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","occurrences":[1620,1621,1622]},{"id":"symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","occurrences":[1015,1016]},{"id":"symbol-479662d70e817846","tex":"k^{2-2s}","aliases":"k^2-2s","display":"k^{2-2s}","kind":"use","occurrences":[1485]},{"id":"symbol-ee47a0653a7c8273","tex":"K^i","aliases":"K^i","display":"K^i","kind":"use","occurrences":[3372]},{"id":"symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","occurrences":[3374,3402,3403]},{"id":"symbol-de6eed3cf1d7e144","tex":"k_a","aliases":"k_a","display":"k_a","kind":"use","occurrences":[86,88,89]},{"id":"symbol-f7f5fb972f2ae455","tex":"k_b","aliases":"k_b","display":"k_b","kind":"use","occurrences":[87,88,89]},{"id":"symbol-5203f1c6488200ed","tex":"K_{ba}","aliases":"K_ba","display":"K_{ba}","kind":"use","occurrences":[2016]},{"id":"symbol-7346537331b57441","tex":"K_{\\mathrm{eff}}","aliases":"K_eff mathrm","display":"K_{\\mathrm{eff}}","kind":"use","occurrences":[2079,2081,2095]},{"id":"symbol-2e49264ff683642e","tex":"K_{\\mathrm{eff}}^*","aliases":"K_eff^* mathrm","display":"K_{\\mathrm{eff}}^*","kind":"use","occurrences":[2079]},{"id":"symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","occurrences":[3779,3786]},{"id":"symbol-69e98d4d896d2853","tex":"k_i^4","aliases":"k_i^4","display":"k_i^4","kind":"use","occurrences":[3789]},{"id":"symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","occurrences":[3228,3229]},{"id":"symbol-d383fcbbf03353b5","tex":"K_M","aliases":"K_M","display":"K_M","kind":"use","occurrences":[638,639,646]},{"id":"symbol-792ddfc072fc3946","tex":"k_m","aliases":"k_m","display":"k_m","kind":"use","occurrences":[503]},{"id":"symbol-76c882ff3b5f883b","tex":"K_{M+1}","aliases":"K_M+1","display":"K_{M+1}","kind":"use","occurrences":[639,644,646]},{"id":"symbol-e9a37605986fa4ad","tex":"\\mathcal K_p","aliases":"K_p mathcal","display":"\\mathcal K_p","kind":"use","occurrences":[1167,1170,1183,1186]},{"id":"symbol-a3b86edec3341206","tex":"K_{p-1}","aliases":"K_p-1","display":"K_{p-1}","kind":"use","occurrences":[1042]},{"id":"symbol-d2422ad0559ddeb1","tex":"\\mathcal K_p^*","aliases":"K_p^* mathcal","display":"\\mathcal K_p^*","kind":"use","occurrences":[1170,1172,1185]},{"id":"symbol-1587c181ffea0832","tex":"\\mathcal K_S","aliases":"K_S mathcal","display":"\\mathcal K_S","kind":"use","occurrences":[1531,1532,1533,1931]},{"id":"symbol-e45ff3c8c2a5a13f","tex":"K_{S,a,T}","aliases":"K_S,a,T","display":"K_{S,a,T}","kind":"use","occurrences":[1395,1398]},{"id":"symbol-8e6a451de88cfbb0","tex":"K_\\mu","aliases":"K_μ mu","display":"K_\\mu","kind":"use","occurrences":[816,817,818]},{"id":"symbol-8929221b063811cf","tex":"K_\\mu^*","aliases":"K_μ^* mu","display":"K_\\mu^*","kind":"use","occurrences":[816,817]},{"id":"symbol-797b703ad526a53a","tex":"K_\\mu^{\\rm poly}","aliases":"K_μ^rm poly mu rm","display":"K_\\mu^{\\rm poly}","kind":"use","occurrences":[818]},{"id":"symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","occurrences":[100]},{"id":"symbol-4bd54056b2ab6858","tex":"\\KK_{\\mathbf r}","aliases":"KK_ r KK mathbf","display":null,"kind":"use","occurrences":[615,616,626]},{"id":"symbol-563d5bb39d3ff859","tex":"\\KK_a","aliases":"KK_a KK","display":null,"kind":"use","occurrences":[2253,2254,2256,2265]},{"id":"symbol-1d66668754c625dc","tex":"\\KK_{\\mathrm{tor}}","aliases":"KK_tor KK mathrm","display":null,"kind":"use","occurrences":[602,603,604,606,610,611]},{"id":"symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","occurrences":[3243,3245,3296,3297,3298,3301,3304,3305,3352,3408,3504,3505,3506]},{"id":"symbol-e3d8f1ac9931ea94","tex":"\\mathfrak L","aliases":"L mathfrak","display":"\\mathfrak L","kind":"use","occurrences":[3793,3795,3800,3823]},{"id":"symbol-5c373cbbb5a44e89","tex":"\\mathrm L","aliases":"L mathrm","display":"\\mathrm L","kind":"use","occurrences":[3348,3349,3353]},{"id":"symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","occurrences":[818,1015,3063,3064,3067,3082,3102,3737,3781]},{"id":"symbol-a44e55170cadffdb","tex":"L^*","aliases":"L^*","display":"L^*","kind":"use","occurrences":[2155]},{"id":"symbol-cc65d4ac028d8c8c","tex":"L^{-1}","aliases":"L^-1","display":"L^{-1}","kind":"use","occurrences":[1212]},{"id":"symbol-d4d9318beb0bea26","tex":"L^{2L}","aliases":"L^2L","display":"L^{2L}","kind":"use","occurrences":[1267]},{"id":"symbol-0387f51f80abc39f","tex":"L^{Lq}","aliases":"L^Lq","display":"L^{Lq}","kind":"use","occurrences":[1221]},{"id":"symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","occurrences":[1141]},{"id":"symbol-63e820edf4b8f186","tex":"L_{ba}^*","aliases":"L_ba^*","display":"L_{ba}^*","kind":"use","occurrences":[2160]},{"id":"symbol-f44072e22348fc64","tex":"\\mathcal L_c","aliases":"L_c mathcal","display":"\\mathcal L_c","kind":"use","occurrences":[3304,3305]},{"id":"symbol-c11b0603a60a2a4c","tex":"L_{CC}","aliases":"L_CC","display":"L_{CC}","kind":"use","occurrences":[3507]},{"id":"symbol-3735436a6c30f5d0","tex":"L_{CD}","aliases":"L_CD","display":"L_{CD}","kind":"use","occurrences":[3507]},{"id":"symbol-d3cc0ec7c5debf23","tex":"\\mathcal L_D","aliases":"L_D mathcal","display":"\\mathcal L_D","kind":"use","occurrences":[3504,3505,3506]},{"id":"symbol-f73fafaae8f13dd0","tex":"\\mathfrak L_D","aliases":"L_D mathfrak","display":"\\mathfrak L_D","kind":"use","occurrences":[3793,3795,3800]},{"id":"symbol-c582ec52f23a0898","tex":"\\mathfrak L_{D_h}","aliases":"L_D_h mathfrak","display":"\\mathfrak L_{D_h}","kind":"use","occurrences":[3823]},{"id":"symbol-bf1c6a9d4287ca7b","tex":"L_{DC}","aliases":"L_DC","display":"L_{DC}","kind":"use","occurrences":[3507]},{"id":"symbol-18907e293d8ed85e","tex":"L_{DD}^{-1}","aliases":"L_DD^-1","display":"L_{DD}^{-1}","kind":"use","occurrences":[3507]},{"id":"symbol-7ef8c7cca2b22b6a","tex":"L_{\\mathrm{eff}}","aliases":"L_eff mathrm","display":"L_{\\mathrm{eff}}","kind":"use","occurrences":[3507]},{"id":"symbol-ff93f51daddcc717","tex":"L_g","aliases":"L_g","display":"L_g","kind":"use","occurrences":[3971,3972,3975,3977]},{"id":"symbol-743ff0ea070acbd8","tex":"L_h","aliases":"L_h","display":"L_h","kind":"use","occurrences":[3521,3528,3538]},{"id":"symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","occurrences":[2613]},{"id":"symbol-777bcedff418f3fc","tex":"L_{p,h}","aliases":"L_p,h","display":"L_{p,h}","kind":"use","occurrences":[268,270]},{"id":"symbol-320283b00a6d4f02","tex":"\\mathcal L_\\eta","aliases":"L_η mathcal eta","display":"\\mathcal L_\\eta","kind":"use","occurrences":[3301,3352]},{"id":"symbol-d760990430efe993","tex":"\\mathcal L_\\eta^{(1/2)}","aliases":"L_η^(1/2) mathcal eta","display":"\\mathcal L_\\eta^{(1/2)}","kind":"use","occurrences":[3408]},{"id":"symbol-48918c8f911d3438","tex":"\\mathcal L_\\xi","aliases":"L_ξ mathcal xi","display":"\\mathcal L_\\xi","kind":"use","occurrences":[3296,3297,3298,3301]},{"id":"symbol-d1f5eb93d8b633b4","tex":"\\mathcal L_\\xi^{(-1)}","aliases":"L_ξ^(-1) mathcal xi","display":"\\mathcal L_\\xi^{(-1)}","kind":"use","occurrences":[3243,3245]},{"id":"symbol-f52ec1f6f7fb5fa2","tex":"\\operatorname{lcp}","aliases":"lcp operatorname","display":"\\operatorname{lcp}","kind":"use","occurrences":[752,753]},{"id":"symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","occurrences":[536,1455,3581]},{"id":"symbol-436d6e601e9eab47","tex":"\\lim_r","aliases":"lim_r lim","display":"\\lim_r","kind":"use","occurrences":[1455]},{"id":"symbol-244cdcc18a0e405f","tex":"\\lim_{r\\downarrow0}","aliases":"lim_rdownarrow0 lim downarrow","display":"\\lim_{r\\downarrow0}","kind":"use","occurrences":[3581]},{"id":"symbol-1416c21d48d701c1","tex":"\\lim_{t\\to0}","aliases":"lim_t→0 lim to","display":"\\lim_{t\\to0}","kind":"use","occurrences":[536]},{"id":"symbol-2a707ccad5a125ef","tex":"\\ll","aliases":"ll","display":null,"kind":"use","occurrences":[3607]},{"id":"symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","occurrences":[596,663,664,1190,1193,1194,1195,1196,1200,2185,2186,2191,2200,2202,2205,2851,2859,3492,3501,4062]},{"id":"symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","occurrences":[663,664]},{"id":"symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","occurrences":[1470]},{"id":"symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","occurrences":[751,1403]},{"id":"symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","occurrences":[2625]},{"id":"symbol-6ab13851c6dc5f1f","tex":"\\mathrm M","aliases":"M mathrm","display":"\\mathrm M","kind":"use","occurrences":[3278,3279,3294,3344,3345,3381]},{"id":"symbol-4977ee063fbe9a0d","tex":"\\mathrm m","aliases":"m mathrm","display":"\\mathrm m","kind":"use","occurrences":[2725,2751,2753,2756]},{"id":"symbol-6ee5820332effab6","tex":"\\mathcal M'","aliases":"M' mathcal","display":"\\mathcal M'","kind":"use","occurrences":[3856,3859,3860]},{"id":"symbol-a4c7fd7f94be16fe","tex":"m'","aliases":"m'","display":"m'","kind":"use","occurrences":[4058,4059,4077,4080]},{"id":"symbol-ca52a161c084be9c","tex":"M^{-1/2}","aliases":"M^-1/2","display":"M^{-1/2}","kind":"use","occurrences":[829]},{"id":"symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","occurrences":[3903,3917,3926,3954,3958]},{"id":"symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","occurrences":[1468,1469,1470,1476,1491,1493,1495,1502,1588,1591,1592,1593,1598,1602,1632,3282,3283,3284,3517,3665]},{"id":"symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","occurrences":[1141,1142,1143,1144,1147,1149,1151,1152,1154,1157,1158,3049,3050,3604,3605,3606,3610,3613,3616,4141]},{"id":"symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","occurrences":[1010,1011,1019]},{"id":"symbol-a0f711a1afb04d3f","tex":"M_0^2","aliases":"M_0^2","display":"M_0^2","kind":"use","occurrences":[1143,1150]},{"id":"symbol-5fa7968cb2b9c241","tex":"m_0^2","aliases":"m_0^2","display":"m_0^2","kind":"use","occurrences":[1015,1016]},{"id":"symbol-ca9e6b144b03a15e","tex":"M_0^j","aliases":"M_0^j","display":"M_0^j","kind":"use","occurrences":[1155]},{"id":"symbol-af2fd5cefc0c8b88","tex":"M_0^n","aliases":"M_0^n","display":"M_0^n","kind":"use","occurrences":[1144,1152,1155]},{"id":"symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","occurrences":[1244,3049,3050]},{"id":"symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","occurrences":[1475]},{"id":"symbol-b9396bdc88291f43","tex":"M_1^j","aliases":"M_1^j","display":"M_1^j","kind":"use","occurrences":[1245]},{"id":"symbol-c6d357730a0838d5","tex":"M_1^L","aliases":"M_1^L","display":"M_1^L","kind":"use","occurrences":[1245]},{"id":"symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"use","occurrences":[1475]},{"id":"symbol-6b34051ea3264f33","tex":"M_D","aliases":"M_D","display":"M_D","kind":"use","occurrences":[3484]},{"id":"symbol-7d6064a1e2fd6d56","tex":"M_G","aliases":"M_G","display":"M_G","kind":"use","occurrences":[3627,3629,3631,3634,3637,3639,3641,3643,3645,3656]},{"id":"symbol-6f65dba0db58365e","tex":"M_{G,h}","aliases":"M_G,h","display":"M_{G,h}","kind":"use","occurrences":[3721,3725]},{"id":"symbol-29c7b0251fbf7073","tex":"m_H","aliases":"m_H","display":"m_H","kind":"use","occurrences":[4151,4152]},{"id":"symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","occurrences":[580,581]},{"id":"symbol-c47f1cec87420595","tex":"M_{p,h}","aliases":"M_p,h","display":"M_{p,h}","kind":"use","occurrences":[207]},{"id":"symbol-8640417ce531dc32","tex":"M_{p,h}^{\\ell}","aliases":"M_p,h^ℓ ell","display":"M_{p,h}^{\\ell}","kind":"use","occurrences":[209]},{"id":"symbol-dbd15cd98c87c0e2","tex":"M_q","aliases":"M_q","display":"M_q","kind":"use","occurrences":[2188,2191,2192,2193]},{"id":"symbol-986d5a7882d8f884","tex":"M_{\\rm enc}","aliases":"M_rm enc rm","display":"M_{\\rm enc}","kind":"use","occurrences":[3582,3605]},{"id":"symbol-ca9432f255742492","tex":"\\mathring","aliases":"mathring","display":null,"kind":"use","occurrences":[3398,3399]},{"id":"symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","occurrences":[1786,3024,4117,4124,4128,4130]},{"id":"symbol-c52aaa85647b7be3","tex":"\\max_r","aliases":"max_r max","display":"\\max_r","kind":"use","occurrences":[1786]},{"id":"symbol-8c09df9956ae55f5","tex":"\\max_{x\\in S}","aliases":"max_x∈ S max in","display":"\\max_{x\\in S}","kind":"use","occurrences":[3024]},{"id":"symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","occurrences":[1170,3085,3086,3089,3091,3092,3096,3100,3111]},{"id":"symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","occurrences":[133,277,414,607,677]},{"id":"symbol-f7ed41372efc24c8","tex":"\\mathrm{mod}","aliases":"mod mathrm","display":"\\mathrm{mod}","kind":"use","occurrences":[129,275,276]},{"id":"symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"use","occurrences":[611,2299,2565]},{"id":"symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","occurrences":[3505,3506,3608,3609]},{"id":"symbol-c947cbb5e27a6a1d","tex":"\\N^{\\mathcal P}","aliases":"N^ P N mathcal","display":null,"kind":"use","occurrences":[611]},{"id":"symbol-f40bbcca7096796b","tex":"\\mathcal N^{-1}","aliases":"N^-1 mathcal","display":"\\mathcal N^{-1}","kind":"use","occurrences":[3505,3506]},{"id":"symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","occurrences":[301]},{"id":"symbol-5a1fe216a765aa7c","tex":"N^{-1/2}","aliases":"N^-1/2","display":"N^{-1/2}","kind":"use","occurrences":[88,94,2409]},{"id":"symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","occurrences":[77,300]},{"id":"symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","occurrences":[2391,2401,2407]},{"id":"symbol-2390514d28436a26","tex":"N^{-3}","aliases":"N^-3","display":"N^{-3}","kind":"use","occurrences":[3227]},{"id":"symbol-acb2c967f2e3c2e2","tex":"N^{-3/2}","aliases":"N^-3/2","display":"N^{-3/2}","kind":"use","occurrences":[2408]},{"id":"symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","occurrences":[2407]},{"id":"symbol-38855ce590112f03","tex":"n^{-s}","aliases":"n^-s","display":"n^{-s}","kind":"use","occurrences":[2323,2331]},{"id":"symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","occurrences":[1274,2406,2453,2458,2460,2461,3655]},{"id":"symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","occurrences":[3225,3240,3241]},{"id":"symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","occurrences":[1810,1812,1816,1829,1912]},{"id":"symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","occurrences":[1520]},{"id":"symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","occurrences":[1520]},{"id":"symbol-b69db358ab95952d","tex":"n_3","aliases":"n_3","display":"n_3","kind":"use","occurrences":[1520]},{"id":"symbol-b9de340bc0b2a993","tex":"n_A","aliases":"n_A","display":"n_A","kind":"use","occurrences":[1810,1812,1819,1832,1913]},{"id":"symbol-64d3fd3e91a16874","tex":"n_a","aliases":"n_a","display":"n_a","kind":"use","occurrences":[84,88]},{"id":"symbol-de5a861749813249","tex":"n_b","aliases":"n_b","display":"n_b","kind":"use","occurrences":[85,88]},{"id":"symbol-71f77540e0e9914c","tex":"n_D","aliases":"n_D","display":"n_D","kind":"use","occurrences":[1810,1812,1822,1835,1914]},{"id":"symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","occurrences":[3744,3745,3746,3748,3751,3762,3774]},{"id":"symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","occurrences":[3517]},{"id":"symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","occurrences":[3748,3751,3762]},{"id":"symbol-946f0d537d4b4f30","tex":"N_{j+1/2}","aliases":"N_j+1/2","display":"N_{j+1/2}","kind":"use","occurrences":[3517]},{"id":"symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","occurrences":[3762]},{"id":"symbol-75599878a7100180","tex":"N_r","aliases":"N_r","display":"N_r","kind":"use","occurrences":[4137,4150,4163,4168]},{"id":"symbol-1dad9952c8e86d69","tex":"\\mathcal N_{\\rm emit}","aliases":"N_rm emit mathcal rm","display":"\\mathcal N_{\\rm emit}","kind":"use","occurrences":[3609]},{"id":"symbol-8e74a14211720a9d","tex":"\\mathcal N_{\\rm obs}","aliases":"N_rm obs mathcal rm","display":"\\mathcal N_{\\rm obs}","kind":"use","occurrences":[3609]},{"id":"symbol-0ea5b5c0e57e64fb","tex":"\\N_{\\geq1}","aliases":"N_≥1 N geq","display":null,"kind":"use","occurrences":[2299]},{"id":"symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","occurrences":[22,1170]},{"id":"symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","occurrences":[286,701,702,1411,1729,1731,1742,1743,1746,1749,1761,1764,2408,2409,2701,2706,2707,2708,2709,2710,2711,2712,2713,2714,2778,2785,2791,2794,2796,2800,2852,2858,2859,2860,2865,3550,3609,3703,3704,3705,3708,3710,3789,3821,3823,3827,3828,3928]},{"id":"symbol-50cad36890440685","tex":"\\mathrm o","aliases":"o mathrm","display":"\\mathrm o","kind":"use","occurrences":[71,72,74,76,78]},{"id":"symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","occurrences":[818,1324,3063,3064,3067,3082,3102,3609,3613,3773,4091,4092]},{"id":"symbol-b77c3e9c22499671","tex":"\\oint","aliases":"oint","display":null,"kind":"use","occurrences":[3558]},{"id":"symbol-95ca3968ddfe525b","tex":"\\oint_{\\partial\\square}","aliases":"oint_∂square oint partial square","display":null,"kind":"use","occurrences":[3558]},{"id":"symbol-754626c55688277f","tex":"\\operatorname{or}","aliases":"or operatorname","display":"\\operatorname{or}","kind":"use","occurrences":[2693,3336]},{"id":"symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"use","occurrences":[82,83]},{"id":"symbol-01eecdf5cc4bca8b","tex":"\\overline A","aliases":"overline A overline","display":"\\overline A","kind":"use","occurrences":[3258]},{"id":"symbol-e83fbd63131abb69","tex":"\\overline f","aliases":"overline f overline","display":"\\overline f","kind":"use","occurrences":[379]},{"id":"symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","occurrences":[321]},{"id":"symbol-5f51cdcfd251a311","tex":"\\overline g","aliases":"overline g overline","display":"\\overline g","kind":"use","occurrences":[421]},{"id":"symbol-69a71f8a9a72cb1e","tex":"\\overline G_{p,w}","aliases":"overline G_p,w overline","display":"\\overline G_{p,w}","kind":"use","occurrences":[321]},{"id":"symbol-dc6c360a7accd337","tex":"\\overline{G_{p,w}}","aliases":"overlineG_p,w overline","display":"\\overline{G_{p,w}}","kind":"use","occurrences":[311]},{"id":"symbol-0a76f689d811f120","tex":"\\overline{\\inner{\\psi_b}{K\\psi_a}}","aliases":"overlineinnerψ_bKψ_a overline inner psi psi","display":null,"kind":"use","occurrences":[2024]},{"id":"symbol-ec76d89d14f0c305","tex":"\\overline{P_+X}","aliases":"overlineP_+X overline","display":"\\overline{P_+X}","kind":"use","occurrences":[3334]},{"id":"symbol-0a2674f534bda381","tex":"\\overline{v_r}","aliases":"overlinev_r overline","display":"\\overline{v_r}","kind":"use","occurrences":[1525]},{"id":"symbol-739fbcd12238ffb3","tex":"\\overline{z_{ba}}","aliases":"overlinez_ba overline","display":"\\overline{z_{ba}}","kind":"use","occurrences":[2114]},{"id":"symbol-0b2f408b058396fe","tex":"\\overline{\\{\\epsilon_m:m\\geq0\\}}","aliases":"overline{ϵ_m:m≥0} overline epsilon geq","display":"\\overline{\\{\\epsilon_m:m\\geq0\\}}","kind":"use","occurrences":[497]},{"id":"symbol-5c3a4a38379b7068","tex":"\\overline\\gamma","aliases":"overlineγ overline gamma","display":"\\overline\\gamma","kind":"use","occurrences":[290]},{"id":"symbol-6e4c3e0b2110e648","tex":"\\overline{\\gamma_{p,w}}","aliases":"overlineγ_p,w overline gamma","display":"\\overline{\\gamma_{p,w}}","kind":"use","occurrences":[307]},{"id":"symbol-55dc76c431a12088","tex":"\\overline\\psi","aliases":"overlineψ overline psi","display":"\\overline\\psi","kind":"use","occurrences":[2727,2753]},{"id":"symbol-74afe80986b7b88b","tex":"\\overline\\psi_j","aliases":"overlineψ_j overline psi","display":"\\overline\\psi_j","kind":"use","occurrences":[2727,2753]},{"id":"symbol-aab687f6768e37ac","tex":"\\overline\\nabla","aliases":"overline∇ overline nabla","display":"\\overline\\nabla","kind":"use","occurrences":[3424,3425,3427,3434,3439,3451,3459,3460,3462]},{"id":"symbol-57fcee80b46c2130","tex":"\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}","aliases":"overline⟨ψ_r,H_rsψ_s⟩ overline langle psi psi rangle","display":"\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}","kind":"use","occurrences":[825]},{"id":"symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","occurrences":[71,72,74,76,78,82,83,601,603,611,616]},{"id":"symbol-627237205a8ef32d","tex":"\\mathrm P","aliases":"P mathrm","display":"\\mathrm P","kind":"use","occurrences":[3388]},{"id":"symbol-394515d30df05582","tex":"p^{\\mathrm c}","aliases":"p^ c mathrm","display":"p^{\\mathrm c}","kind":"use","occurrences":[2053,2065]},{"id":"symbol-9b5512f13515252c","tex":"p^{(1-2\\alpha)w}","aliases":"p^(1-2α)w alpha","display":"p^{(1-2\\alpha)w}","kind":"use","occurrences":[341]},{"id":"symbol-5e3c49b631652076","tex":"p^{(1/2-\\alpha)w}","aliases":"p^(1/2-α)w alpha","display":"p^{(1/2-\\alpha)w}","kind":"use","occurrences":[341]},{"id":"symbol-d0b51e771199536c","tex":"p^{(m+1)/2}","aliases":"p^(m+1)/2","display":"p^{(m+1)/2}","kind":"use","occurrences":[395]},{"id":"symbol-cfe2afc69712b120","tex":"p^{(m-1)/2}","aliases":"p^(m-1)/2","display":"p^{(m-1)/2}","kind":"use","occurrences":[468]},{"id":"symbol-6feb1c26b70e6413","tex":"P^+_A","aliases":"P^+_A","display":"P^+_A","kind":"use","occurrences":[2659]},{"id":"symbol-d61400e021ea8cb0","tex":"p^{-(h-r)}","aliases":"p^-(h-r)","display":"p^{-(h-r)}","kind":"use","occurrences":[129,275,276]},{"id":"symbol-0e19dd0107059fb7","tex":"p^{-(h-r+1)}","aliases":"p^-(h-r+1)","display":"p^{-(h-r+1)}","kind":"use","occurrences":[276]},{"id":"symbol-050b78fe0bbad3ec","tex":"p^{-(r+1)/2}","aliases":"p^-(r+1)/2","display":"p^{-(r+1)/2}","kind":"use","occurrences":[954]},{"id":"symbol-aedd6029b023e559","tex":"p^{-(r-1)/2}","aliases":"p^-(r-1)/2","display":"p^{-(r-1)/2}","kind":"use","occurrences":[957]},{"id":"symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","occurrences":[170,187,211,472]},{"id":"symbol-d718e07c40dc0d2e","tex":"p^{-1/2}","aliases":"p^-1/2","display":"p^{-1/2}","kind":"use","occurrences":[184,186,188,217,395,402,853]},{"id":"symbol-054866e9b11e6acb","tex":"p^{-2w}","aliases":"p^-2w","display":"p^{-2w}","kind":"use","occurrences":[373]},{"id":"symbol-8dec76303c90ddfc","tex":"P^-_A","aliases":"P^-_A","display":"P^-_A","kind":"use","occurrences":[2659]},{"id":"symbol-570d98a38290a283","tex":"p^{-h}","aliases":"p^-h","display":"p^{-h}","kind":"use","occurrences":[276]},{"id":"symbol-d044418aad4b95c8","tex":"p^{-h/2}","aliases":"p^-h/2","display":"p^{-h/2}","kind":"use","occurrences":[162]},{"id":"symbol-8a1b7e2cb35dff5a","tex":"p^{-m}","aliases":"p^-m","display":"p^{-m}","kind":"use","occurrences":[377,397,404,471]},{"id":"symbol-cfebc46ff80f9645","tex":"p^{-r/2}","aliases":"p^-r/2","display":"p^{-r/2}","kind":"use","occurrences":[949,954,957]},{"id":"symbol-46695d5ac294b8b0","tex":"p^{-s(M+1)}","aliases":"p^-s(M+1)","display":"p^{-s(M+1)}","kind":"use","occurrences":[481,550]},{"id":"symbol-fd6c590f56fc7427","tex":"p^{-s/2}","aliases":"p^-s/2","display":"p^{-s/2}","kind":"use","occurrences":[972]},{"id":"symbol-d2c543a54857b35f","tex":"p^{-w}","aliases":"p^-w","display":"p^{-w}","kind":"use","occurrences":[303,361,363,366,368,372,675,703,704,705,708]},{"id":"symbol-4618987e95ac91f8","tex":"p^{-w/2}","aliases":"p^-w/2","display":"p^{-w/2}","kind":"use","occurrences":[192,193,302,360,365]},{"id":"symbol-40ea66b34b55a33f","tex":"p^{-\\alpha w}","aliases":"p^-α w alpha","display":"p^{-\\alpha w}","kind":"use","occurrences":[339]},{"id":"symbol-3100e6c13d0c66d3","tex":"p^{2s(M+1)}","aliases":"p^2s(M+1)","display":"p^{2s(M+1)}","kind":"use","occurrences":[484]},{"id":"symbol-0000c447057fb504","tex":"p^{2sm}","aliases":"p^2sm","display":"p^{2sm}","kind":"use","occurrences":[476,484]},{"id":"symbol-df7ab5771f41cb6f","tex":"p^A","aliases":"p^A","display":"p^A","kind":"use","occurrences":[2145]},{"id":"symbol-6c2132e901148066","tex":"p^B","aliases":"p^B","display":"p^B","kind":"use","occurrences":[2145]},{"id":"symbol-7f4968674b99b5fb","tex":"p^b","aliases":"p^b","display":"p^b","kind":"use","occurrences":[989,990,997]},{"id":"symbol-0edaee8e5b48a0f0","tex":"p^h","aliases":"p^h","display":"p^h","kind":"use","occurrences":[12,18,29,52,53,66,70,123,184,188,195,265,266,863,1140]},{"id":"symbol-c872c2116fdfaa92","tex":"p^{h+1}","aliases":"p^h+1","display":"p^{h+1}","kind":"use","occurrences":[5,12,19,67]},{"id":"symbol-9952deedd172b29f","tex":"p^{h-1}","aliases":"p^h-1","display":"p^{h-1}","kind":"use","occurrences":[196,197,203,207]},{"id":"symbol-8afb4c38411cb8b1","tex":"p^{h-2}","aliases":"p^h-2","display":"p^{h-2}","kind":"use","occurrences":[206]},{"id":"symbol-220bbebb58967375","tex":"p^{h-k-1}","aliases":"p^h-k-1","display":"p^{h-k-1}","kind":"use","occurrences":[281]},{"id":"symbol-be94f0538ad303b4","tex":"p^{h-r}","aliases":"p^h-r","display":"p^{h-r}","kind":"use","occurrences":[126]},{"id":"symbol-f19099897aa002b9","tex":"p^{h-r+1}","aliases":"p^h-r+1","display":"p^{h-r+1}","kind":"use","occurrences":[168]},{"id":"symbol-351cda8cd86b95b8","tex":"p^{h_p}","aliases":"p^h_p","display":"p^{h_p}","kind":"use","occurrences":[74,76,78]},{"id":"symbol-b9a485297aa071aa","tex":"p^{h_p+1}","aliases":"p^h_p+1","display":"p^{h_p+1}","kind":"use","occurrences":[72]},{"id":"symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","occurrences":[16]},{"id":"symbol-f21691df4ee7e60b","tex":"p^{k+1}","aliases":"p^k+1","display":"p^{k+1}","kind":"use","occurrences":[21]},{"id":"symbol-dc82ec356cf9d9b6","tex":"p^{k+2}","aliases":"p^k+2","display":"p^{k+2}","kind":"use","occurrences":[24]},{"id":"symbol-4927a5c0083e2327","tex":"p^{m-1}","aliases":"p^m-1","display":"p^{m-1}","kind":"use","occurrences":[464,466,468,471,593]},{"id":"symbol-ab24a79a4655574e","tex":"p^{m/2}","aliases":"p^m/2","display":"p^{m/2}","kind":"use","occurrences":[384,389,395,403,405]},{"id":"symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","occurrences":[35,36]},{"id":"symbol-c020cb0886702b18","tex":"p^{n+1}","aliases":"p^n+1","display":"p^{n+1}","kind":"use","occurrences":[37]},{"id":"symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","occurrences":[125,129,132,148,275,276,720,951]},{"id":"symbol-1fc0b92fec83c631","tex":"p^{\\rm c}_\\beta","aliases":"p^rm c_β rm beta","display":"p^{\\rm c}_\\beta","kind":"use","occurrences":[2251]},{"id":"symbol-878e0df7265da180","tex":"p^{w/2}","aliases":"p^w/2","display":"p^{w/2}","kind":"use","occurrences":[342,354,370]},{"id":"symbol-9377ed70d686e166","tex":"p^{\\alpha m}","aliases":"p^α m alpha","display":"p^{\\alpha m}","kind":"use","occurrences":[597]},{"id":"symbol-832eb479480de132","tex":"\\mathcal P_{\\mathrm o}","aliases":"P_ o mathcal mathrm","display":"\\mathcal P_{\\mathrm o}","kind":"use","occurrences":[71,72,74,76,78]},{"id":"symbol-7cd48fb48ab38484","tex":"p_*","aliases":"p_*","display":"p_*","kind":"use","occurrences":[3954,3957]},{"id":"symbol-25581a5c9fb5bdcb","tex":"p_*^2","aliases":"p_*^2","display":"p_*^2","kind":"use","occurrences":[3952]},{"id":"symbol-83f42125c2adbdfd","tex":"p_*^3","aliases":"p_*^3","display":"p_*^3","kind":"use","occurrences":[3954,3958]},{"id":"symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","occurrences":[3450]},{"id":"symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","occurrences":[2036,2096,2170,2171,3450]},{"id":"symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","occurrences":[2036,2097,2170,2179]},{"id":"symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"use","occurrences":[1960,1962,1984,3150,3151,3153,3154]},{"id":"symbol-e7d1a3be90d27945","tex":"p_A","aliases":"p_A","display":"p_A","kind":"use","occurrences":[1987]},{"id":"symbol-fc3eca2c0cc0fb84","tex":"p_A^{\\mathrm c}","aliases":"p_A^ c mathrm","display":"p_A^{\\mathrm c}","kind":"use","occurrences":[2048,2175,2176]},{"id":"symbol-4338c025955f4745","tex":"P_a^{(0)}","aliases":"P_a^(0)","display":"P_a^{(0)}","kind":"use","occurrences":[2073,2075,2076,2094]},{"id":"symbol-50041424806fdeac","tex":"P_A^+","aliases":"P_A^+","display":"P_A^+","kind":"use","occurrences":[2660,2665,2666]},{"id":"symbol-7931d33253166dfc","tex":"P_A^-","aliases":"P_A^-","display":"P_A^-","kind":"use","occurrences":[2660,2665,2666]},{"id":"symbol-d2ca5ff6fec3e166","tex":"p_a^A","aliases":"p_a^A","display":"p_a^A","kind":"use","occurrences":[2135,2136,2143]},{"id":"symbol-8b2da815d2e052f3","tex":"p_a^q","aliases":"p_a^q","display":"p_a^q","kind":"use","occurrences":[2188]},{"id":"symbol-1ff79702d0f06146","tex":"p_a^{q-1}","aliases":"p_a^q-1","display":"p_a^{q-1}","kind":"use","occurrences":[2189]},{"id":"symbol-484d493d93b99218","tex":"p_{a\\mu}","aliases":"p_aμ mu","display":"p_{a\\mu}","kind":"use","occurrences":[2135,2143,2145]},{"id":"symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","occurrences":[1961,1962,1984]},{"id":"symbol-d072d8979775d522","tex":"p_B","aliases":"p_B","display":"p_B","kind":"use","occurrences":[1988]},{"id":"symbol-d04253126b67c473","tex":"p_b","aliases":"p_b","display":"p_b","kind":"use","occurrences":[2021,2157,2158,2184,2186,2336,3791]},{"id":"symbol-43452d7cc9435d59","tex":"p_B^{\\mathrm c}","aliases":"p_B^ c mathrm","display":"p_B^{\\mathrm c}","kind":"use","occurrences":[2176]},{"id":"symbol-bb071e0da3d1f51a","tex":"P_b^2","aliases":"P_b^2","display":"P_b^2","kind":"use","occurrences":[2004]},{"id":"symbol-0f1c833f254f57e3","tex":"p_b^{q-1}","aliases":"p_b^q-1","display":"p_b^{q-1}","kind":"use","occurrences":[2189]},{"id":"symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","occurrences":[1959,2003,2523]},{"id":"symbol-325fba8fb10630f7","tex":"\\mathcal P_{\\mathrm{in}}","aliases":"P_in mathcal mathrm","display":"\\mathcal P_{\\mathrm{in}}","kind":"use","occurrences":[82,83]},{"id":"symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","occurrences":[3675]},{"id":"symbol-045a1b8959b0bef9","tex":"p_j^2","aliases":"p_j^2","display":"p_j^2","kind":"use","occurrences":[1593,1598]},{"id":"symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","occurrences":[1595]},{"id":"symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","occurrences":[889,890]},{"id":"symbol-82a6b3a669544b39","tex":"p_{L,0}","aliases":"p_L,0","display":"p_{L,0}","kind":"use","occurrences":[1331]},{"id":"symbol-f3065e30b0c8c4bc","tex":"p_{L,j}","aliases":"p_L,j","display":"p_{L,j}","kind":"use","occurrences":[1330,1332,1334,1339]},{"id":"symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","occurrences":[3148,3149]},{"id":"symbol-081e137663ec699f","tex":"\\mathcal P_{\\mathrm{out}}","aliases":"P_out mathcal mathrm","display":"\\mathcal P_{\\mathrm{out}}","kind":"use","occurrences":[82]},{"id":"symbol-0f540fdbe066e4cf","tex":"\\mathcal P_{\\mathrm{out}}^*","aliases":"P_out^* mathcal mathrm","display":"\\mathcal P_{\\mathrm{out}}^*","kind":"use","occurrences":[83]},{"id":"symbol-d4876c09442ebddb","tex":"P_p","aliases":"P_p","display":"P_p","kind":"use","occurrences":[841,844,854,857,862,3058]},{"id":"symbol-e83884e0b8fb5b18","tex":"P_p^*","aliases":"P_p^*","display":"P_p^*","kind":"use","occurrences":[857]},{"id":"symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","occurrences":[3058]},{"id":"symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","occurrences":[1298,1299,1300,3850,3851,3852,3853,3861,3875,3876,3884,3888,3894,3914,3941,3954,4001,4056,4057,4061]},{"id":"symbol-824f428da635dd8f","tex":"p_R","aliases":"p_R","display":"p_R","kind":"use","occurrences":[3889]},{"id":"symbol-b35e522aef4bea0d","tex":"P_{R,j}","aliases":"P_R,j","display":"P_{R,j}","kind":"use","occurrences":[3943]},{"id":"symbol-010f131ed455d9fc","tex":"P_S","aliases":"P_S","display":"P_S","kind":"use","occurrences":[1359,1362,1383,1393,1395,1398,1405,1460,1465]},{"id":"symbol-27ac1d6f9a948981","tex":"p_x","aliases":"p_x","display":"p_x","kind":"use","occurrences":[1561,1562]},{"id":"symbol-398dcf493d87760a","tex":"p_{xy}","aliases":"p_xy","display":"p_{xy}","kind":"use","occurrences":[2951,2952]},{"id":"symbol-a1a9970aad3792ab","tex":"p_{XYZ}","aliases":"p_XYZ","display":"p_{XYZ}","kind":"use","occurrences":[2957,2963,2970,2983,3059]},{"id":"symbol-0ba60bcf1db4d4e8","tex":"p_{XZY}","aliases":"p_XZY","display":"p_{XZY}","kind":"use","occurrences":[2958,2963,3059]},{"id":"symbol-17584b33b8ab7132","tex":"p_y","aliases":"p_y","display":"p_y","kind":"use","occurrences":[1561,1562]},{"id":"symbol-eecfb231ea3c5eb1","tex":"p_{yx}","aliases":"p_yx","display":"p_{yx}","kind":"use","occurrences":[2951,2952]},{"id":"symbol-611887b9ab141940","tex":"p_{YXZ}","aliases":"p_YXZ","display":"p_{YXZ}","kind":"use","occurrences":[2959,2963,2984,3059]},{"id":"symbol-567a8517d9d15c5f","tex":"p_{YZX}","aliases":"p_YZX","display":"p_{YZX}","kind":"use","occurrences":[2960,2963,3059]},{"id":"symbol-91513c11f34221ca","tex":"P_z","aliases":"P_z","display":"P_z","kind":"use","occurrences":[3056]},{"id":"symbol-60af386a48855ec5","tex":"p_z","aliases":"p_z","display":"p_z","kind":"use","occurrences":[1561,1562]},{"id":"symbol-f61bdb6306d6c2c5","tex":"p_{ZXY}","aliases":"p_ZXY","display":"p_{ZXY}","kind":"use","occurrences":[2961,2963,3059]},{"id":"symbol-783bdc5dbbc88b2c","tex":"p_{ZYX}","aliases":"p_ZYX","display":"p_{ZYX}","kind":"use","occurrences":[2962,2963,2971,3059]},{"id":"symbol-3298ad8090debb2c","tex":"P_\\pm^*","aliases":"P_±^* pm","display":"P_\\pm^*","kind":"use","occurrences":[1607]},{"id":"symbol-66044690d086e578","tex":"P_\\pm^2","aliases":"P_±^2 pm","display":"P_\\pm^2","kind":"use","occurrences":[1608,1621]},{"id":"symbol-58fb42807c0a2ca7","tex":"P_\\theta","aliases":"P_θ theta","display":"P_\\theta","kind":"use","occurrences":[3263,3623,3627,3631,3637,3638,3656]},{"id":"symbol-1a282e80877df43d","tex":"P_\\theta^2","aliases":"P_θ^2 theta","display":"P_\\theta^2","kind":"use","occurrences":[3261,3621,3627]},{"id":"symbol-b0351f4b03177b3e","tex":"P_{\\theta\\theta}","aliases":"P_θθ theta theta","display":"P_{\\theta\\theta}","kind":"use","occurrences":[3623,3637,3655]},{"id":"symbol-68c3342b398c78bd","tex":"P_\\Lambda'","aliases":"P_Λ' Lambda","display":"P_\\Lambda'","kind":"use","occurrences":[3851,3860,3884,3894,4057,4058,4074]},{"id":"symbol-0a2084aa04022a3d","tex":"P_{\\Lambda,j}","aliases":"P_Λ,j Lambda","display":"P_{\\Lambda,j}","kind":"use","occurrences":[3943]},{"id":"symbol-2b4b53c02eca0902","tex":"P_{\\Lambda,j}^2","aliases":"P_Λ,j^2 Lambda","display":"P_{\\Lambda,j}^2","kind":"use","occurrences":[3943]},{"id":"symbol-6aaf97ce7bc4622c","tex":"P_\\Lambda^2","aliases":"P_Λ^2 Lambda","display":"P_\\Lambda^2","kind":"use","occurrences":[3852,3860,3894,3948,4058]},{"id":"symbol-df000ea46aaa7e45","tex":"p_\\mu^B","aliases":"p_μ^B mu","display":"p_\\mu^B","kind":"use","occurrences":[2135,2143]},{"id":"symbol-c17f246de5f52801","tex":"P_\\tau","aliases":"P_τ tau","display":"P_\\tau","kind":"use","occurrences":[3623,3627,3635,3637,3656]},{"id":"symbol-db675a504cfbee47","tex":"P_\\tau^2","aliases":"P_τ^2 tau","display":"P_\\tau^2","kind":"use","occurrences":[3621,3627]},{"id":"symbol-0aa9f14ee49222c6","tex":"P_{\\tau\\theta}","aliases":"P_τθ tau theta","display":"P_{\\tau\\theta}","kind":"use","occurrences":[3635]},{"id":"symbol-308d77c0d16ee668","tex":"P_{\\tau\\tau}","aliases":"P_ττ tau tau","display":"P_{\\tau\\tau}","kind":"use","occurrences":[3623,3637]},{"id":"symbol-cfc2554aee2ed59e","tex":"P_\\varphi","aliases":"P_φ varphi","display":"P_\\varphi","kind":"use","occurrences":[4043,4044,4048,4057]},{"id":"symbol-519ea6b75332fc0d","tex":"P_\\varphi^2","aliases":"P_φ^2 varphi","display":"P_\\varphi^2","kind":"use","occurrences":[4044]},{"id":"symbol-8a243f3b87b7dca3","tex":"\\mathrm{par}","aliases":"par mathrm","display":"\\mathrm{par}","kind":"use","occurrences":[3300,3301]},{"id":"symbol-fe8f749b378cd86e","tex":"\\mathrm{phase}","aliases":"phase mathrm","display":"\\mathrm{phase}","kind":"use","occurrences":[95,100,635]},{"id":"symbol-53743af9d1feb8d1","tex":"\\mathrm{phys}","aliases":"phys mathrm","display":"\\mathrm{phys}","kind":"use","occurrences":[1355,1496,1498,1499,1502]},{"id":"symbol-eda0fef223ad56b8","tex":"\\propto","aliases":"propto","display":null,"kind":"use","occurrences":[3491]},{"id":"symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","occurrences":[789]},{"id":"symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","occurrences":[3851,3860,3875,3885]},{"id":"symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","occurrences":[169,170,472]},{"id":"symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","occurrences":[2707]},{"id":"symbol-e4b91cbee49fa788","tex":"Q^4","aliases":"Q^4","display":"Q^4","kind":"use","occurrences":[3917]},{"id":"symbol-d014cc8d6971e544","tex":"q^{ab}","aliases":"q^ab","display":"q^{ab}","kind":"use","occurrences":[3366]},{"id":"symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","occurrences":[566]},{"id":"symbol-8622e3ac0dd5438d","tex":"q^{M+1}","aliases":"q^M+1","display":"q^{M+1}","kind":"use","occurrences":[568,573]},{"id":"symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","occurrences":[3449]},{"id":"symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","occurrences":[3449]},{"id":"symbol-506afac37903bdfd","tex":"Q_A","aliases":"Q_A","display":"Q_A","kind":"use","occurrences":[2045,2047,2048]},{"id":"symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","occurrences":[3133,3135,3138,3791,3836]},{"id":"symbol-d8eb43a9c9648747","tex":"Q_A^n","aliases":"Q_A^n","display":"Q_A^n","kind":"use","occurrences":[2652]},{"id":"symbol-975da5ad2f8f771b","tex":"Q_A^{n+1}","aliases":"Q_A^n+1","display":"Q_A^{n+1}","kind":"use","occurrences":[2652]},{"id":"symbol-4ac081124caea3b8","tex":"q_{AB}","aliases":"q_AB","display":"q_{AB}","kind":"use","occurrences":[2176]},{"id":"symbol-44b1f051975da19b","tex":"q_{ab}","aliases":"q_ab","display":"q_{ab}","kind":"use","occurrences":[3277,3359,3362,3381]},{"id":"symbol-b6cbebd4328bd504","tex":"Q_B","aliases":"Q_B","display":"Q_B","kind":"use","occurrences":[2048]},{"id":"symbol-3653386b19dd38cf","tex":"q_{BA}","aliases":"q_BA","display":"q_{BA}","kind":"use","occurrences":[2174,2175,2176]},{"id":"symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","occurrences":[3061,3062,3063,3067]},{"id":"symbol-ae104bef53b5518c","tex":"q_{cd}","aliases":"q_cd","display":"q_{cd}","kind":"use","occurrences":[3366]},{"id":"symbol-688d5aa3a341b160","tex":"Q_f","aliases":"Q_f","display":"Q_f","kind":"use","occurrences":[3060,3062,3063,3066,3067]},{"id":"symbol-393f3636d9c3ffef","tex":"q_h","aliases":"q_h","display":"q_h","kind":"use","occurrences":[1571,1572,1592,1736,1738,1815,1863]},{"id":"symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","occurrences":[1326,1327,3943]},{"id":"symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"use","occurrences":[1570,1573,1701,1742,1865,2753]},{"id":"symbol-acc408ad72dcbd3f","tex":"Q_j^{(L)}","aliases":"Q_j^(L)","display":"Q_j^{(L)}","kind":"use","occurrences":[1324,1325,1330]},{"id":"symbol-e88cbadb72581711","tex":"q_j^*","aliases":"q_j^*","display":"q_j^*","kind":"use","occurrences":[2754,2755]},{"id":"symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","occurrences":[1326]},{"id":"symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"use","occurrences":[789]},{"id":"symbol-237bdbcaf2851c9d","tex":"Q_r^{(L)}","aliases":"Q_r^(L)","display":"Q_r^{(L)}","kind":"use","occurrences":[1349]},{"id":"symbol-c07593a62404ec46","tex":"Q_s^{(L)}","aliases":"Q_s^(L)","display":"Q_s^{(L)}","kind":"use","occurrences":[1349]},{"id":"symbol-65fa07eabd1080a1","tex":"q_x^a","aliases":"q_x^a","display":"q_x^a","kind":"use","occurrences":[2812]},{"id":"symbol-5e6c82058fa3275d","tex":"q_x^b","aliases":"q_x^b","display":"q_x^b","kind":"use","occurrences":[2813]},{"id":"symbol-629e3cebbe7000f8","tex":"q_x^{g,a}","aliases":"q_x^g,a","display":"q_x^{g,a}","kind":"use","occurrences":[2813]},{"id":"symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"use","occurrences":[2349,2350,2351,2352,2353,2358,2531,2532]},{"id":"symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"use","occurrences":[2353,2363,2532]},{"id":"symbol-fbcb007cb50c0f2b","tex":"Q_\\theta","aliases":"Q_θ theta","display":"Q_\\theta","kind":"use","occurrences":[3263,3623,3625,3627,3631,3637]},{"id":"symbol-9310f52f93902981","tex":"Q_\\theta^2","aliases":"Q_θ^2 theta","display":"Q_\\theta^2","kind":"use","occurrences":[3261,3621,3623,3627,3637,3638]},{"id":"symbol-0b28e92c9075b032","tex":"Q_{\\theta\\theta}","aliases":"Q_θθ theta theta","display":"Q_{\\theta\\theta}","kind":"use","occurrences":[3623,3637]},{"id":"symbol-6f7058a34b152234","tex":"Q_\\mu","aliases":"Q_μ mu","display":"Q_\\mu","kind":"use","occurrences":[2130,2131]},{"id":"symbol-9e65ea2a28eb600e","tex":"Q_\\tau","aliases":"Q_τ tau","display":"Q_\\tau","kind":"use","occurrences":[3623,3625,3627,3636,3637]},{"id":"symbol-76f99c7b5823d720","tex":"Q_\\tau^2","aliases":"Q_τ^2 tau","display":"Q_\\tau^2","kind":"use","occurrences":[3621,3623,3627,3637]},{"id":"symbol-c341e0f3ba60fc73","tex":"Q_{\\tau\\theta}","aliases":"Q_τθ tau theta","display":"Q_{\\tau\\theta}","kind":"use","occurrences":[3636]},{"id":"symbol-1fc923810ce585fe","tex":"Q_{\\tau\\tau}","aliases":"Q_ττ tau tau","display":"Q_{\\tau\\tau}","kind":"use","occurrences":[3623,3637]},{"id":"symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","occurrences":[525,635,828,874,944,948,1011,1282,1289,1292,1296,2396,2686,2717,2722,3619]},{"id":"symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","occurrences":[1516,1560,1587,1735,3219,3420,3441]},{"id":"symbol-8ac168ec1a0ac655","tex":"\\mathbf r","aliases":"r mathbf","display":"\\mathbf r","kind":"use","occurrences":[611,613,614,615,616,617,622,626,627,630,631,635]},{"id":"symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","occurrences":[198,201,202,3338,3340,3345,3352,4161]},{"id":"symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"use","occurrences":[1208,1210,1258]},{"id":"symbol-0c8820d81c53436b","tex":"R''","aliases":"R''","display":"R''","kind":"use","occurrences":[3852,3860,3894]},{"id":"symbol-cdb814bc7d9fc317","tex":"R'^2","aliases":"R'^2","display":"R'^2","kind":"use","occurrences":[3852,3860]},{"id":"symbol-37387465eb47206f","tex":"R'^3","aliases":"R'^3","display":"R'^3","kind":"use","occurrences":[3860]},{"id":"symbol-e672c832eb007691","tex":"R^{\\mathsf T}","aliases":"R^ T mathsf","display":"R^{\\mathsf T}","kind":"use","occurrences":[2053,2055]},{"id":"symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","occurrences":[2259]},{"id":"symbol-072e7fe97b551a6a","tex":"r^{-2}","aliases":"r^-2","display":"r^{-2}","kind":"use","occurrences":[4063]},{"id":"symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","occurrences":[3581,3582,3585,3604,3605,4048,4052,4058,4059,4063,4072,4075,4077,4078,4079,4080,4087,4090,4109]},{"id":"symbol-061c21e92840dde5","tex":"\\mathbb R^{2n}","aliases":"R^2n mathbb","display":"\\mathbb R^{2n}","kind":"use","occurrences":[3420]},{"id":"symbol-508222661b47a325","tex":"R^3","aliases":"R^3","display":"R^3","kind":"use","occurrences":[3894]},{"id":"symbol-15bd74f3637fba8d","tex":"\\mathbb R^3","aliases":"R^3 mathbb","display":"\\mathbb R^3","kind":"use","occurrences":[1560,1587]},{"id":"symbol-c6163472e649a229","tex":"\\mathbb R^4","aliases":"R^4 mathbb","display":"\\mathbb R^4","kind":"use","occurrences":[1735]},{"id":"symbol-37e1261ad0123242","tex":"R^6","aliases":"R^6","display":"R^6","kind":"use","occurrences":[3917]},{"id":"symbol-3a803a9bd1627024","tex":"\\mathbb R^6","aliases":"R^6 mathbb","display":"\\mathbb R^6","kind":"use","occurrences":[3441]},{"id":"symbol-d6d881fd83e7a2b0","tex":"R^7","aliases":"R^7","display":"R^7","kind":"use","occurrences":[3917]},{"id":"symbol-be22a5b4e4f47e1b","tex":"R^8","aliases":"R^8","display":"R^8","kind":"use","occurrences":[3917]},{"id":"symbol-89e5eada6032fa59","tex":"R^{IJ}","aliases":"R^IJ","display":"R^{IJ}","kind":"use","occurrences":[3272]},{"id":"symbol-92a95a3f59d5b018","tex":"R^{k-1}","aliases":"R^k-1","display":"R^{k-1}","kind":"use","occurrences":[1771,1772,1874]},{"id":"symbol-19ee12c051e5e943","tex":"R^{KL}","aliases":"R^KL","display":"R^{KL}","kind":"use","occurrences":[3279]},{"id":"symbol-f1a40c4569184e2d","tex":"\\mathcal R^{KL}","aliases":"R^KL mathcal","display":"\\mathcal R^{KL}","kind":"use","occurrences":[3345]},{"id":"symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","occurrences":[3952]},{"id":"symbol-7f4928b93fd91121","tex":"r_+","aliases":"r_+","display":"r_+","kind":"use","occurrences":[3920,3925,3927,3928]},{"id":"symbol-7defeb26f73dee49","tex":"r_{+,h}","aliases":"r_+,h","display":"r_{+,h}","kind":"use","occurrences":[4040]},{"id":"symbol-2d4a4709819c91da","tex":"r_+^2","aliases":"r_+^2","display":"r_+^2","kind":"use","occurrences":[3927,3928]},{"id":"symbol-bcc60ff453e6985c","tex":"r_+^{\\rm exact}","aliases":"r_+^rm exact rm","display":"r_+^{\\rm exact}","kind":"use","occurrences":[4040]},{"id":"symbol-7335b8673a12eddc","tex":"r_-","aliases":"r_-","display":"r_-","kind":"use","occurrences":[3921]},{"id":"symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","occurrences":[2836,2840,3586,3589]},{"id":"symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","occurrences":[2836,2840]},{"id":"symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","occurrences":[1081,1201]},{"id":"symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","occurrences":[1202]},{"id":"symbol-5dbe454ca813d1d3","tex":"R_5","aliases":"R_5","display":"R_5","kind":"use","occurrences":[1203]},{"id":"symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","occurrences":[2913,2931,2935,2936]},{"id":"symbol-f79643ae14df8c41","tex":"R_{Aa}","aliases":"R_Aa","display":"R_{Aa}","kind":"use","occurrences":[2051]},{"id":"symbol-12288fef14aab090","tex":"R_{a\\mu}","aliases":"R_aμ mu","display":"R_{a\\mu}","kind":"use","occurrences":[2131,2132,2135]},{"id":"symbol-849a3daf5b3e8a7a","tex":"R_{b\\nu}","aliases":"R_bν nu","display":"R_{b\\nu}","kind":"use","occurrences":[2132]},{"id":"symbol-0f7be7718d9eb684","tex":"R_C","aliases":"R_C","display":"R_C","kind":"use","occurrences":[3489,3490,3492]},{"id":"symbol-71e0f18ed7179e19","tex":"R_d","aliases":"R_d","display":"R_d","kind":"use","occurrences":[209,212,213,214]},{"id":"symbol-bfb439ae05415aec","tex":"R_d^*","aliases":"R_d^*","display":"R_d^*","kind":"use","occurrences":[213]},{"id":"symbol-55c658da2911c141","tex":"R_e","aliases":"R_e","display":"R_e","kind":"use","occurrences":[212]},{"id":"symbol-c9377ce9ef801522","tex":"R_F","aliases":"R_F","display":"R_F","kind":"use","occurrences":[3482]},{"id":"symbol-3a7065b026fde28d","tex":"r_H","aliases":"r_H","display":"r_H","kind":"use","occurrences":[4140,4149,4151,4152]},{"id":"symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","occurrences":[2165,2167,2168,3587,3590,3593,3595,4091,4092]},{"id":"symbol-d0a12b6c6c489d99","tex":"r_i^2","aliases":"r_i^2","display":"r_i^2","kind":"use","occurrences":[4092]},{"id":"symbol-229e4596f8cda462","tex":"R_{i\\eta}","aliases":"R_iη eta","display":"R_{i\\eta}","kind":"use","occurrences":[2481]},{"id":"symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","occurrences":[1865,2167,2168]},{"id":"symbol-1e373259488dec7c","tex":"\\mathfrak R_{L,j}","aliases":"R_L,j mathfrak","display":"\\mathfrak R_{L,j}","kind":"use","occurrences":[1208,1210,1258]},{"id":"symbol-a951830e0ee3b4ef","tex":"\\mathfrak R_{L,k}","aliases":"R_L,k mathfrak","display":"\\mathfrak R_{L,k}","kind":"use","occurrences":[1210]},{"id":"symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","occurrences":[1118,1128,1133,1135]},{"id":"symbol-091fb9d0a9551219","tex":"r_{\\max}","aliases":"r_max max","display":"r_{\\max}","kind":"use","occurrences":[4117,4128,4130]},{"id":"symbol-a75d4101f5ae9880","tex":"R_{mn}","aliases":"R_mn","display":"R_{mn}","kind":"use","occurrences":[1118,1126,1128]},{"id":"symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","occurrences":[1034,1101,1118,1128,1135,1152]},{"id":"symbol-fd828431f1a05a6f","tex":"r_o","aliases":"r_o","display":"r_o","kind":"use","occurrences":[4091,4092]},{"id":"symbol-5a71c171d2d1861c","tex":"r_o^2","aliases":"r_o^2","display":"r_o^2","kind":"use","occurrences":[4092]},{"id":"symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"use","occurrences":[611]},{"id":"symbol-18f41d2d86df50af","tex":"R_p'","aliases":"R_p'","display":"R_p'","kind":"use","occurrences":[1190,1192]},{"id":"symbol-473eb620029bf88c","tex":"\\mathcal R_{p,0}","aliases":"R_p,0 mathcal","display":"\\mathcal R_{p,0}","kind":"use","occurrences":[201]},{"id":"symbol-646f864dabf7bf1e","tex":"\\mathcal R_{p,h}","aliases":"R_p,h mathcal","display":"\\mathcal R_{p,h}","kind":"use","occurrences":[198,202]},{"id":"symbol-1a4a0bc6234fa1f9","tex":"\\mathcal R_{p,h-1}","aliases":"R_p,h-1 mathcal","display":"\\mathcal R_{p,h-1}","kind":"use","occurrences":[202]},{"id":"symbol-39702065bb040b60","tex":"r_{\\rm exc}","aliases":"r_rm exc rm","display":"r_{\\rm exc}","kind":"use","occurrences":[4116,4128,4129]},{"id":"symbol-59eb0e6ce7c7f455","tex":"R_{\\rm turn}","aliases":"R_rm turn rm","display":"R_{\\rm turn}","kind":"use","occurrences":[3912,3922]},{"id":"symbol-257791562a17d424","tex":"R_v","aliases":"R_v","display":"R_v","kind":"use","occurrences":[731,732]},{"id":"symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","occurrences":[2813,2815]},{"id":"symbol-f0e893f7b0a8cb70","tex":"r_x","aliases":"r_x","display":"r_x","kind":"use","occurrences":[1842,1846,1858]},{"id":"symbol-84d9e04e97e949b2","tex":"R_x^{\\mathsf T}","aliases":"R_x^ T mathsf","display":"R_x^{\\mathsf T}","kind":"use","occurrences":[2815]},{"id":"symbol-06cb8b88ad7472f6","tex":"r_x^2","aliases":"r_x^2","display":"r_x^2","kind":"use","occurrences":[1856,1858]},{"id":"symbol-5d6e45e73e872e8e","tex":"r_y","aliases":"r_y","display":"r_y","kind":"use","occurrences":[1842,1846,1858]},{"id":"symbol-607cd9691e7de5b0","tex":"r_y^2","aliases":"r_y^2","display":"r_y^2","kind":"use","occurrences":[1856,1858]},{"id":"symbol-e4805d2243d9f7ec","tex":"R_z","aliases":"R_z","display":"R_z","kind":"use","occurrences":[2472,2475,2478,2479,2480,2481]},{"id":"symbol-22b0ec1ebb0d2020","tex":"r_z","aliases":"r_z","display":"r_z","kind":"use","occurrences":[1842,1846]},{"id":"symbol-98b9fa48a3b1a5b4","tex":"r_z^2","aliases":"r_z^2","display":"r_z^2","kind":"use","occurrences":[1856]},{"id":"symbol-8c6f7b1421c8035e","tex":"r_\\pm","aliases":"r_± pm","display":"r_\\pm","kind":"use","occurrences":[3903]},{"id":"symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran","display":null,"kind":"use","occurrences":[147,150,167,230,231,233,250,258,285,294,964,965,1299,1992,1994]},{"id":"symbol-040c923c57f7a5f9","tex":"\\operatorname{Ran}","aliases":"Ran operatorname","display":"\\operatorname{Ran}","kind":"use","occurrences":[461,470,486,564,615]},{"id":"symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","occurrences":[100,136]},{"id":"symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","occurrences":[824,1342,2016,2022,2033,2048,2237,2238,2243,2246,2280,2389,2626,2686,2692,2707,2730,2732,2738,2754]},{"id":"symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","occurrences":[818,1015,1063,1064,1083,1324,2251,2353,2511,2974,3006,3063,3064,3067,3082,3102,3582,3605,3609,3613,3723,3724,3733,3734,3735,3737,3740,3742,3748,3749,3759,3773,3775,3777,3781,3912,3922,4040,4042,4044,4116,4128,4129]},{"id":"symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","occurrences":[1003]},{"id":"symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","occurrences":[2625]},{"id":"symbol-78e04f6069787e74","tex":"\\rVert^2","aliases":"rVert^2 rVert","display":"\\rVert^2","kind":"use","occurrences":[2625]},{"id":"symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","occurrences":[870,872,873,876,879,882,890,904,905,906,914,916,917,1062,1063,1070,1095,2726]},{"id":"symbol-fe39e36813ced2eb","tex":"S^{\\mathsf T}","aliases":"S^ T mathsf","display":"S^{\\mathsf T}","kind":"use","occurrences":[2055]},{"id":"symbol-d7305573ca2e697d","tex":"S^*","aliases":"S^*","display":"S^*","kind":"use","occurrences":[736,739,828]},{"id":"symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","occurrences":[737]},{"id":"symbol-bc919570ec8558a8","tex":"S^a","aliases":"S^a","display":"S^a","kind":"use","occurrences":[3243,3245,3248,3366]},{"id":"symbol-dcc980c36e2b8074","tex":"S^\\ell","aliases":"S^ℓ ell","display":"S^\\ell","kind":"use","occurrences":[861]},{"id":"symbol-c7fcb94025e0dcdc","tex":"S_{\\mathrm D}","aliases":"S_ D mathrm","display":"S_{\\mathrm D}","kind":"use","occurrences":[3278,3279]},{"id":"symbol-d510ede8513110af","tex":"S_{\\mathrm G}","aliases":"S_ G mathrm","display":"S_{\\mathrm G}","kind":"use","occurrences":[3278,3279]},{"id":"symbol-b14c6b717dca7bbf","tex":"S_{\\mathrm M}^{(1)}","aliases":"S_ M^(1) mathrm","display":"S_{\\mathrm M}^{(1)}","kind":"use","occurrences":[3278,3279,3294]},{"id":"symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","occurrences":[3278,3319,3321,3324,3326]},{"id":"symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","occurrences":[3179,3183,3185,3186,3190,3195,3196,3197,3198]},{"id":"symbol-7cad08312c1a5a2b","tex":"s_{0,C}","aliases":"s_0,C","display":"s_{0,C}","kind":"use","occurrences":[3182,3183,3188,3197,3205,3216]},{"id":"symbol-1782caccc889225f","tex":"s_0^2","aliases":"s_0^2","display":"s_0^2","kind":"use","occurrences":[3180]},{"id":"symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","occurrences":[3972,3977]},{"id":"symbol-92f500208fc7568a","tex":"S_{A,1,-1}","aliases":"S_A,1,-1","display":"S_{A,1,-1}","kind":"use","occurrences":[3015]},{"id":"symbol-241b8b95f6be5201","tex":"S_{A,1,1}","aliases":"S_A,1,1","display":"S_{A,1,1}","kind":"use","occurrences":[3015]},{"id":"symbol-860b396d57115c52","tex":"S_{A,\\sigma,(a,b)}^2","aliases":"S_A,σ,(a,b)^2 sigma","display":"S_{A,\\sigma,(a,b)}^2","kind":"use","occurrences":[3006]},{"id":"symbol-da78c73c79172728","tex":"S_{A,\\sigma,v}","aliases":"S_A,σ,v sigma","display":"S_{A,\\sigma,v}","kind":"use","occurrences":[2942]},{"id":"symbol-1832780214718152","tex":"S_{A,\\sigma,v}^{\\,2}","aliases":"S_A,σ,v^ 2 sigma","display":"S_{A,\\sigma,v}^{\\,2}","kind":"use","occurrences":[2920,2940]},{"id":"symbol-19bf8b5638e7fe88","tex":"S_{A,\\tau,u}^{-1}","aliases":"S_A,τ,u^-1 tau","display":"S_{A,\\tau,u}^{-1}","kind":"use","occurrences":[2666]},{"id":"symbol-f2070054f7f973f8","tex":"S_{A,\\tau,u^g}","aliases":"S_A,τ,u^g tau","display":"S_{A,\\tau,u^g}","kind":"use","occurrences":[2671]},{"id":"symbol-300f3cadafb9f8db","tex":"S_{A,\\tau,v}","aliases":"S_A,τ,v tau","display":"S_{A,\\tau,v}","kind":"use","occurrences":[3006,3024]},{"id":"symbol-5df9b8f58b7161eb","tex":"s_b","aliases":"s_b","display":"s_b","kind":"use","occurrences":[2241,2247]},{"id":"symbol-971eac196f4fcd05","tex":"S_{\\mathrm{BV}}","aliases":"S_BV mathrm","display":"S_{\\mathrm{BV}}","kind":"use","occurrences":[3319,3323]},{"id":"symbol-92045ce8b164570b","tex":"S_C","aliases":"S_C","display":"S_C","kind":"use","occurrences":[3489,3490,3501,3505,3506]},{"id":"symbol-a04e13a6c34c8c88","tex":"s_C","aliases":"s_C","display":"s_C","kind":"use","occurrences":[3178,3188,3190,3195,3205,3216]},{"id":"symbol-59f247a519a27877","tex":"S_{C,\\hbar}","aliases":"S_C,ℏ hbar","display":"S_{C,\\hbar}","kind":"use","occurrences":[3501]},{"id":"symbol-4d4661ecebad12fb","tex":"s_C^2","aliases":"s_C^2","display":"s_C^2","kind":"use","occurrences":[3178,3191,3202,3204]},{"id":"symbol-a1170f96f1f38668","tex":"S_G","aliases":"S_G","display":"S_G","kind":"use","occurrences":[3621]},{"id":"symbol-6e8fe4e26c95819b","tex":"S_h","aliases":"S_h","display":"S_h","kind":"use","occurrences":[3590,3592]},{"id":"symbol-64eeff7a0fb8b5f2","tex":"\\mathcal S_h","aliases":"S_h mathcal","display":"\\mathcal S_h","kind":"use","occurrences":[2726]},{"id":"symbol-612081e3dc461c18","tex":"S_h^{(2)}","aliases":"S_h^(2)","display":"S_h^{(2)}","kind":"use","occurrences":[3745]},{"id":"symbol-c6c3417bdce5077d","tex":"S_{i,\\mathrm{coarse}}","aliases":"S_i,coarse mathrm","display":"S_{i,\\mathrm{coarse}}","kind":"use","occurrences":[2998]},{"id":"symbol-d71adb22cfa23dea","tex":"S_{i,\\mathrm{fine}}^2","aliases":"S_i,fine^2 mathrm","display":"S_{i,\\mathrm{fine}}^2","kind":"use","occurrences":[2998]},{"id":"symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","occurrences":[3525,3526,3528,3536,3541]},{"id":"symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","occurrences":[2316,2317,2395,2398,2407,2410,2413,2416,2418,2420,2421,2422,2424,2428,2430,2434,2436,2437,2440,2443,2445,2448,2459,2462,3224,3230]},{"id":"symbol-74d9bae07e421353","tex":"S_N^*","aliases":"S_N^*","display":"S_N^*","kind":"use","occurrences":[2437]},{"id":"symbol-25afd0ab9b73955e","tex":"\\mathcal S_p","aliases":"S_p mathcal","display":"\\mathcal S_p","kind":"use","occurrences":[1062,1063,1070,1095]},{"id":"symbol-7994b21f9b979579","tex":"S_{p,h}","aliases":"S_p,h","display":"S_{p,h}","kind":"use","occurrences":[264,267,268]},{"id":"symbol-2ee7713fc89d10bf","tex":"S_{p,h+1}","aliases":"S_p,h+1","display":"S_{p,h+1}","kind":"use","occurrences":[267]},{"id":"symbol-f12cd5af3f8bce51","tex":"S_{p,h}^*","aliases":"S_p,h^*","display":"S_{p,h}^*","kind":"use","occurrences":[268]},{"id":"symbol-5b939a36dfaae506","tex":"s_q","aliases":"s_q","display":"s_q","kind":"use","occurrences":[1864,1865]},{"id":"symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"use","occurrences":[1845,1846,1856,1864,1865]},{"id":"symbol-c1c5554e3132b956","tex":"S_{\\mathrm{sign}}","aliases":"S_sign mathrm","display":"S_{\\mathrm{sign}}","kind":"use","occurrences":[663]},{"id":"symbol-b387b1687e20541b","tex":"S_u","aliases":"S_u","display":"S_u","kind":"use","occurrences":[3025]},{"id":"symbol-2f57d1ab5cc8f2ce","tex":"S_v","aliases":"S_v","display":"S_v","kind":"use","occurrences":[3025]},{"id":"symbol-50b3713c204305be","tex":"S_W","aliases":"S_W","display":"S_W","kind":"use","occurrences":[2686,2688]},{"id":"symbol-6a81d4e3e02ea338","tex":"\\mathrm{sign}","aliases":"sign mathrm","display":"\\mathrm{sign}","kind":"use","occurrences":[66,74,103,107,647,655,656,663]},{"id":"symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","occurrences":[1262,3563]},{"id":"symbol-e02ae8115a0a024c","tex":"\\mathrm{Site}","aliases":"Site mathrm","display":"\\mathrm{Site}","kind":"use","occurrences":[2987]},{"id":"symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","occurrences":[384]},{"id":"symbol-72d39f9d2160995b","tex":"\\mathrm{std}","aliases":"std mathrm","display":"\\mathrm{std}","kind":"use","occurrences":[197,202]},{"id":"symbol-f90138132573de84","tex":"\\mathfrak{su}","aliases":"su mathfrak","display":"\\mathfrak{su}","kind":"use","occurrences":[2844]},{"id":"symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","occurrences":[496,527,577,1902,2448,2453,2460,2461]},{"id":"symbol-0341617620bd99b1","tex":"\\sup_m","aliases":"sup_m sup","display":"\\sup_m","kind":"use","occurrences":[496,527]},{"id":"symbol-b7e2fcded366f165","tex":"\\sup_{m>M}","aliases":"sup_m>M sup","display":"\\sup_{m>M}","kind":"use","occurrences":[577]},{"id":"symbol-9ee1f451cebb509d","tex":"\\sup_{n>N}","aliases":"sup_n>N sup","display":"\\sup_{n>N}","kind":"use","occurrences":[2448,2453,2460,2461]},{"id":"symbol-8fdddc59d2c468c6","tex":"\\sup_{|t|\\le T}","aliases":"sup_|t|≤ T sup le","display":"\\sup_{|t|\\le T}","kind":"use","occurrences":[1902]},{"id":"symbol-c161d17e61a95e63","tex":"\\T","aliases":"T","display":null,"kind":"use","occurrences":[1414,1500,3618,3632]},{"id":"symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","occurrences":[1516,1518,1526,1527,1664,1671,1764,3219,3230]},{"id":"symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","occurrences":[219,221,262,3220,3221,3222,3223,3228,3236,3242,3250,3327]},{"id":"symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","occurrences":[880,1236,1984,1985,1986,2053,2055,2815,3669,3798]},{"id":"symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"use","occurrences":[1062,1064,1070,1083,2870]},{"id":"symbol-27360346a415cf2f","tex":"\\T^3","aliases":"T^3 T","display":null,"kind":"use","occurrences":[1414,1500]},{"id":"symbol-b1c3e8d8054aff08","tex":"\\mathbb T^3","aliases":"T^3 mathbb","display":"\\mathbb T^3","kind":"use","occurrences":[1516,1518,1526,1527,1664,1671,1764,3219,3230]},{"id":"symbol-78de04bec91c6824","tex":"T^g","aliases":"T^g","display":"T^g","kind":"use","occurrences":[2647]},{"id":"symbol-ceeef154f3220be7","tex":"T^I","aliases":"T^I","display":"T^I","kind":"use","occurrences":[3397]},{"id":"symbol-4ff31b43de1984cc","tex":"t_*","aliases":"t_*","display":"t_*","kind":"use","occurrences":[665]},{"id":"symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","occurrences":[531]},{"id":"symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","occurrences":[2594,2596]},{"id":"symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","occurrences":[2594,2596,2598]},{"id":"symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","occurrences":[2594,2596]},{"id":"symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"use","occurrences":[635,636]},{"id":"symbol-50b87ddd607eac6e","tex":"T_a^*","aliases":"T_a^*","display":"T_a^*","kind":"use","occurrences":[635]},{"id":"symbol-0b61846000086dfe","tex":"T_h","aliases":"T_h","display":"T_h","kind":"use","occurrences":[3742,3745]},{"id":"symbol-8428b354040d2207","tex":"t_H","aliases":"t_H","display":"t_H","kind":"use","occurrences":[4139,4149,4151,4154]},{"id":"symbol-5c6f45d72806f285","tex":"T_j^n","aliases":"T_j^n","display":"T_j^n","kind":"use","occurrences":[2634,2638,2644]},{"id":"symbol-d9a3064def648911","tex":"\\mathcal T_K","aliases":"T_K mathcal","display":"\\mathcal T_K","kind":"use","occurrences":[3220,3221,3222,3223,3236,3242,3250]},{"id":"symbol-27d2210af02484ac","tex":"\\mathcal T_{K+L}","aliases":"T_K+L mathcal","display":"\\mathcal T_{K+L}","kind":"use","occurrences":[3221]},{"id":"symbol-0874c28c34a14314","tex":"\\mathcal T_K^{(4)}","aliases":"T_K^(4) mathcal","display":"\\mathcal T_K^{(4)}","kind":"use","occurrences":[3327]},{"id":"symbol-6fc5e3fb71882a80","tex":"\\mathcal T_{K_j}","aliases":"T_K_j mathcal","display":"\\mathcal T_{K_j}","kind":"use","occurrences":[3228]},{"id":"symbol-fae727c1986a152d","tex":"\\mathcal T_L","aliases":"T_L mathcal","display":"\\mathcal T_L","kind":"use","occurrences":[3221]},{"id":"symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","occurrences":[1034,1035,1153,1169]},{"id":"symbol-807e619b0b36adc6","tex":"T_p","aliases":"T_p","display":"T_p","kind":"use","occurrences":[1071,1072,1107]},{"id":"symbol-1a4b381b4644d7ce","tex":"t_P","aliases":"t_P","display":"t_P","kind":"use","occurrences":[3868,3872]},{"id":"symbol-efd63624efc4bcf3","tex":"\\mathcal T_{p,h}","aliases":"T_p,h mathcal","display":"\\mathcal T_{p,h}","kind":"use","occurrences":[219,221,262]},{"id":"symbol-40affe5513361680","tex":"\\mathcal T_{p,h}^*","aliases":"T_p,h^* mathcal","display":"\\mathcal T_{p,h}^*","kind":"use","occurrences":[221,262]},{"id":"symbol-9ef06c1f67b393ad","tex":"t_P^2","aliases":"t_P^2","display":"t_P^2","kind":"use","occurrences":[3868]},{"id":"symbol-bcbfdf0d1b59ea6b","tex":"t_{\\rm Shapiro}","aliases":"t_rm Shapiro rm","display":"t_{\\rm Shapiro}","kind":"use","occurrences":[3613]},{"id":"symbol-e88803809ed376b6","tex":"T_{y\\leftarrow x}","aliases":"T_y← x leftarrow","display":"T_{y\\leftarrow x}","kind":"use","occurrences":[2610,2611,2871,2875,2876]},{"id":"symbol-d52dba8cafc2212f","tex":"T_{z\\leftarrow y}","aliases":"T_z← y leftarrow","display":"T_{z\\leftarrow y}","kind":"use","occurrences":[2876]},{"id":"symbol-a084b0e913e1579b","tex":"\\mathrm{tor}","aliases":"tor mathrm","display":"\\mathrm{tor}","kind":"use","occurrences":[602,603,604,606,610,611]},{"id":"symbol-8dd594b4bce51346","tex":"\\mathrm{tors}","aliases":"tors mathrm","display":"\\mathrm{tors}","kind":"use","occurrences":[66,74]},{"id":"symbol-89161acad9d67f8d","tex":"\\operatorname{Tr}_B","aliases":"Tr_B operatorname","display":"\\operatorname{Tr}_B","kind":"use","occurrences":[2136]},{"id":"symbol-7ded009c618f2421","tex":"\\operatorname{Tr}_{\\mathrm{sign}}","aliases":"Tr_sign operatorname mathrm","display":"\\operatorname{Tr}_{\\mathrm{sign}}","kind":"use","occurrences":[103]},{"id":"symbol-6e37ab0f79926296","tex":"\\mathrm U","aliases":"U mathrm","display":"\\mathrm U","kind":"use","occurrences":[3346,3347,3348,3349,3353,3405]},{"id":"symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","occurrences":[3585,3586]},{"id":"symbol-11b41534cdf61584","tex":"u''","aliases":"u''","display":"u''","kind":"use","occurrences":[3585]},{"id":"symbol-df5c58f22a65049f","tex":"u^{-1}","aliases":"u^-1","display":"u^{-1}","kind":"use","occurrences":[3026]},{"id":"symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","occurrences":[1712]},{"id":"symbol-12b5175db95756e4","tex":"U^g","aliases":"U^g","display":"U^g","kind":"use","occurrences":[2618,2619,2647,2688]},{"id":"symbol-da5e436b748b454f","tex":"u^p","aliases":"u^p","display":"u^p","kind":"use","occurrences":[23]},{"id":"symbol-7c05ecd0a14b319f","tex":"u^T","aliases":"u^T","display":"u^T","kind":"use","occurrences":[3984]},{"id":"symbol-7d05da6d886b7035","tex":"u_{(2\\pi,0,0)}","aliases":"u_(2π,0,0) pi","display":"u_{(2\\pi,0,0)}","kind":"use","occurrences":[1852,1866]},{"id":"symbol-5274c3aacd45a0cc","tex":"u_{(2\\pi,2\\pi,2\\pi)}","aliases":"u_(2π,2π,2π) pi pi pi","display":"u_{(2\\pi,2\\pi,2\\pi)}","kind":"use","occurrences":[1853,1866]},{"id":"symbol-4b0b57f70ac14529","tex":"u_{(6,0,0)}","aliases":"u_(6,0,0)","display":"u_{(6,0,0)}","kind":"use","occurrences":[1866]},{"id":"symbol-8b335f28186fa8e0","tex":"u_{(6,6,6)}","aliases":"u_(6,6,6)","display":"u_{(6,6,6)}","kind":"use","occurrences":[1866]},{"id":"symbol-df73ec13d71ee514","tex":"u_*","aliases":"u_*","display":"u_*","kind":"use","occurrences":[1017]},{"id":"symbol-e124bcd45f61b489","tex":"U_{-1}","aliases":"U_-1","display":"U_{-1}","kind":"use","occurrences":[1037,1099,1145]},{"id":"symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","occurrences":[1038]},{"id":"symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","occurrences":[924,1847,3591,3600]},{"id":"symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","occurrences":[1082]},{"id":"symbol-e94ce7ed00675a9a","tex":"U_{2,h}","aliases":"U_2,h","display":"U_{2,h}","kind":"use","occurrences":[7,14,66]},{"id":"symbol-432c14b23419ac6d","tex":"u_A","aliases":"u_A","display":"u_A","kind":"use","occurrences":[1848,1850,1851]},{"id":"symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","occurrences":[2196,2199,2200,2202,2205,2208]},{"id":"symbol-77eae673cef2716c","tex":"U_{AB}","aliases":"U_AB","display":"U_{AB}","kind":"use","occurrences":[3209]},{"id":"symbol-d776df64b15c1825","tex":"U_{a\\to b}","aliases":"U_a→ b to","display":"U_{a\\to b}","kind":"use","occurrences":[3572]},{"id":"symbol-05e45e2c62b065b7","tex":"U_{a\\to b}'","aliases":"U_a→ b' to","display":"U_{a\\to b}'","kind":"use","occurrences":[3572]},{"id":"symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","occurrences":[2199,2200,2202,2205,2208]},{"id":"symbol-1f20e1d7a367ca68","tex":"\\mathcal U_{\\mathrm{CRT}}","aliases":"U_CRT mathcal mathrm","display":"\\mathcal U_{\\mathrm{CRT}}","kind":"use","occurrences":[74,75,78]},{"id":"symbol-344059846d20b215","tex":"\\mathcal U_{\\mathrm{CRT}}^*","aliases":"U_CRT^* mathcal mathrm","display":"\\mathcal U_{\\mathrm{CRT}}^*","kind":"use","occurrences":[78]},{"id":"symbol-b7a598e95d01e562","tex":"u_D","aliases":"u_D","display":"u_D","kind":"use","occurrences":[1849,1850,1851]},{"id":"symbol-cb69d506a7bd35e5","tex":"U_e","aliases":"U_e","display":"U_e","kind":"use","occurrences":[3557]},{"id":"symbol-f72fa924cc22eb23","tex":"U_H","aliases":"U_H","display":"U_H","kind":"use","occurrences":[1603,1612,1615,1645,1646,1647,1651,1654,1657,1660,1934]},{"id":"symbol-f59a4109ae21c00d","tex":"U_{H(1,r)}","aliases":"U_H(1,r)","display":"U_{H(1,r)}","kind":"use","occurrences":[1846]},{"id":"symbol-6387f759e35b4e7d","tex":"U_{h,S}","aliases":"U_h,S","display":"U_{h,S}","kind":"use","occurrences":[1686,1930]},{"id":"symbol-f51e0c71f0f6f66a","tex":"U_{H_{h,n}}","aliases":"U_H_h,n","display":"U_{H_{h,n}}","kind":"use","occurrences":[1684,1687,1694,1696,1761,1766,1778,1899,1905]},{"id":"symbol-1e4c058a9b28e970","tex":"U_{H_n}","aliases":"U_H_n","display":"U_{H_n}","kind":"use","occurrences":[1684,1687,1761,1766,1778,1899,1905]},{"id":"symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","occurrences":[2996,2997,3051,3065,3070,3075,3590,3593,3594,3595]},{"id":"symbol-423beb526b282cb9","tex":"u_{i+1}","aliases":"u_i+1","display":"u_{i+1}","kind":"use","occurrences":[3590,3593,3595]},{"id":"symbol-37b4eb196aed5935","tex":"u_{i-1}","aliases":"u_i-1","display":"u_{i-1}","kind":"use","occurrences":[3593,3595]},{"id":"symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","occurrences":[2723,2725,2730,2731,2759,2802,2881,2882,2885,3675]},{"id":"symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","occurrences":[523,524,583,4118]},{"id":"symbol-d776fc575cc5531d","tex":"U_j'","aliases":"U_j'","display":"U_j'","kind":"use","occurrences":[2885]},{"id":"symbol-79c73d373618f65a","tex":"u_{j+1}","aliases":"u_j+1","display":"u_{j+1}","kind":"use","occurrences":[3934,4118]},{"id":"symbol-3ea98bddbefa3e22","tex":"u_{j+2}","aliases":"u_j+2","display":"u_{j+2}","kind":"use","occurrences":[3934,4118]},{"id":"symbol-2ef4170d2ceda737","tex":"U_{j-1}","aliases":"U_j-1","display":"U_{j-1}","kind":"use","occurrences":[2741]},{"id":"symbol-276fce3b16629317","tex":"u_{j-1}","aliases":"u_j-1","display":"u_{j-1}","kind":"use","occurrences":[3934,4118]},{"id":"symbol-e4ddf66f1da3ef2c","tex":"U_{j-1}^*","aliases":"U_j-1^*","display":"U_{j-1}^*","kind":"use","occurrences":[2730]},{"id":"symbol-ae1b72e4bc4c00c0","tex":"U_{j-1}^n","aliases":"U_j-1^n","display":"U_{j-1}^n","kind":"use","occurrences":[2638]},{"id":"symbol-969a2f5751eb71e7","tex":"u_{j-2}","aliases":"u_j-2","display":"u_{j-2}","kind":"use","occurrences":[3934,4118]},{"id":"symbol-b2a7e488064e7e3e","tex":"U_j^n","aliases":"U_j^n","display":"U_j^n","kind":"use","occurrences":[2633,2638,2644]},{"id":"symbol-c8e7ba9b6fa73cc6","tex":"U_L","aliases":"U_L","display":"U_L","kind":"use","occurrences":[3477,3478]},{"id":"symbol-a49e958a93a3404a","tex":"U_M","aliases":"U_M","display":"U_M","kind":"use","occurrences":[78]},{"id":"symbol-c05f0648abbf982d","tex":"u_M","aliases":"u_M","display":"u_M","kind":"use","occurrences":[3591]},{"id":"symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","occurrences":[564,566]},{"id":"symbol-65a04898bfc53b4d","tex":"U_{m-1}","aliases":"U_m-1","display":"U_{m-1}","kind":"use","occurrences":[1118,1131,1134]},{"id":"symbol-139443253ce41588","tex":"U_M^{\\mathrm{emb}}","aliases":"U_M^emb mathrm","display":"U_M^{\\mathrm{emb}}","kind":"use","occurrences":[542,547,552,558,563]},{"id":"symbol-15cdb597eea20cc8","tex":"U_{mn-1}","aliases":"U_mn-1","display":"U_{mn-1}","kind":"use","occurrences":[1118,1126,1131,1134]},{"id":"symbol-2b181e5223e87432","tex":"U_{n+1}","aliases":"U_n+1","display":"U_{n+1}","kind":"use","occurrences":[1039]},{"id":"symbol-8acdf74b89e87442","tex":"U_{n-2}","aliases":"U_n-2","display":"U_{n-2}","kind":"use","occurrences":[1097,1144,1153]},{"id":"symbol-d17a28f5e5468c9f","tex":"U_p","aliases":"U_p","display":"U_p","kind":"use","occurrences":[1072]},{"id":"symbol-279939f80579ae9e","tex":"U_{p,h}","aliases":"U_p,h","display":"U_{p,h}","kind":"use","occurrences":[5,13,53,66,274,276]},{"id":"symbol-9fa59f1be311c29b","tex":"U_{p,h+1,N}","aliases":"U_p,h+1,N","display":"U_{p,h+1,N}","kind":"use","occurrences":[770,772]},{"id":"symbol-600ccddcc17ae38b","tex":"U_{p,h,N}","aliases":"U_p,h,N","display":"U_{p,h,N}","kind":"use","occurrences":[738,739,771,772]},{"id":"symbol-c62e6bd466c701a3","tex":"U_{p,h,N}^*","aliases":"U_p,h,N^*","display":"U_{p,h,N}^*","kind":"use","occurrences":[739]},{"id":"symbol-15c0f31655cd03d4","tex":"\\mathcal U_{p,h_p}","aliases":"U_p,h_p mathcal","display":"\\mathcal U_{p,h_p}","kind":"use","occurrences":[604]},{"id":"symbol-3e789f6b6ba49589","tex":"\\mathcal U_{p,M}","aliases":"U_p,M mathcal","display":"\\mathcal U_{p,M}","kind":"use","occurrences":[444,452]},{"id":"symbol-4a0208673d51f84e","tex":"\\mathcal U_{p,m}","aliases":"U_p,m mathcal","display":"\\mathcal U_{p,m}","kind":"use","occurrences":[384,385,387,391,394]},{"id":"symbol-6001a276c34996c2","tex":"\\mathcal U_{p,m+1}","aliases":"U_p,m+1 mathcal","display":"\\mathcal U_{p,m+1}","kind":"use","occurrences":[394]},{"id":"symbol-63cfb5dda4ee754e","tex":"\\mathcal U_{p,M}^*","aliases":"U_p,M^* mathcal","display":"\\mathcal U_{p,M}^*","kind":"use","occurrences":[444]},{"id":"symbol-e1387ffa6db2b83b","tex":"U_{p-1}","aliases":"U_p-1","display":"U_{p-1}","kind":"use","occurrences":[1061,1062,1064,1067,1068,1069,1070,1071,1084,1095]},{"id":"symbol-d340f681877ced98","tex":"U_{p-2}","aliases":"U_p-2","display":"U_{p-2}","kind":"use","occurrences":[1068,1070,1071,1072]},{"id":"symbol-0835809a6850acb4","tex":"u_q","aliases":"u_q","display":"u_q","kind":"use","occurrences":[1865]},{"id":"symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","occurrences":[1846,1856,1865]},{"id":"symbol-2c0c595b21c3bb2f","tex":"U_{rs}","aliases":"U_rs","display":"U_{rs}","kind":"use","occurrences":[812]},{"id":"symbol-ec1736aa4eb4331a","tex":"U_{rs}^*","aliases":"U_rs^*","display":"U_{rs}^*","kind":"use","occurrences":[812]},{"id":"symbol-27d332518a7e4b64","tex":"U_{rt}","aliases":"U_rt","display":"U_{rt}","kind":"use","occurrences":[812]},{"id":"symbol-0b4d1c17b7c34773","tex":"U_S","aliases":"U_S","display":"U_S","kind":"use","occurrences":[1686,1930]},{"id":"symbol-737d443707dfaa85","tex":"U_{ts}^*","aliases":"U_ts^*","display":"U_{ts}^*","kind":"use","occurrences":[812]},{"id":"symbol-ad63d401332166ba","tex":"u_x","aliases":"u_x","display":"u_x","kind":"use","occurrences":[3048]},{"id":"symbol-72ae5158c8aa8c59","tex":"U_{x+\\widehat\\mu,\\nu}","aliases":"U_x+widehatμ,ν widehat mu nu","display":"U_{x+\\widehat\\mu,\\nu}","kind":"use","occurrences":[2678]},{"id":"symbol-66ca98fe046c552c","tex":"U_{x+\\widehat\\nu,\\mu}^{-1}","aliases":"U_x+widehatν,μ^-1 widehat nu mu","display":"U_{x+\\widehat\\nu,\\mu}^{-1}","kind":"use","occurrences":[2678]},{"id":"symbol-2ca8103aa551294d","tex":"U_{x,\\mu}","aliases":"U_x,μ mu","display":"U_{x,\\mu}","kind":"use","occurrences":[2606,2609,2617,2626,2678,2804,2806]},{"id":"symbol-3127b0bda4adc1a6","tex":"u_{x,\\mu}","aliases":"u_x,μ mu","display":"u_{x,\\mu}","kind":"use","occurrences":[2612]},{"id":"symbol-9681ef253d45ed2b","tex":"U_{x,\\mu}^*","aliases":"U_x,μ^* mu","display":"U_{x,\\mu}^*","kind":"use","occurrences":[2606]},{"id":"symbol-ed02f96136bb7a27","tex":"U_{x,\\mu}^{-1}","aliases":"U_x,μ^-1 mu","display":"U_{x,\\mu}^{-1}","kind":"use","occurrences":[2612]},{"id":"symbol-e53279255a429197","tex":"U_{x,\\mu}^g","aliases":"U_x,μ^g mu","display":"U_{x,\\mu}^g","kind":"use","occurrences":[2609]},{"id":"symbol-776573e0476b4656","tex":"u_{x,\\mu}^g","aliases":"u_x,μ^g mu","display":"u_{x,\\mu}^g","kind":"use","occurrences":[2612]},{"id":"symbol-643d4c9f3c093afb","tex":"U_{x,\\nu}^{-1}","aliases":"U_x,ν^-1 nu","display":"U_{x,\\nu}^{-1}","kind":"use","occurrences":[2678]},{"id":"symbol-1174cbd293ed0592","tex":"U_{x-\\widehat\\mu,\\mu}^{-1}","aliases":"U_x-widehatμ,μ^-1 widehat mu mu","display":"U_{x-\\widehat\\mu,\\mu}^{-1}","kind":"use","occurrences":[2617,2622]},{"id":"symbol-7dded304aadbda87","tex":"U_{x-\\widehat\\mu,\\mu}^g","aliases":"U_x-widehatμ,μ^g widehat mu mu","display":"U_{x-\\widehat\\mu,\\mu}^g","kind":"use","occurrences":[2622]},{"id":"symbol-c6df7a94e294a06b","tex":"u_y","aliases":"u_y","display":"u_y","kind":"use","occurrences":[3048]},{"id":"symbol-07a46937ac588412","tex":"u_z","aliases":"u_z","display":"u_z","kind":"use","occurrences":[3048]},{"id":"symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","occurrences":[384,386,412,423,424,432,434,441,444,445,446,491,540,543,544,637,640,641,643,645,646,3237,3239]},{"id":"symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","occurrences":[4086,4095,4122]},{"id":"symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","occurrences":[226,229,231,234,235,239,240,242,667,777,778,2258,2259,2264,2270,2271,2274,2278]},{"id":"symbol-cafb796619c588dd","tex":"v^{-1}","aliases":"v^-1","display":"v^{-1}","kind":"use","occurrences":[3026,3137]},{"id":"symbol-d4bbf0f17c40b804","tex":"v^{-1/2}","aliases":"v^-1/2","display":"v^{-1/2}","kind":"use","occurrences":[3136]},{"id":"symbol-759397de55065787","tex":"V^2","aliases":"V^2","display":"V^2","kind":"use","occurrences":[3674]},{"id":"symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","occurrences":[4051,4074,4078,4088]},{"id":"symbol-baf0eae4627f5125","tex":"V^\\ell","aliases":"V^ℓ ell","display":"V^\\ell","kind":"use","occurrences":[1324]},{"id":"symbol-1edd83621988d298","tex":"\\mathcal V_0","aliases":"V_0 mathcal","display":"\\mathcal V_0","kind":"use","occurrences":[637,645]},{"id":"symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","occurrences":[750,924,925,926,936,1245,1544,3597]},{"id":"symbol-7f891d9184f7892b","tex":"\\mathcal V_1","aliases":"V_1 mathcal","display":"\\mathcal V_1","kind":"use","occurrences":[637]},{"id":"symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","occurrences":[1544]},{"id":"symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","occurrences":[54,1544]},{"id":"symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","occurrences":[1544]},{"id":"symbol-a478c8ae160947ff","tex":"V_a","aliases":"V_a","display":"V_a","kind":"use","occurrences":[2254,2255,2257,2264,2266,2269]},{"id":"symbol-a23b8f8fe4c89a4d","tex":"\\mathcal V_a","aliases":"V_a mathcal","display":"\\mathcal V_a","kind":"use","occurrences":[3237,3239]},{"id":"symbol-56405f2e425e038c","tex":"V_a^*","aliases":"V_a^*","display":"V_a^*","kind":"use","occurrences":[2255,2258,2264,2268]},{"id":"symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","occurrences":[2268]},{"id":"symbol-3c64cfa0416300f6","tex":"V_b^*","aliases":"V_b^*","display":"V_b^*","kind":"use","occurrences":[2269]},{"id":"symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","occurrences":[3727,3732]},{"id":"symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","occurrences":[2994,2997,3051,3065,3598]},{"id":"symbol-283debf6bf3b04eb","tex":"v_{i+1}","aliases":"v_i+1","display":"v_{i+1}","kind":"use","occurrences":[3598]},{"id":"symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","occurrences":[3675]},{"id":"symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","occurrences":[3517,3521,3522,3525]},{"id":"symbol-7b6a271c4eb9d3c3","tex":"v_{j-1}","aliases":"v_j-1","display":"v_{j-1}","kind":"use","occurrences":[1238]},{"id":"symbol-9381c9616ec2e3a3","tex":"\\mathcal V_L","aliases":"V_L mathcal","display":"\\mathcal V_L","kind":"use","occurrences":[491]},{"id":"symbol-05a897b8ae55039b","tex":"v_{L-1}","aliases":"v_L-1","display":"v_{L-1}","kind":"use","occurrences":[1237]},{"id":"symbol-f960dcf025809c6a","tex":"v_{Lq+j}","aliases":"v_Lq+j","display":"v_{Lq+j}","kind":"use","occurrences":[1245]},{"id":"symbol-db3b5f8ecfc5c837","tex":"\\mathcal V_M","aliases":"V_M mathcal","display":"\\mathcal V_M","kind":"use","occurrences":[434,444,445,540,543,640,643,646]},{"id":"symbol-eb7cc70f89201b8d","tex":"\\mathcal V_m","aliases":"V_m mathcal","display":"\\mathcal V_m","kind":"use","occurrences":[384,386,412,423,424,432,441,446]},{"id":"symbol-20cd50e2bdb5b105","tex":"v_M","aliases":"v_M","display":"v_M","kind":"use","occurrences":[3597]},{"id":"symbol-c5e6ae3f2110675e","tex":"\\mathcal V_{M+1}","aliases":"V_M+1 mathcal","display":"\\mathcal V_{M+1}","kind":"use","occurrences":[641]},{"id":"symbol-24a238a771b3be94","tex":"\\mathcal V_M^\\perp","aliases":"V_M^⊥ mathcal perp","display":"\\mathcal V_M^\\perp","kind":"use","occurrences":[544,641]},{"id":"symbol-ac8b12063b2b1211","tex":"V_p","aliases":"V_p","display":"V_p","kind":"use","occurrences":[609]},{"id":"symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","occurrences":[16,277]},{"id":"symbol-ecc628efee7ceee2","tex":"V_{p,h}","aliases":"V_p,h","display":"V_{p,h}","kind":"use","occurrences":[184,187,189,190,192,260,267]},{"id":"symbol-8da5c99bd0752f70","tex":"V_{p,h+w-1}","aliases":"V_p,h+w-1","display":"V_{p,h+w-1}","kind":"use","occurrences":[192]},{"id":"symbol-1ce4eff1892c6bcc","tex":"V_{p,h-1}^*","aliases":"V_p,h-1^*","display":"V_{p,h-1}^*","kind":"use","occurrences":[218]},{"id":"symbol-f8bbef892e55e829","tex":"V_{p,h}^*","aliases":"V_p,h^*","display":"V_{p,h}^*","kind":"use","occurrences":[188,189]},{"id":"symbol-ada632c6fceb6663","tex":"V_{p,m}","aliases":"V_p,m","display":"V_{p,m}","kind":"use","occurrences":[392,394,439,443]},{"id":"symbol-310efc759ac5294c","tex":"V_{p,M-1}","aliases":"V_p,M-1","display":"V_{p,M-1}","kind":"use","occurrences":[443]},{"id":"symbol-f0b72f4f5ef32b81","tex":"V_{p,m\\to M}","aliases":"V_p,m→ M to","display":"V_{p,m\\to M}","kind":"use","occurrences":[443,444]},{"id":"symbol-e96f90baddd90c65","tex":"V_{p,m\\to M}^*","aliases":"V_p,m→ M^* to","display":"V_{p,m\\to M}^*","kind":"use","occurrences":[444]},{"id":"symbol-b70c2d293c03377d","tex":"V_q","aliases":"V_q","display":"V_q","kind":"use","occurrences":[609]},{"id":"symbol-724afc19e77e7bfd","tex":"v_q","aliases":"v_q","display":"v_q","kind":"use","occurrences":[852,854,856]},{"id":"symbol-2e67d412be824541","tex":"v_R","aliases":"v_R","display":"v_R","kind":"use","occurrences":[3890]},{"id":"symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","occurrences":[1525,1786,1788,2576]},{"id":"symbol-c580baf0833ca721","tex":"v_x","aliases":"v_x","display":"v_x","kind":"use","occurrences":[3047]},{"id":"symbol-5b59d34e2b2271f4","tex":"v_y","aliases":"v_y","display":"v_y","kind":"use","occurrences":[3047]},{"id":"symbol-f88941ebb3c7a6bd","tex":"v_z","aliases":"v_z","display":"v_z","kind":"use","occurrences":[3047]},{"id":"symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"use","occurrences":[2200]},{"id":"symbol-e355f29d98c97c06","tex":"\\operatorname{vol}","aliases":"vol operatorname","display":"\\operatorname{vol}","kind":"use","occurrences":[3272,3279,3286,3287,3294,3401]},{"id":"symbol-c6a2a0e38dc26bdf","tex":"\\operatorname{vol}_e","aliases":"vol_e operatorname","display":"\\operatorname{vol}_e","kind":"use","occurrences":[3272,3279,3286,3287,3294,3401]},{"id":"symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","occurrences":[215,218,891,892,2646,2647]},{"id":"symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","occurrences":[4096]},{"id":"symbol-5638fa9691fa3e47","tex":"W^{(2)}","aliases":"W^(2)","display":"W^{(2)}","kind":"use","occurrences":[2836]},{"id":"symbol-f6df49aeb91e7b72","tex":"\\mathcal W^*","aliases":"W^* mathcal","display":"\\mathcal W^*","kind":"use","occurrences":[892]},{"id":"symbol-11a0326400d8d7a5","tex":"W^{1,\\infty}","aliases":"W^1,∞ infty","display":"W^{1,\\infty}","kind":"use","occurrences":[3960,3989]},{"id":"symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","occurrences":[2389]},{"id":"symbol-b2aee1add211ab2b","tex":"w_+","aliases":"w_+","display":"w_+","kind":"use","occurrences":[4093,4095,4130]},{"id":"symbol-7bb383dea36e9371","tex":"w_+'","aliases":"w_+'","display":"w_+'","kind":"use","occurrences":[4095]},{"id":"symbol-5d0598ac5861dddc","tex":"w_+^2","aliases":"w_+^2","display":"w_+^2","kind":"use","occurrences":[4101,4103]},{"id":"symbol-9182875dbc779918","tex":"w_-","aliases":"w_-","display":"w_-","kind":"use","occurrences":[4094,4095,4129]},{"id":"symbol-c0eb6b90de5e504c","tex":"w_-'","aliases":"w_-'","display":"w_-'","kind":"use","occurrences":[4095]},{"id":"symbol-e48a71f06564da0e","tex":"w_-^2","aliases":"w_-^2","display":"w_-^2","kind":"use","occurrences":[4101,4103]},{"id":"symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","occurrences":[2849]},{"id":"symbol-59e63012bc6d4f87","tex":"w_0","aliases":"w_0","display":"w_0","kind":"use","occurrences":[1155]},{"id":"symbol-84ca84b36e79ec9b","tex":"W_f","aliases":"W_f","display":"W_f","kind":"use","occurrences":[2694]},{"id":"symbol-057eae80a542fec9","tex":"W_f^{\\operatorname{or}(f,c)}","aliases":"W_f^or(f,c) operatorname","display":"W_f^{\\operatorname{or}(f,c)}","kind":"use","occurrences":[2693]},{"id":"symbol-db94573662ebfd05","tex":"W_h","aliases":"W_h","display":"W_h","kind":"use","occurrences":[2845,2851,2858,2860]},{"id":"symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","occurrences":[3517,3521,3522,3523,3525]},{"id":"symbol-ebdf9f205f201a52","tex":"w_k","aliases":"w_k","display":"w_k","kind":"use","occurrences":[1154]},{"id":"symbol-1d116e67893836b3","tex":"w_{k+1}","aliases":"w_k+1","display":"w_{k+1}","kind":"use","occurrences":[1154]},{"id":"symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","occurrences":[1369]},{"id":"symbol-9b94508bc7508bf1","tex":"\\mathcal W_n","aliases":"W_n mathcal","display":"\\mathcal W_n","kind":"use","occurrences":[2646,2647]},{"id":"symbol-850addc7bcb89021","tex":"w_{nq+j}","aliases":"w_nq+j","display":"w_{nq+j}","kind":"use","occurrences":[1155]},{"id":"symbol-05ee5dd14eda9a8e","tex":"\\mathcal W_{p,h}","aliases":"W_p,h mathcal","display":"\\mathcal W_{p,h}","kind":"use","occurrences":[215,218]},{"id":"symbol-42b5720616e472f2","tex":"\\mathcal W_{p,h}^*","aliases":"W_p,h^* mathcal","display":"\\mathcal W_{p,h}^*","kind":"use","occurrences":[218]},{"id":"symbol-dfa15e280d3611a8","tex":"w_r","aliases":"w_r","display":"w_r","kind":"use","occurrences":[821,823,826,1525]},{"id":"symbol-75d29d71343727f9","tex":"W_\\square","aliases":"W_square square","display":"W_\\square","kind":"use","occurrences":[3558,3569,3578]},{"id":"symbol-28b939ffc4f05e71","tex":"W_u","aliases":"W_u","display":"W_u","kind":"use","occurrences":[3052,3063,3066,3067,3068]},{"id":"symbol-377769d6a9c94ee0","tex":"W_v","aliases":"W_v","display":"W_v","kind":"use","occurrences":[3052,3063,3067]},{"id":"symbol-a1354be5a54b8730","tex":"W_{x,\\mu\\nu}","aliases":"W_x,μν mu nu","display":"W_{x,\\mu\\nu}","kind":"use","occurrences":[2678,2682,2686]},{"id":"symbol-782a1381512fbead","tex":"W_{x,\\mu\\nu}^g","aliases":"W_x,μν^g mu nu","display":"W_{x,\\mu\\nu}^g","kind":"use","occurrences":[2682]},{"id":"symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","occurrences":[3272,3279,3286,3287,3290,3291,3336,3338,3340,3341,3342,3345,3374,3380,3397,3555,3556,3561]},{"id":"symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","occurrences":[3985]},{"id":"symbol-35728400f785376e","tex":"\\widehat D_4","aliases":"widehat D_4 widehat","display":"\\widehat D_4","kind":"use","occurrences":[3985]},{"id":"symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","occurrences":[1501,1502,3220,3682,3683,3699,3701]},{"id":"symbol-9f20b388cdc4505b","tex":"\\widehat f_h","aliases":"widehat f_h widehat","display":"\\widehat f_h","kind":"use","occurrences":[3682,3683,3699,3701]},{"id":"symbol-4aae4b9331147cb6","tex":"\\widehat f_k","aliases":"widehat f_k widehat","display":"\\widehat f_k","kind":"use","occurrences":[3220]},{"id":"symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"use","occurrences":[1426,1429,1432,1433,1434,1435,1436,1437,1449,1450,1452,1458,1460,1462,1465,1466,1470,1474,1475,1490,1491,1492,1496,1498,1508]},{"id":"symbol-49b14a998ffa2541","tex":"\\widehat H_m","aliases":"widehat H_m widehat","display":"\\widehat H_m","kind":"use","occurrences":[1426,1429,1432,1433,1434,1437,1449,1450,1452,1458,1460,1462,1465,1466,1470,1474,1490,1491,1492,1496,1498,1508]},{"id":"symbol-df544400edfb9943","tex":"\\widehat H_m^*","aliases":"widehat H_m^* widehat","display":"\\widehat H_m^*","kind":"use","occurrences":[1433,1435,1436,1450]},{"id":"symbol-3546643ac017f642","tex":"\\widehat H_{m_1}","aliases":"widehat H_m_1 widehat","display":"\\widehat H_{m_1}","kind":"use","occurrences":[1475]},{"id":"symbol-937384cfab5dedd5","tex":"\\widehat H_{m_2}","aliases":"widehat H_m_2 widehat","display":"\\widehat H_{m_2}","kind":"use","occurrences":[1475]},{"id":"symbol-4a6775720ae1e169","tex":"\\widehat N","aliases":"widehat N widehat","display":"\\widehat N","kind":"use","occurrences":[1322,1323,1354]},{"id":"symbol-5ec76de4222f5aa9","tex":"\\widehat U","aliases":"widehat U widehat","display":"\\widehat U","kind":"use","occurrences":[2799,2800]},{"id":"symbol-bf98495fc78739ad","tex":"\\widehat U_h","aliases":"widehat U_h widehat","display":"\\widehat U_h","kind":"use","occurrences":[2799,2800]},{"id":"symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","occurrences":[2603,2606,2609,2612,2617,2620,2622,2626,2678,2806,2809]},{"id":"symbol-81743d091025fb84","tex":"\\widehat\\nu","aliases":"widehatν widehat nu","display":"\\widehat\\nu","kind":"use","occurrences":[2678,2680]},{"id":"symbol-8d678effd8898b69","tex":"\\widetilde d","aliases":"widetilde d widetilde","display":"\\widetilde d","kind":"use","occurrences":[3688,3689,3691]},{"id":"symbol-44074f8fd9fb43af","tex":"\\widetilde d_h","aliases":"widetilde d_h widetilde","display":"\\widetilde d_h","kind":"use","occurrences":[3688,3689,3691]},{"id":"symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"use","occurrences":[3371,3374,3402,3403]},{"id":"symbol-5cb9657258ebae5f","tex":"\\widetilde E^a","aliases":"widetilde E^a widetilde","display":"\\widetilde E^a","kind":"use","occurrences":[3371,3374,3402,3403]},{"id":"symbol-089f71382a559dc8","tex":"\\widetilde g","aliases":"widetilde g widetilde","display":"\\widetilde g","kind":"use","occurrences":[3687,3689,3692]},{"id":"symbol-434fbfd6e1ff4264","tex":"\\widetilde g_U","aliases":"widetilde g_U widetilde","display":"\\widetilde g_U","kind":"use","occurrences":[3687,3689,3692]},{"id":"symbol-34052e77aaaf9a0d","tex":"\\widetilde g_V","aliases":"widetilde g_V widetilde","display":"\\widetilde g_V","kind":"use","occurrences":[3687,3689,3692]},{"id":"symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","occurrences":[556,558,560,561,587]},{"id":"symbol-c74e69495f6cc411","tex":"\\widetilde H_M","aliases":"widetilde H_M widetilde","display":"\\widetilde H_M","kind":"use","occurrences":[556,558,560,561,587]},{"id":"symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","occurrences":[2278,2279,2280]},{"id":"symbol-152380ca5fee3168","tex":"\\widetilde k","aliases":"widetilde k widetilde","display":"\\widetilde k","kind":"use","occurrences":[3779,3781]},{"id":"symbol-a3f611dfe324685e","tex":"\\widetilde{\\mathbf k}","aliases":"widetilde k widetilde mathbf","display":"\\widetilde{\\mathbf k}","kind":"use","occurrences":[3780,3781]},{"id":"symbol-4e03157a81c6cd1b","tex":"\\widetilde k_i","aliases":"widetilde k_i widetilde","display":"\\widetilde k_i","kind":"use","occurrences":[3779,3781]},{"id":"symbol-b7d1e1357771f82a","tex":"\\widetilde k_j","aliases":"widetilde k_j widetilde","display":"\\widetilde k_j","kind":"use","occurrences":[3781]},{"id":"symbol-939a02f74ff6b85e","tex":"\\widetilde n","aliases":"widetilde n widetilde","display":"\\widetilde n","kind":"use","occurrences":[3360,3361,3362,3363]},{"id":"symbol-271a4979a4fd2464","tex":"\\widetilde n^I","aliases":"widetilde n^I widetilde","display":"\\widetilde n^I","kind":"use","occurrences":[3363]},{"id":"symbol-5e250d9bc1a41414","tex":"\\widetilde n_I","aliases":"widetilde n_I widetilde","display":"\\widetilde n_I","kind":"use","occurrences":[3360,3361,3362]},{"id":"symbol-c51909f438842cb3","tex":"\\widetilde n_J","aliases":"widetilde n_J widetilde","display":"\\widetilde n_J","kind":"use","occurrences":[3362]},{"id":"symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","occurrences":[3686,3689]},{"id":"symbol-ed8c63057cfee2e2","tex":"\\widetilde Q","aliases":"widetilde Q widetilde","display":"\\widetilde Q","kind":"use","occurrences":[3686,3689]},{"id":"symbol-a1dfb1836c1fd75b","tex":"\\widetilde R","aliases":"widetilde R widetilde","display":"\\widetilde R","kind":"use","occurrences":[3972,3974,3975]},{"id":"symbol-23b8f5bb7ff61960","tex":"\\widetilde R_j","aliases":"widetilde R_j widetilde","display":"\\widetilde R_j","kind":"use","occurrences":[3975]},{"id":"symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"use","occurrences":[2790,3686,3689]},{"id":"symbol-dc68d486952395d8","tex":"\\widetilde U_h","aliases":"widetilde U_h widetilde","display":"\\widetilde U_h","kind":"use","occurrences":[2790]},{"id":"symbol-60cd197d59c08b6e","tex":"\\widetilde V","aliases":"widetilde V widetilde","display":"\\widetilde V","kind":"use","occurrences":[3686,3689]},{"id":"symbol-ea2a8d1df2130cc8","tex":"\\widetilde X","aliases":"widetilde X widetilde","display":"\\widetilde X","kind":"use","occurrences":[2270]},{"id":"symbol-86c1d03c1a864e81","tex":"\\widetilde z","aliases":"widetilde z widetilde","display":"\\widetilde z","kind":"use","occurrences":[3686,3689,3691,3693]},{"id":"symbol-2e6d3d996f44d01d","tex":"\\mathbf x","aliases":"x mathbf","display":"\\mathbf x","kind":"use","occurrences":[3607]},{"id":"symbol-8aac0daabbebf28a","tex":"\\mathbf x^2","aliases":"x^2 mathbf","display":"\\mathbf x^2","kind":"use","occurrences":[3607]},{"id":"symbol-f3522098af4f1d38","tex":"x^a","aliases":"x^a","display":"x^a","kind":"use","occurrences":[3370]},{"id":"symbol-bad8ba0dd09b4f7d","tex":"X^k","aliases":"X^k","display":"X^k","kind":"use","occurrences":[1768,1771]},{"id":"symbol-f15ad8081e22a9d1","tex":"X^{k-1-r}","aliases":"X^k-1-r","display":"X^{k-1-r}","kind":"use","occurrences":[1768,1769]},{"id":"symbol-7c235d1ebb9c7923","tex":"X^{k-r}","aliases":"X^k-r","display":"X^{k-r}","kind":"use","occurrences":[1769]},{"id":"symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","occurrences":[45]},{"id":"symbol-efe34588925c4a06","tex":"x^{p-1}","aliases":"x^p-1","display":"x^{p-1}","kind":"use","occurrences":[31]},{"id":"symbol-f8239c285a2b08c9","tex":"x^{p-2}","aliases":"x^p-2","display":"x^{p-2}","kind":"use","occurrences":[32]},{"id":"symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","occurrences":[751]},{"id":"symbol-ff79273075ae7ad3","tex":"X_{0,j+1}^{-1}","aliases":"X_0,j+1^-1","display":"X_{0,j+1}^{-1}","kind":"use","occurrences":[2837]},{"id":"symbol-c4a48827c05d4509","tex":"X_{00}","aliases":"X_00","display":"X_{00}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-3f15eebc2170872f","tex":"X_{01}","aliases":"X_01","display":"X_{01}","kind":"use","occurrences":[2840,2841]},{"id":"symbol-5d877f50bbc2ba84","tex":"X_{01}^{-1}","aliases":"X_01^-1","display":"X_{01}^{-1}","kind":"use","occurrences":[2840,2841]},{"id":"symbol-2b1877e0f569623b","tex":"X_{02}^{-1}","aliases":"X_02^-1","display":"X_{02}^{-1}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","occurrences":[1850,1855,3481,3482,3487,3489,3492]},{"id":"symbol-bd1d17514be24765","tex":"X_{1,j+1}^{-1}","aliases":"X_1,j+1^-1","display":"X_{1,j+1}^{-1}","kind":"use","occurrences":[2837]},{"id":"symbol-805649c1b3a8b1e8","tex":"X_{10}","aliases":"X_10","display":"X_{10}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-f17c0dba15962adb","tex":"X_{11}","aliases":"X_11","display":"X_{11}","kind":"use","occurrences":[2840,2842]},{"id":"symbol-5ec4a4d89f255491","tex":"X_{11}^{-1}","aliases":"X_11^-1","display":"X_{11}^{-1}","kind":"use","occurrences":[2840,2842]},{"id":"symbol-7a61a0fe64865421","tex":"X_{12}^{-1}","aliases":"X_12^-1","display":"X_{12}^{-1}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-3674278fc192e035","tex":"x_1^2","aliases":"x_1^2","display":"x_1^2","kind":"use","occurrences":[3484,3486]},{"id":"symbol-48fbe887931fba9d","tex":"X_{1j}","aliases":"X_1j","display":"X_{1j}","kind":"use","occurrences":[2837]},{"id":"symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","occurrences":[2438]},{"id":"symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","occurrences":[1850,1855,3481,3482,3487,3489]},{"id":"symbol-6f4fed7ff48f6423","tex":"X_2^2","aliases":"X_2^2","display":"X_2^2","kind":"use","occurrences":[2433]},{"id":"symbol-8d0a8687ec2ae46a","tex":"x_2^2","aliases":"x_2^2","display":"x_2^2","kind":"use","occurrences":[3489]},{"id":"symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","occurrences":[1850,1855]},{"id":"symbol-77b2927cea561ce1","tex":"X_{ba}","aliases":"X_ba","display":"X_{ba}","kind":"use","occurrences":[1991,1992,1996,2521]},{"id":"symbol-7a2d99f30c0cc9af","tex":"X_{ba}^{V}","aliases":"X_ba^V","display":"X_{ba}^{V}","kind":"use","occurrences":[2268,2269]},{"id":"symbol-e3e9d53e5f23ac1e","tex":"X_{bc}","aliases":"X_bc","display":"X_{bc}","kind":"use","occurrences":[2524]},{"id":"symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","occurrences":[2703,2706,2707]},{"id":"symbol-24f9a3fe39a3dae6","tex":"X_{ca}","aliases":"X_ca","display":"X_{ca}","kind":"use","occurrences":[2524]},{"id":"symbol-14a3d06f2604bc2a","tex":"X_{cb}","aliases":"X_cb","display":"X_{cb}","kind":"use","occurrences":[1996]},{"id":"symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","occurrences":[1114]},{"id":"symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","occurrences":[1133,1134]},{"id":"symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","occurrences":[2301,2302,2303,2304]},{"id":"symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","occurrences":[34,36]},{"id":"symbol-77180a1ffcf8a2ca","tex":"x_{p,q}","aliases":"x_p,q","display":"x_{p,q}","kind":"use","occurrences":[1106]},{"id":"symbol-facd8502ae1cad01","tex":"X_{rs}","aliases":"X_rs","display":"X_{rs}","kind":"use","occurrences":[805,806]},{"id":"symbol-58812c8c592a182f","tex":"Y^*","aliases":"Y^*","display":"Y^*","kind":"use","occurrences":[2429]},{"id":"symbol-61c288ea4029d6a5","tex":"y^{-1}","aliases":"y^-1","display":"y^{-1}","kind":"use","occurrences":[906]},{"id":"symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","occurrences":[904,913,3495]},{"id":"symbol-827a2f52108af912","tex":"y^j","aliases":"y^j","display":"y^j","kind":"use","occurrences":[1267]},{"id":"symbol-13920cdfdee3eee1","tex":"Y^k","aliases":"Y^k","display":"Y^k","kind":"use","occurrences":[1768,1771]},{"id":"symbol-10e9863564d53a0d","tex":"y^L","aliases":"y^L","display":"y^L","kind":"use","occurrences":[1267]},{"id":"symbol-d0d8e30ac786064e","tex":"y^{Lq+j}","aliases":"y^Lq+j","display":"y^{Lq+j}","kind":"use","occurrences":[1266]},{"id":"symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","occurrences":[1263]},{"id":"symbol-90d98d4bd026d21c","tex":"Y^r","aliases":"Y^r","display":"Y^r","kind":"use","occurrences":[1768,1769]},{"id":"symbol-95314123831de1c4","tex":"Y^{r+1}","aliases":"Y^r+1","display":"Y^{r+1}","kind":"use","occurrences":[1769]},{"id":"symbol-211b18dbfa6b49f3","tex":"y^{rq+j}","aliases":"y^rq+j","display":"y^{rq+j}","kind":"use","occurrences":[3661]},{"id":"symbol-e5f6938d2e45504d","tex":"y_*","aliases":"y_*","display":"y_*","kind":"use","occurrences":[3487,3488]},{"id":"symbol-d50f75767567fffe","tex":"Y_{00}","aliases":"Y_00","display":"Y_{00}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-dadd0c0a7ac5ceef","tex":"Y_{00}^{-1}","aliases":"Y_00^-1","display":"Y_{00}^{-1}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-8c837e3015731152","tex":"Y_{01}^{-1}","aliases":"Y_01^-1","display":"Y_{01}^{-1}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-344ce98a533836a9","tex":"Y_{0j}^{-1}","aliases":"Y_0j^-1","display":"Y_{0j}^{-1}","kind":"use","occurrences":[2837]},{"id":"symbol-9ece1fcef8e48fb6","tex":"Y_{1j}","aliases":"Y_1j","display":"Y_{1j}","kind":"use","occurrences":[2837,2839]},{"id":"symbol-00758ebfad58b574","tex":"Y_{1j}^{-1}","aliases":"Y_1j^-1","display":"Y_{1j}^{-1}","kind":"use","occurrences":[2837,2839]},{"id":"symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","occurrences":[2439]},{"id":"symbol-f5ca61ea6e5f00f5","tex":"Y_{20}","aliases":"Y_20","display":"Y_{20}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-9e5725576b0ba70d","tex":"Y_{21}","aliases":"Y_21","display":"Y_{21}","kind":"use","occurrences":[2836,2840]},{"id":"symbol-9597b8a338d4d116","tex":"Y_{2j}","aliases":"Y_2j","display":"Y_{2j}","kind":"use","occurrences":[2837]},{"id":"symbol-7be44d74687569ce","tex":"y_a","aliases":"y_a","display":"y_a","kind":"use","occurrences":[2258]},{"id":"symbol-4c1e8c1c8cca56ba","tex":"Y_{ba}","aliases":"Y_ba","display":"Y_{ba}","kind":"use","occurrences":[1996]},{"id":"symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","occurrences":[2312]},{"id":"symbol-bbfc803998827b8a","tex":"\\mathrm{YM}","aliases":"YM mathrm","display":"\\mathrm{YM}","kind":"use","occurrences":[2686]},{"id":"symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","occurrences":[107,647,655,656,2859,2860,2957,2958,2959,2960,2961,2962,2963,2966,2967,2968,2969,2970,2971,2983,2984,2999,3050,3059,3150,3151,3153,3154,3155,3156,3162]},{"id":"symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","occurrences":[790,1516,1520,1528,3219]},{"id":"symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","occurrences":[560,588]},{"id":"symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","occurrences":[2859]},{"id":"symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","occurrences":[3481]},{"id":"symbol-d425233dded4b47a","tex":"\\Z^3","aliases":"Z^3 Z","display":null,"kind":"use","occurrences":[1356,1358,1501]},{"id":"symbol-7fb250c40766c45f","tex":"\\mathbb Z^3","aliases":"Z^3 mathbb","display":"\\mathbb Z^3","kind":"use","occurrences":[1520,1528]},{"id":"symbol-5b75520135285cf1","tex":"Z^A","aliases":"Z^A","display":"Z^A","kind":"use","occurrences":[3150,3151,3153,3154]},{"id":"symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","occurrences":[1802]},{"id":"symbol-0bb5e126e766a7f8","tex":"z^{Lq+j}","aliases":"z^Lq+j","display":"z^{Lq+j}","kind":"use","occurrences":[1210,1216]},{"id":"symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","occurrences":[1206,1211]},{"id":"symbol-10c7caeba58a1aa9","tex":"z_*","aliases":"z_*","display":"z_*","kind":"use","occurrences":[3488]},{"id":"symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","occurrences":[3424,3425,3434,3436,3437,3439,3459,3460,3461,3462,3470,3474]},{"id":"symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","occurrences":[3424,3425,3434,3436,3437,3439,3459,3460,3461,3462,3470,3471]},{"id":"symbol-642b65364417d9f0","tex":"z_{ab}","aliases":"z_ab","display":"z_{ab}","kind":"use","occurrences":[2114,2115]},{"id":"symbol-32babd86fec042f0","tex":"z_{ba}","aliases":"z_ba","display":"z_{ba}","kind":"use","occurrences":[2113,2115]},{"id":"symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","occurrences":[1085,1089,2731,2732,2760]},{"id":"symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","occurrences":[2312,3474,3685,3689,3690,3693]},{"id":"symbol-a1bb8bb52da2ab42","tex":"z_{n+1}","aliases":"z_n+1","display":"z_{n+1}","kind":"use","occurrences":[3689,3690,3691]},{"id":"symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"use","occurrences":[790]},{"id":"symbol-31d760e606ff3dad","tex":"\\Z_p^\\times","aliases":"Z_p^× Z times","display":null,"kind":"use","occurrences":[1307]},{"id":"symbol-99e9f1953d7e02c1","tex":"Z_{\\mathrm{sign}}","aliases":"Z_sign mathrm","display":"Z_{\\mathrm{sign}}","kind":"use","occurrences":[107,647,655,656]},{"id":"symbol-3338e0334cca7109","tex":"z_{x,\\mu}","aliases":"z_x,μ mu","display":"z_{x,\\mu}","kind":"use","occurrences":[2806,2810]},{"id":"symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","occurrences":[560,577]},{"id":"symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","occurrences":[286,1716,1721,2601]},{"id":"symbol-66f78e7a5c255442","tex":"|^5","aliases":"|^5","display":"|^5","kind":"use","occurrences":[1744]},{"id":"symbol-a2e3fd70ad04e933","tex":"|^6","aliases":"|^6","display":"|^6","kind":"use","occurrences":[1702]},{"id":"symbol-06b81a055469dbd1","tex":"|^p","aliases":"|^p","display":"|^p","kind":"use","occurrences":[2579,2582,2587]},{"id":"symbol-18ec6838a96d8568","tex":"|_{\\mathcal V_M}","aliases":"|_ V_M mathcal","display":"|_{\\mathcal V_M}","kind":"use","occurrences":[444,646]},{"id":"symbol-ffbdd0e2e2408112","tex":"|_{Q\\HH}","aliases":"|_QHH HH","display":null,"kind":"use","occurrences":[678,679,681]},{"id":"symbol-2db86546abe159fd","tex":"|_{\\Ran P}","aliases":"|_Ran P Ran","display":null,"kind":"use","occurrences":[231]},{"id":"symbol-b7302e1b720e88ef","tex":"|_{\\Ran Q}","aliases":"|_Ran Q Ran","display":null,"kind":"use","occurrences":[231,294]},{"id":"symbol-9152fcee04370f9c","tex":"|_{\\varphi}","aliases":"|_φ varphi","display":"|_{\\varphi}","kind":"use","occurrences":[3294]},{"id":"symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","occurrences":[5,7,11,12,39,74,833,1307,2026,2359,2674,2820,2944,2987,3019,3243,3245,3256,3335,3465,3466,3467,3618,3632]},{"id":"symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","occurrences":[339,340,341,344,346,347,350,362,368,597,598,850,853,854,855,1107,1217,1218,1219,1255,1329,1330,1513,1542,1543,1544,1545,1548,1549,1556,1557,1558,1561,1565,1581,1593,1594,1595,1701,2251,2348,2349,2350,2351,2352,2353,2358,2359,2531,2532,2718,2725,2730,2731,2741,2742,3287,3613,3614]},{"id":"symbol-6846e04eb85319e1","tex":"\\alpha^*","aliases":"α^* alpha","display":"\\alpha^*","kind":"use","occurrences":[2718,2742]},{"id":"symbol-34a0d2bb0218af80","tex":"\\alpha^n","aliases":"α^n alpha","display":"\\alpha^n","kind":"use","occurrences":[1329]},{"id":"symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","occurrences":[1513,1545,1548,1549,1556,1558,1565,1581,1594,1595,1701]},{"id":"symbol-70124300c10527d0","tex":"\\alpha_j^*","aliases":"α_j^* alpha","display":"\\alpha_j^*","kind":"use","occurrences":[1548,1556]},{"id":"symbol-8ae91c89408a2a13","tex":"\\alpha_j^2","aliases":"α_j^2 alpha","display":"\\alpha_j^2","kind":"use","occurrences":[1548,1557,1593]},{"id":"symbol-993a80257a1ce549","tex":"\\alpha_k","aliases":"α_k alpha","display":"\\alpha_k","kind":"use","occurrences":[1549,1556,1595]},{"id":"symbol-48b4dca7f4ffcdf4","tex":"\\alpha_q","aliases":"α_q alpha","display":"\\alpha_q","kind":"use","occurrences":[850,853,854,855]},{"id":"symbol-c26b0386db57f0c4","tex":"\\alpha_x","aliases":"α_x alpha","display":"\\alpha_x","kind":"use","occurrences":[1542,1561]},{"id":"symbol-58f0a9598ff4ff43","tex":"\\alpha_y","aliases":"α_y alpha","display":"\\alpha_y","kind":"use","occurrences":[1542,1544,1561]},{"id":"symbol-f60ac066d843ff0e","tex":"\\alpha_z","aliases":"α_z alpha","display":"\\alpha_z","kind":"use","occurrences":[1543,1561]},{"id":"symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","occurrences":[247,249,255,590,591,592,593,596,599,1513,1543,1545,1547,1548,1549,1550,1558,1561,1565,1581,1593,1594,2250,2251,2353,2359,2363,2532,2719,2725,2730,2739,3287]},{"id":"symbol-bd66c98b32c8ec9d","tex":"\\beta^*","aliases":"β^* beta","display":"\\beta^*","kind":"use","occurrences":[1548,2719]},{"id":"symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","occurrences":[1548,1593]},{"id":"symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","occurrences":[2073,2079,2084,2086,2087,2088,2089,2090,2095,3157,3162,3258,3373,3378,3399]},{"id":"symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","occurrences":[290,291,292,306,307,308,309,339,947,948,952,954,957,962,975,1000,1002,1005,1013,1014,1022,1026,2153,2155,2158,2160,2161,2165,2169,2174,2195,2197,2994,2995,2996,3273,3274,3275,3279,3281,3282,3298,3305,3309,3342,3345,3346,3383,3384,3395,3396,3620,3621,3623,3625,3627,3637,3638,3674]},{"id":"symbol-03569cbd604e9e67","tex":"\\Gamma^*","aliases":"Γ^* Gamma","display":"\\Gamma^*","kind":"use","occurrences":[2079,2084]},{"id":"symbol-628f274efa0c1a48","tex":"\\gamma^0","aliases":"γ^0 gamma","display":"\\gamma^0","kind":"use","occurrences":[3383,3384,3395]},{"id":"symbol-660830aacf9a8a38","tex":"\\gamma^1","aliases":"γ^1 gamma","display":"\\gamma^1","kind":"use","occurrences":[3395]},{"id":"symbol-4569df46706329ae","tex":"\\gamma^2","aliases":"γ^2 gamma","display":"\\gamma^2","kind":"use","occurrences":[3395]},{"id":"symbol-03799c3dc7cb9b3c","tex":"\\gamma^3","aliases":"γ^3 gamma","display":"\\gamma^3","kind":"use","occurrences":[3395]},{"id":"symbol-f455ccf4bfbfc11e","tex":"\\Gamma^i","aliases":"Γ^i Gamma","display":"\\Gamma^i","kind":"use","occurrences":[3373]},{"id":"symbol-768b2242190301f3","tex":"\\gamma^I","aliases":"γ^I gamma","display":"\\gamma^I","kind":"use","occurrences":[3273,3279,3281,3282,3342,3346,3396]},{"id":"symbol-23236d6d32bb1087","tex":"\\gamma^J","aliases":"γ^J gamma","display":"\\gamma^J","kind":"use","occurrences":[3273,3282,3345]},{"id":"symbol-e26c0ed5d9be65cf","tex":"\\gamma^K","aliases":"γ^K gamma","display":"\\gamma^K","kind":"use","occurrences":[3346]},{"id":"symbol-2771b61f3f3f5974","tex":"\\gamma_0","aliases":"γ_0 gamma","display":"\\gamma_0","kind":"use","occurrences":[962,1013,1022]},{"id":"symbol-fd17a77e6abbf292","tex":"\\gamma_0^2","aliases":"γ_0^2 gamma","display":"\\gamma_0^2","kind":"use","occurrences":[1014,1026]},{"id":"symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","occurrences":[962]},{"id":"symbol-e27dee4a4efd5ee2","tex":"\\gamma_{12}","aliases":"γ_12 gamma","display":"\\gamma_{12}","kind":"use","occurrences":[2169]},{"id":"symbol-e896a0b8b00bb35e","tex":"\\gamma_{21}","aliases":"γ_21 gamma","display":"\\gamma_{21}","kind":"use","occurrences":[2169]},{"id":"symbol-7fd85439665ac62c","tex":"\\gamma_5","aliases":"γ_5 gamma","display":"\\gamma_5","kind":"use","occurrences":[3395,3396]},{"id":"symbol-ea15a4cf61a5a234","tex":"\\Gamma_a","aliases":"Γ_a Gamma","display":"\\Gamma_a","kind":"use","occurrences":[3399]},{"id":"symbol-1c4421c45efc7807","tex":"\\gamma_{a+1}","aliases":"γ_a+1 gamma","display":"\\gamma_{a+1}","kind":"use","occurrences":[975]},{"id":"symbol-0e0f0609597bb3e8","tex":"\\gamma_{ab}","aliases":"γ_ab gamma","display":"\\gamma_{ab}","kind":"use","occurrences":[2158,2195]},{"id":"symbol-e593d31e966a0e06","tex":"\\gamma_{ba}","aliases":"γ_ba gamma","display":"\\gamma_{ba}","kind":"use","occurrences":[2153,2155,2158,2160,2161,2174,2195,2197]},{"id":"symbol-54218a765dc34dd7","tex":"\\gamma_{bi}","aliases":"γ_bi gamma","display":"\\gamma_{bi}","kind":"use","occurrences":[2165]},{"id":"symbol-9e99730a6fe995c5","tex":"\\gamma_{h-1}","aliases":"γ_h-1 gamma","display":"\\gamma_{h-1}","kind":"use","occurrences":[962,975]},{"id":"symbol-8e6227ccf23c3c25","tex":"\\gamma_I","aliases":"γ_I gamma","display":"\\gamma_I","kind":"use","occurrences":[3274]},{"id":"symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","occurrences":[2994,2995,2996]},{"id":"symbol-eb6169646c2f7354","tex":"\\gamma_{IJ}","aliases":"γ_IJ gamma","display":"\\gamma_{IJ}","kind":"use","occurrences":[3274,3275,3298,3305,3309,3346]},{"id":"symbol-322e5c7da80d563a","tex":"\\gamma_J","aliases":"γ_J gamma","display":"\\gamma_J","kind":"use","occurrences":[3274]},{"id":"symbol-b8f615ebae18b9ca","tex":"\\gamma_{p,w}","aliases":"γ_p,w gamma","display":"\\gamma_{p,w}","kind":"use","occurrences":[306,307,308,309,339]},{"id":"symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","occurrences":[947,948,952,954,1005]},{"id":"symbol-e850f2bbe26688ef","tex":"\\gamma_{r-1}","aliases":"γ_r-1 gamma","display":"\\gamma_{r-1}","kind":"use","occurrences":[952,957]},{"id":"symbol-311a62961865d104","tex":"\\gamma_r^2","aliases":"γ_r^2 gamma","display":"\\gamma_r^2","kind":"use","occurrences":[1000,1002]},{"id":"symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","occurrences":[1701,1705,1706,1707,1724,1725,1726,1907,1935,2617,2619,2625,2627,3428,3429,3430,3431,3432,3434,3435,3438,3439,3445,3452,3459,3464,3469,3472,3502,3503,3506,3520,3521,3522,3525,3531,3534,3546,3613,3668,3674,3703,3704,3705,3708,3710,3711,3713,3714,3715,3987,3993,4141]},{"id":"symbol-02c68d637b81b90c","tex":"\\delta^3","aliases":"δ^3 delta","display":"\\delta^3","kind":"use","occurrences":[3388,3393]},{"id":"symbol-8d634cbf29cf1a83","tex":"\\delta_{\\mathcal F_{ij}}","aliases":"δ_ F_ij delta mathcal","display":"\\delta_{\\mathcal F_{ij}}","kind":"use","occurrences":[2517]},{"id":"symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","occurrences":[3728]},{"id":"symbol-dac3f880e6e3670a","tex":"\\delta_2","aliases":"δ_2 delta","display":"\\delta_2","kind":"use","occurrences":[3728]},{"id":"symbol-020f6e328243f9c3","tex":"\\delta_3","aliases":"δ_3 delta","display":"\\delta_3","kind":"use","occurrences":[3728]},{"id":"symbol-08ab92c996841336","tex":"\\delta_{\\left[U,V\\right]_{\\mathrm{par}}}","aliases":"δ_[U,V]_par delta left right mathrm","display":"\\delta_{\\left[U,V\\right]_{\\mathrm{par}}}","kind":"use","occurrences":[3300]},{"id":"symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","occurrences":[3226,3230,3237,3243,3245]},{"id":"symbol-3ff78dbd4ffd8b65","tex":"\\delta_{AB}","aliases":"δ_AB delta","display":"\\delta_{AB}","kind":"use","occurrences":[3388,3393]},{"id":"symbol-845f98ac1e4d0fbd","tex":"\\delta_b","aliases":"δ_b delta","display":"\\delta_b","kind":"use","occurrences":[3237,3243]},{"id":"symbol-8510827ccfbd978d","tex":"\\delta_{B_i}","aliases":"δ_B_i delta","display":"\\delta_{B_i}","kind":"use","occurrences":[2517]},{"id":"symbol-0e4773cb2ae1266d","tex":"\\delta_{B_j}","aliases":"δ_B_j delta","display":"\\delta_{B_j}","kind":"use","occurrences":[2517]},{"id":"symbol-60edcde84b590e7e","tex":"\\Delta_C","aliases":"Δ_C Delta","display":"\\Delta_C","kind":"use","occurrences":[3503,3506]},{"id":"symbol-c6a395594f216319","tex":"\\Delta_D","aliases":"Δ_D Delta","display":"\\Delta_D","kind":"use","occurrences":[3503,3506]},{"id":"symbol-9437cea347b38d28","tex":"\\delta_{de}","aliases":"δ_de delta","display":"\\delta_{de}","kind":"use","occurrences":[211,212]},{"id":"symbol-9e14435a4d85c517","tex":"\\Delta_F","aliases":"Δ_F Delta","display":"\\Delta_F","kind":"use","occurrences":[3502,3503]},{"id":"symbol-608581f0be0c4b00","tex":"\\Delta_h","aliases":"Δ_h Delta","display":"\\Delta_h","kind":"use","occurrences":[3668,3674]},{"id":"symbol-990016db1ae37714","tex":"\\Delta_{h,n}","aliases":"Δ_h,n Delta","display":"\\Delta_{h,n}","kind":"use","occurrences":[1701,1705,1706,1707,1724,1726,1907,1935]},{"id":"symbol-eaf44d5b23935330","tex":"\\Delta_{h,n}^*","aliases":"Δ_h,n^* Delta","display":"\\Delta_{h,n}^*","kind":"use","occurrences":[1726,1907]},{"id":"symbol-83b61b3d00ad9b6e","tex":"\\Delta_{h,n}^2","aliases":"Δ_h,n^2 Delta","display":"\\Delta_{h,n}^2","kind":"use","occurrences":[1725]},{"id":"symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"use","occurrences":[3051,3371,3729,3735,3737,3739,3748,3751,3752,3761,3762,3765,3769,3770]},{"id":"symbol-d900de86d0f4ee16","tex":"\\delta_i^2","aliases":"δ_i^2 delta","display":"\\delta_i^2","kind":"use","occurrences":[3729]},{"id":"symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","occurrences":[3735,3748,3749,3765,3769,3781]},{"id":"symbol-288a046779bf284f","tex":"\\Delta_j","aliases":"Δ_j Delta","display":"\\Delta_j","kind":"use","occurrences":[3520,3522,3525,3531]},{"id":"symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","occurrences":[1701,1721,3729,3735,3736,3737,3738,3739,3748,3751,3761,3762,3765,3769,3770,3775]},{"id":"symbol-147b73e82dc15b8b","tex":"\\Delta_{j-1}","aliases":"Δ_j-1 Delta","display":"\\Delta_{j-1}","kind":"use","occurrences":[3522]},{"id":"symbol-5c2d3273fd73d5e9","tex":"\\Delta_{j-1}^2","aliases":"Δ_j-1^2 Delta","display":"\\Delta_{j-1}^2","kind":"use","occurrences":[3525]},{"id":"symbol-e4b6e196f224dc32","tex":"\\Delta_j^2","aliases":"Δ_j^2 Delta","display":"\\Delta_j^2","kind":"use","occurrences":[3521,3525,3534,3546]},{"id":"symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","occurrences":[1210,1326]},{"id":"symbol-f101f26c86293154","tex":"\\delta_K","aliases":"δ_K delta","display":"\\delta_K","kind":"use","occurrences":[3831,3832,3833,3834]},{"id":"symbol-9601f08c373fe8e4","tex":"\\delta_k","aliases":"δ_k delta","display":"\\delta_k","kind":"use","occurrences":[3735,3737,3740,3762,3771,3777,3778]},{"id":"symbol-3f3790d3553b608d","tex":"\\delta_l","aliases":"δ_l delta","display":"\\delta_l","kind":"use","occurrences":[3737]},{"id":"symbol-e8b18ae884be8df2","tex":"\\delta_{mn}","aliases":"δ_mn delta","display":"\\delta_{mn}","kind":"use","occurrences":[453]},{"id":"symbol-ea2a7f3a25fa545c","tex":"\\delta_{pn,m}","aliases":"δ_pn,m delta","display":"\\delta_{pn,m}","kind":"use","occurrences":[1185]},{"id":"symbol-1d9e0b50c7e8925a","tex":"\\delta_{qq'}","aliases":"δ_qq' delta","display":"\\delta_{qq'}","kind":"use","occurrences":[170,472]},{"id":"symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","occurrences":[136,798]},{"id":"symbol-b0a780cb0a0d5ad7","tex":"\\delta_{rt}","aliases":"δ_rt delta","display":"\\delta_{rt}","kind":"use","occurrences":[812]},{"id":"symbol-e32d7eec9c371dea","tex":"\\delta_{st}","aliases":"δ_st delta","display":"\\delta_{st}","kind":"use","occurrences":[812]},{"id":"symbol-024a046ea306ad65","tex":"\\Delta_U","aliases":"Δ_U Delta","display":"\\Delta_U","kind":"use","occurrences":[2617,2619,2625,2627]},{"id":"symbol-4a73afb10ef0b959","tex":"\\delta_U","aliases":"δ_U delta","display":"\\delta_U","kind":"use","occurrences":[3300]},{"id":"symbol-b35f037ae1ca06a1","tex":"\\delta_u","aliases":"δ_u delta","display":"\\delta_u","kind":"use","occurrences":[3406,3407]},{"id":"symbol-7a9a9e1ad8d08502","tex":"\\Delta_{U^g}","aliases":"Δ_U^g Delta","display":"\\Delta_{U^g}","kind":"use","occurrences":[2619]},{"id":"symbol-32a0678296fa0cc7","tex":"\\delta_V","aliases":"δ_V delta","display":"\\delta_V","kind":"use","occurrences":[3300]},{"id":"symbol-43ab2fa7aeb228e6","tex":"\\delta_x","aliases":"δ_x delta","display":"\\delta_x","kind":"use","occurrences":[3052,3064,3103]},{"id":"symbol-4bff5300fdcd34eb","tex":"\\delta_y","aliases":"δ_y delta","display":"\\delta_y","kind":"use","occurrences":[3052,3064,3103]},{"id":"symbol-bb8ce6bd7aafe6f1","tex":"\\delta_z","aliases":"δ_z delta","display":"\\delta_z","kind":"use","occurrences":[3052,3064,3103]},{"id":"symbol-141577db886abda4","tex":"\\Delta_\\Delta","aliases":"Δ_Δ Delta Delta","display":"\\Delta_\\Delta","kind":"use","occurrences":[3428,3431,3435,3452]},{"id":"symbol-536a24f4cca16e8d","tex":"\\Delta_\\Delta^{-1}","aliases":"Δ_Δ^-1 Delta Delta","display":"\\Delta_\\Delta^{-1}","kind":"use","occurrences":[3429]},{"id":"symbol-ee986ad8011bd9d2","tex":"\\delta_{\\mu\\nu}","aliases":"δ_μν delta mu nu","display":"\\delta_{\\mu\\nu}","kind":"use","occurrences":[2132]},{"id":"symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","occurrences":[433,435,438,1401,1403,1405,1407,1408,1463,1465,1732,1734,2427,2431,2495,2496,2579,2580,2582,2599,2600,2601,3118,3119,3865,3867,3937,3940,3953]},{"id":"symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","occurrences":[2495]},{"id":"symbol-e3e5a993e2ec8e09","tex":"\\epsilon^{abc}","aliases":"ϵ^abc epsilon","display":"\\epsilon^{abc}","kind":"use","occurrences":[3360,3371,3379,3381]},{"id":"symbol-fbcb4249d64b915d","tex":"\\epsilon^I","aliases":"ϵ^I epsilon","display":"\\epsilon^I","kind":"use","occurrences":[3397]},{"id":"symbol-ab7cfeb562562c69","tex":"\\epsilon^i","aliases":"ϵ^i epsilon","display":"\\epsilon^i","kind":"use","occurrences":[3237,3243,3373]},{"id":"symbol-afe3d484b60c1007","tex":"\\epsilon^{IJ}","aliases":"ϵ^IJ epsilon","display":"\\epsilon^{IJ}","kind":"use","occurrences":[3397]},{"id":"symbol-6dfce6e7e372892a","tex":"\\epsilon^{ij}","aliases":"ϵ^ij epsilon","display":"\\epsilon^{ij}","kind":"use","occurrences":[3237]},{"id":"symbol-d1776e7e5d78a172","tex":"\\epsilon^{ijk}","aliases":"ϵ^ijk epsilon","display":"\\epsilon^{ijk}","kind":"use","occurrences":[3238]},{"id":"symbol-009aa60286314a39","tex":"\\epsilon_{0123}","aliases":"ϵ_0123 epsilon","display":"\\epsilon_{0123}","kind":"use","occurrences":[3266]},{"id":"symbol-da012babef541427","tex":"\\epsilon_1","aliases":"ϵ_1 epsilon","display":"\\epsilon_1","kind":"use","occurrences":[962]},{"id":"symbol-c7ffdc6ef6b651ea","tex":"\\epsilon_2","aliases":"ϵ_2 epsilon","display":"\\epsilon_2","kind":"use","occurrences":[962]},{"id":"symbol-e61b7ea9a3581771","tex":"\\epsilon_{a+1}","aliases":"ϵ_a+1 epsilon","display":"\\epsilon_{a+1}","kind":"use","occurrences":[974]},{"id":"symbol-de18910f25a2ed52","tex":"\\epsilon_{a,\\mathbf r}","aliases":"ϵ_a, r epsilon mathbf","display":"\\epsilon_{a,\\mathbf r}","kind":"use","occurrences":[635]},{"id":"symbol-62e4a848980609ac","tex":"\\epsilon_{abc}","aliases":"ϵ_abc epsilon","display":"\\epsilon_{abc}","kind":"use","occurrences":[3238]},{"id":"symbol-1f74d9a4539571bc","tex":"\\epsilon_h","aliases":"ϵ_h epsilon","display":"\\epsilon_h","kind":"use","occurrences":[962,974,1000]},{"id":"symbol-a93d9c9ebca43b37","tex":"\\epsilon_{ij}","aliases":"ϵ_ij epsilon","display":"\\epsilon_{ij}","kind":"use","occurrences":[3237]},{"id":"symbol-973f701abab21f48","tex":"\\epsilon_{ijk}","aliases":"ϵ_ijk epsilon","display":"\\epsilon_{ijk}","kind":"use","occurrences":[3371,3374]},{"id":"symbol-1ab16a16a710a778","tex":"\\epsilon_{IJKL}","aliases":"ϵ_IJKL epsilon","display":"\\epsilon_{IJKL}","kind":"use","occurrences":[3272,3279,3341,3345,3360]},{"id":"symbol-6f89c09ead72a18a","tex":"\\epsilon_{JIKL}","aliases":"ϵ_JIKL epsilon","display":"\\epsilon_{JIKL}","kind":"use","occurrences":[3345]},{"id":"symbol-778fb0127de2ea78","tex":"\\epsilon_{KLIJ}","aliases":"ϵ_KLIJ epsilon","display":"\\epsilon_{KLIJ}","kind":"use","occurrences":[3341]},{"id":"symbol-0d161a854cc2f845","tex":"\\epsilon_m","aliases":"ϵ_m epsilon","display":"\\epsilon_m","kind":"use","occurrences":[493,494,496,503,505,515,517,519,521,526,527,528,531,542,552,553,555,556,566,577,579,586,592,593,594,597]},{"id":"symbol-5e17754e3e2e4c50","tex":"\\epsilon_{m_j}","aliases":"ϵ_m_j epsilon","display":"\\epsilon_{m_j}","kind":"use","occurrences":[581]},{"id":"symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","occurrences":[3118,3119]},{"id":"symbol-a28ffe0adef622b5","tex":"\\epsilon_{p,0}","aliases":"ϵ_p,0 epsilon","display":"\\epsilon_{p,0}","kind":"use","occurrences":[276]},{"id":"symbol-b3b3cfb9deb4c4e2","tex":"\\epsilon_{p,h+1}","aliases":"ϵ_p,h+1 epsilon","display":"\\epsilon_{p,h+1}","kind":"use","occurrences":[261]},{"id":"symbol-1a16cbc91fb1fd77","tex":"\\epsilon_{p,r}","aliases":"ϵ_p,r epsilon","display":"\\epsilon_{p,r}","kind":"use","occurrences":[256,257,259,262,274,276]},{"id":"symbol-70bb170c716771c3","tex":"\\epsilon_r","aliases":"ϵ_r epsilon","display":"\\epsilon_r","kind":"use","occurrences":[944,952,1000,1005]},{"id":"symbol-c5bef5ad97898be2","tex":"\\epsilon_{r+1}","aliases":"ϵ_r+1 epsilon","display":"\\epsilon_{r+1}","kind":"use","occurrences":[1002]},{"id":"symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","occurrences":[9,12,40,47,67,3353,3355,3363]},{"id":"symbol-1fde43544e1e7fe7","tex":"\\zeta^I","aliases":"ζ^I zeta","display":"\\zeta^I","kind":"use","occurrences":[3363]},{"id":"symbol-3a7b183402eb6a5c","tex":"\\zeta_{p,h}","aliases":"ζ_p,h zeta","display":"\\zeta_{p,h}","kind":"use","occurrences":[9,40,67]},{"id":"symbol-fa38a4e2bce94527","tex":"\\zeta_{p,h+1}","aliases":"ζ_p,h+1 zeta","display":"\\zeta_{p,h+1}","kind":"use","occurrences":[67]},{"id":"symbol-44c728f80de640dd","tex":"\\zeta_{p,h}^{\\,t}","aliases":"ζ_p,h^ t zeta","display":"\\zeta_{p,h}^{\\,t}","kind":"use","occurrences":[12,47]},{"id":"symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","occurrences":[2451,2452,2453,2481,2638,2649,2653,2654,2655,2656,3029,3243,3245,3265,3273,3282,3301,3351,3352,3359,3362,3363,3365,3408,3643,3756,3794,3795,3799,3801,3815,3817,3818,3819,3824]},{"id":"symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","occurrences":[2453,2481]},{"id":"symbol-4024c59fd0a358ce","tex":"\\eta^a","aliases":"η^a eta","display":"\\eta^a","kind":"use","occurrences":[3243,3363,3408]},{"id":"symbol-e0f19d7866e16390","tex":"\\eta^b","aliases":"η^b eta","display":"\\eta^b","kind":"use","occurrences":[3243]},{"id":"symbol-aae04eeae6b07750","tex":"\\eta^{IJ}","aliases":"η^IJ eta","display":"\\eta^{IJ}","kind":"use","occurrences":[3273,3282,3362]},{"id":"symbol-50d5e65dfebdbedc","tex":"\\eta_a","aliases":"η_a eta","display":"\\eta_a","kind":"use","occurrences":[3795,3799]},{"id":"symbol-18a46bf826b0cbf4","tex":"\\eta_{IJ}","aliases":"η_IJ eta","display":"\\eta_{IJ}","kind":"use","occurrences":[3265,3359]},{"id":"symbol-1dd0a87bdefa7845","tex":"\\eta_j","aliases":"η_j eta","display":"\\eta_j","kind":"use","occurrences":[2638,2655,2656]},{"id":"symbol-d0ab24b9a250f9cb","tex":"\\eta_{j+1}^-","aliases":"η_j+1^- eta","display":"\\eta_{j+1}^-","kind":"use","occurrences":[2638,2649,2654]},{"id":"symbol-a3e03897b69d9ea1","tex":"\\eta_{j-1}^+","aliases":"η_j-1^+ eta","display":"\\eta_{j-1}^+","kind":"use","occurrences":[2638,2654]},{"id":"symbol-5e88b9488b9fcd67","tex":"\\eta_j^+","aliases":"η_j^+ eta","display":"\\eta_j^+","kind":"use","occurrences":[2649,2653]},{"id":"symbol-2d58b3c78f282ec7","tex":"\\eta_j^-","aliases":"η_j^- eta","display":"\\eta_j^-","kind":"use","occurrences":[2653]},{"id":"symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","occurrences":[1271,1272,1275,3095,3096,3342,3387,3393,3403]},{"id":"symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","occurrences":[830,840,841,843,844,846,847,848,850,854,855,859,862,1073,1074,1084,1106,1111,1127,1128,1130,1131,1161,1164,1174,1191,1192,1193,1195,1196,1198,2093,2095,2096,2097,2098,2690,2691,2694,2695,2697,2698,2706,2708,2709,3070,3074,3077,3091,3095,3113,3261,3263,3619,3621,3623,3625,3627,3629,3631,3634,3635,3636,3637,3638,3639,3641,3643,3644,3652,3655,3656,3985,4112]},{"id":"symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"use","occurrences":[3663,3664]},{"id":"symbol-772c337813af41af","tex":"\\Theta^2","aliases":"Θ^2 Theta","display":"\\Theta^2","kind":"use","occurrences":[1272,1275]},{"id":"symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","occurrences":[2708]},{"id":"symbol-8481a76e553d57b9","tex":"\\vartheta^2","aliases":"ϑ^2 vartheta","display":"\\vartheta^2","kind":"use","occurrences":[3664]},{"id":"symbol-c112a1cdbb689680","tex":"\\theta^4","aliases":"θ^4 theta","display":"\\theta^4","kind":"use","occurrences":[2708]},{"id":"symbol-a37bcd8a9273f437","tex":"\\Theta_{\\mathrm D}","aliases":"Θ_ D Theta mathrm","display":"\\Theta_{\\mathrm D}","kind":"use","occurrences":[3387,3393]},{"id":"symbol-0a9a14fec4bd0562","tex":"\\theta_+","aliases":"θ_+ theta","display":"\\theta_+","kind":"use","occurrences":[4112]},{"id":"symbol-0123f2e7368df612","tex":"\\theta_-","aliases":"θ_- theta","display":"\\theta_-","kind":"use","occurrences":[4112]},{"id":"symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","occurrences":[1193]},{"id":"symbol-e543323c4045f446","tex":"\\theta_f","aliases":"θ_f theta","display":"\\theta_f","kind":"use","occurrences":[2694,2695]},{"id":"symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"use","occurrences":[3070,3074,3077]},{"id":"symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","occurrences":[830,840,843,848,859,3091]},{"id":"symbol-998cf8e699fcb189","tex":"\\Theta_{\\mathrm{red}}","aliases":"Θ_red Theta mathrm","display":"\\Theta_{\\mathrm{red}}","kind":"use","occurrences":[3403]},{"id":"symbol-66f2af2663fe43f3","tex":"\\theta_x","aliases":"θ_x theta","display":"\\theta_x","kind":"use","occurrences":[3095]},{"id":"symbol-878cb89cae859a1d","tex":"\\Theta_{xy}","aliases":"Θ_xy Theta","display":"\\Theta_{xy}","kind":"use","occurrences":[3095,3096]},{"id":"symbol-6602dff3fef9759e","tex":"\\theta_y","aliases":"θ_y theta","display":"\\theta_y","kind":"use","occurrences":[3095,3113]},{"id":"symbol-4f43ba5cea859e71","tex":"\\Theta_\\Sigma","aliases":"Θ_Σ Theta Sigma","display":"\\Theta_\\Sigma","kind":"use","occurrences":[3342]},{"id":"symbol-769672901a15b39c","tex":"\\iota_c","aliases":"ι_c iota","display":"\\iota_c","kind":"use","occurrences":[3353,3355]},{"id":"symbol-df590a2d29cde0af","tex":"\\iota_L","aliases":"ι_L iota","display":"\\iota_L","kind":"use","occurrences":[867,873,889,890,892]},{"id":"symbol-9a81ea7f53a2a389","tex":"\\iota_L^*","aliases":"ι_L^* iota","display":"\\iota_L^*","kind":"use","occurrences":[873,889,890]},{"id":"symbol-62f48d52cf8d451c","tex":"\\iota_\\eta","aliases":"ι_η iota eta","display":"\\iota_\\eta","kind":"use","occurrences":[3352]},{"id":"symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","occurrences":[269,270,828,841,847,857,860,1028,1034,1042,1062,1064,1068,1069,1070,1076,1079,1083,1085,1095,1126,1133,1134,1135,2985,3062,3064,3107,3279,3341,3345,3374,3397,3399,3401,3403]},{"id":"symbol-0fc1259c08b990e9","tex":"\\kappa^2","aliases":"κ^2 kappa","display":"\\kappa^2","kind":"use","occurrences":[1083]},{"id":"symbol-f2aa78ec120c23cd","tex":"\\kappa^{p-1}","aliases":"κ^p-1 kappa","display":"\\kappa^{p-1}","kind":"use","occurrences":[1095]},{"id":"symbol-f4d1001288e3b575","tex":"\\kappa_m","aliases":"κ_m kappa","display":"\\kappa_m","kind":"use","occurrences":[1133,1134]},{"id":"symbol-39929c2ef6bc800a","tex":"\\kappa_{m,n}","aliases":"κ_m,n kappa","display":"\\kappa_{m,n}","kind":"use","occurrences":[1134,1135]},{"id":"symbol-46fe9f573328c1ae","tex":"\\kappa_{mn}","aliases":"κ_mn kappa","display":"\\kappa_{mn}","kind":"use","occurrences":[1126]},{"id":"symbol-810ceeda3892b7a8","tex":"\\kappa_p","aliases":"κ_p kappa","display":"\\kappa_p","kind":"use","occurrences":[269,270,1064,1076,1079]},{"id":"symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","occurrences":[673,674,675,677,678,679,681,682,685,690,691,693,694,698,700,701,702,703,706,707,897,898,899,912,913,934,1020,1021,1280,1282,1283,1286,1288,1296,1297,2305,2306,2307,2308,2309,2310,2312,2315,2317,2318,2319,2320,2321,2322,2324,2325,2365,2366,2374,2375,2376,2377,2379,2380,2384,2386,2388,2389,2390,2397,2398,2399,2403,2407,2447,2448,2449,2450,2459,2460,2461,3233,3239,3243,3245,3295,3296,3298,3301,3348,3628,3629,3633,3640,3641,3643,3657]},{"id":"symbol-b20ef177d3714eb3","tex":"\\Lambda'","aliases":"Λ' Lambda","display":"\\Lambda'","kind":"use","occurrences":[3852,3860,3894]},{"id":"symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","occurrences":[702]},{"id":"symbol-2669d46cbfa6c78a","tex":"\\Lambda^3","aliases":"Λ^3 Lambda","display":"\\Lambda^3","kind":"use","occurrences":[3860]},{"id":"symbol-d356f71025f33224","tex":"\\lambda^i","aliases":"λ^i lambda","display":"\\lambda^i","kind":"use","occurrences":[3233,3239]},{"id":"symbol-9f563c9de566dc8a","tex":"\\lambda^{IJ}","aliases":"λ^IJ lambda","display":"\\lambda^{IJ}","kind":"use","occurrences":[3298]},{"id":"symbol-75946f9ee1ce0aa6","tex":"\\lambda^j","aliases":"λ^j lambda","display":"\\lambda^j","kind":"use","occurrences":[3243]},{"id":"symbol-8fa5b25f3a5291e2","tex":"\\Lambda^{\\rm TT}_{ij,kl}","aliases":"Λ^rm TT_ij,kl Lambda rm","display":"\\Lambda^{\\rm TT}_{ij,kl}","kind":"use","occurrences":[3781]},{"id":"symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","occurrences":[3643]},{"id":"symbol-129252f1935f1ee1","tex":"\\Lambda_h","aliases":"Λ_h Lambda","display":"\\Lambda_h","kind":"use","occurrences":[3164,3682,3683]},{"id":"symbol-8890f23babb77fdb","tex":"\\Lambda_j","aliases":"Λ_j Lambda","display":"\\Lambda_j","kind":"use","occurrences":[3943]},{"id":"symbol-75281f36da6b8ff7","tex":"\\lambda_n","aliases":"λ_n lambda","display":"\\lambda_n","kind":"use","occurrences":[2305,2306,2312,2318,2319,2322,2325,2365,2366,2377,2379,2384,2399,2448,2450]},{"id":"symbol-3d5f1cc7f79d67ce","tex":"\\lambda_n^2","aliases":"λ_n^2 lambda","display":"\\lambda_n^2","kind":"use","occurrences":[2306,2317,2380,2386]},{"id":"symbol-ecdf3ba0683d5eab","tex":"\\lambda_\\theta","aliases":"λ_θ lambda theta","display":"\\lambda_\\theta","kind":"use","occurrences":[3629,3641]},{"id":"symbol-deb6b1daac660ba3","tex":"\\lambda_\\tau","aliases":"λ_τ lambda tau","display":"\\lambda_\\tau","kind":"use","occurrences":[3628]},{"id":"symbol-c26c6e94e2af784f","tex":"\\mu^k","aliases":"μ^k mu","display":"\\mu^k","kind":"use","occurrences":[3243]},{"id":"symbol-2ffb97b3578a68df","tex":"\\mu_m","aliases":"μ_m mu","display":"\\mu_m","kind":"use","occurrences":[1133]},{"id":"symbol-72a42d4b5c7b8dbd","tex":"\\mu_{m,n}","aliases":"μ_m,n mu","display":"\\mu_{m,n}","kind":"use","occurrences":[1135]},{"id":"symbol-fe764fd9990b4bf7","tex":"\\mu_{mn}","aliases":"μ_mn mu","display":"\\mu_{mn}","kind":"use","occurrences":[1126]},{"id":"symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","occurrences":[427,428,429,1160,1166,1168,1174,1176,1188,1190,2132,2678,2682,2686,2698,2700,2701,2706,2707,3304,3363,3364,3366,3609]},{"id":"symbol-c773963d9aaa806d","tex":"\\nu_{\\rm emit}","aliases":"ν_rm emit nu rm","display":"\\nu_{\\rm emit}","kind":"use","occurrences":[3609]},{"id":"symbol-5f40876cacfe4bfb","tex":"\\nu_{\\rm obs}","aliases":"ν_rm obs nu rm","display":"\\nu_{\\rm obs}","kind":"use","occurrences":[3609]},{"id":"symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"use","occurrences":[3316,3317,3318,3319,3325]},{"id":"symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","occurrences":[2638,3164,3234,3239,3243,3245,3295,3296,3297,3298,3301,3754,3756,3757,3769,3770,3771,3793,3794,3795,3796,3797,3799,3801,3814,3819,3824]},{"id":"symbol-3b20e380f56b440c","tex":"\\xi^a","aliases":"ξ^a xi","display":"\\xi^a","kind":"use","occurrences":[3234,3239,3243,3245]},{"id":"symbol-cf3f0e1455e18ba2","tex":"\\xi^b","aliases":"ξ^b xi","display":"\\xi^b","kind":"use","occurrences":[3243]},{"id":"symbol-31dd9a2ca27f8848","tex":"\\Xi^\\ddagger","aliases":"Ξ^ddagger Xi ddagger","display":null,"kind":"use","occurrences":[3317,3319]},{"id":"symbol-59bb53c8543f7887","tex":"\\xi_a","aliases":"ξ_a xi","display":"\\xi_a","kind":"use","occurrences":[3793,3795,3799]},{"id":"symbol-273ca48755e332d5","tex":"\\xi_i","aliases":"ξ_i xi","display":"\\xi_i","kind":"use","occurrences":[3754,3769,3770]},{"id":"symbol-1deb1e97057da782","tex":"\\xi_j","aliases":"ξ_j xi","display":"\\xi_j","kind":"use","occurrences":[2638,3769,3770]},{"id":"symbol-d631fd84ebdad115","tex":"\\xi_j^+","aliases":"ξ_j^+ xi","display":"\\xi_j^+","kind":"use","occurrences":[2638]},{"id":"symbol-d0850d2b3c9c36b7","tex":"\\xi_j^-","aliases":"ξ_j^- xi","display":"\\xi_j^-","kind":"use","occurrences":[2638]},{"id":"symbol-430363506ce958cc","tex":"\\xi_k","aliases":"ξ_k xi","display":"\\xi_k","kind":"use","occurrences":[3771]},{"id":"symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","occurrences":[220,221,262,2263,2264,2275,2276,2280,3261,3263,3341,3342,3346,3379,3380,3781,3783,3784,3785,3940,3965,4048,4049,4050,4059,4061,4062,4063,4066,4068,4070,4071,4075,4077,4078,4084,4087,4090,4093,4094,4105,4131,4161]},{"id":"symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"use","occurrences":[3208,3209]},{"id":"symbol-d465d49b065aadd9","tex":"\\Pi'","aliases":"Π' Pi","display":"\\Pi'","kind":"use","occurrences":[4084]},{"id":"symbol-f6c04fa92bb658c0","tex":"\\Pi^2","aliases":"Π^2 Pi","display":"\\Pi^2","kind":"use","occurrences":[3784,4059,4075,4077,4078,4105,4161]},{"id":"symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","occurrences":[1345,1485,1833,1836]},{"id":"symbol-0c9f6eacc84d6080","tex":"\\pi^3","aliases":"π^3 pi","display":"\\pi^3","kind":"use","occurrences":[1915,1921]},{"id":"symbol-a23e3c8d3f2c871d","tex":"\\Pi_a","aliases":"Π_a Pi","display":"\\Pi_a","kind":"use","occurrences":[2263,2264,2275,2276,2280]},{"id":"symbol-6983098f86ea86cc","tex":"\\pi_a","aliases":"π_a pi","display":"\\pi_a","kind":"use","occurrences":[2194,2195,2196,2197,2200]},{"id":"symbol-9bf697610fc96b35","tex":"\\varpi_A","aliases":"ϖ_A varpi","display":"\\varpi_A","kind":"use","occurrences":[3208]},{"id":"symbol-eeab2b59b76e99ba","tex":"\\Pi_b","aliases":"Π_b Pi","display":"\\Pi_b","kind":"use","occurrences":[2280]},{"id":"symbol-15b2d314605b1489","tex":"\\pi_b","aliases":"π_b pi","display":"\\pi_b","kind":"use","occurrences":[2195]},{"id":"symbol-26ac1047af452808","tex":"\\Pi_{bc}","aliases":"Π_bc Pi","display":"\\Pi_{bc}","kind":"use","occurrences":[3379]},{"id":"symbol-744abce417454055","tex":"\\varpi_C","aliases":"ϖ_C varpi","display":"\\varpi_C","kind":"use","occurrences":[3209]},{"id":"symbol-7a72c1d968673c18","tex":"\\Pi_h","aliases":"Π_h Pi","display":"\\Pi_h","kind":"use","occurrences":[3940,3965,4050]},{"id":"symbol-5fe4316a85529609","tex":"\\Pi_{ij}","aliases":"Π_ij Pi","display":"\\Pi_{ij}","kind":"use","occurrences":[3781]},{"id":"symbol-bc3ccd852974b115","tex":"\\pi_{ij}","aliases":"π_ij pi","display":"\\pi_{ij}","kind":"use","occurrences":[3731,3735,3748,3749,3751,3761,3775]},{"id":"symbol-45ec2a0e8556ad15","tex":"\\Pi_{ik}","aliases":"Π_ik Pi","display":"\\Pi_{ik}","kind":"use","occurrences":[3781]},{"id":"symbol-9670b7adc5d4414b","tex":"\\Pi_{il}","aliases":"Π_il Pi","display":"\\Pi_{il}","kind":"use","occurrences":[3781]},{"id":"symbol-364fc2af609ec6e0","tex":"\\pi_{ji}","aliases":"π_ji pi","display":"\\pi_{ji}","kind":"use","occurrences":[3731]},{"id":"symbol-e203f46ce227724d","tex":"\\Pi_{jk}","aliases":"Π_jk Pi","display":"\\Pi_{jk}","kind":"use","occurrences":[3781]},{"id":"symbol-ea3a8b556b9dfae9","tex":"\\Pi_{jl}","aliases":"Π_jl Pi","display":"\\Pi_{jl}","kind":"use","occurrences":[3781]},{"id":"symbol-bbe3cb6f07086c40","tex":"\\pi_{kk}","aliases":"π_kk pi","display":"\\pi_{kk}","kind":"use","occurrences":[3734]},{"id":"symbol-66df9abdcaa5244e","tex":"\\Pi_{kl}","aliases":"Π_kl Pi","display":"\\Pi_{kl}","kind":"use","occurrences":[3781]},{"id":"symbol-fcdf8b96328dc85b","tex":"\\Pi_P","aliases":"Π_P Pi","display":"\\Pi_P","kind":"use","occurrences":[3263]},{"id":"symbol-1aedf1628657716c","tex":"\\Pi_P^2","aliases":"Π_P^2 Pi","display":"\\Pi_P^2","kind":"use","occurrences":[3261]},{"id":"symbol-fc4c702d6f1c7544","tex":"\\Pi_Q","aliases":"Π_Q Pi","display":"\\Pi_Q","kind":"use","occurrences":[3263]},{"id":"symbol-10341aa4e026fcc1","tex":"\\Pi_Q^2","aliases":"Π_Q^2 Pi","display":"\\Pi_Q^2","kind":"use","occurrences":[3261]},{"id":"symbol-d2eb2d4e3fc3398a","tex":"\\Pi_r","aliases":"Π_r Pi","display":"\\Pi_r","kind":"use","occurrences":[220,221,262]},{"id":"symbol-4d66a630bbdba50b","tex":"\\pi_{\\rm tr}","aliases":"π_rm tr pi rm","display":"\\pi_{\\rm tr}","kind":"use","occurrences":[3734,3742,3748,3749,3759,3775]},{"id":"symbol-b484755dd099904d","tex":"\\pi_\\chi","aliases":"π_χ pi chi","display":"\\pi_\\chi","kind":"use","occurrences":[3349]},{"id":"symbol-a49eb9b81d375c75","tex":"\\rho_{(x,\\mathbf d)}^n","aliases":"ρ_(x, d)^n rho mathbf","display":"\\rho_{(x,\\mathbf d)}^n","kind":"use","occurrences":[751]},{"id":"symbol-6fd3bc029e9c79bd","tex":"\\rho_+","aliases":"ρ_+ rho","display":"\\rho_+","kind":"use","occurrences":[2122,2177]},{"id":"symbol-80922e0e24a01913","tex":"\\rho_-","aliases":"ρ_- rho","display":"\\rho_-","kind":"use","occurrences":[2122,2178]},{"id":"symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","occurrences":[2122]},{"id":"symbol-cf67cfa9d97d052e","tex":"\\rho_{12}","aliases":"ρ_12 rho","display":"\\rho_{12}","kind":"use","occurrences":[2171]},{"id":"symbol-362dfb6c4b153cec","tex":"\\rho_A","aliases":"ρ_A rho","display":"\\rho_A","kind":"use","occurrences":[2136,2139]},{"id":"symbol-acbabae96acb9647","tex":"\\rho_{AB}","aliases":"ρ_AB rho","display":"\\rho_{AB}","kind":"use","occurrences":[2134,2135,2136,2139,2142]},{"id":"symbol-adbd7f547dc7bde2","tex":"\\rho_{ab}","aliases":"ρ_ab rho","display":"\\rho_{ab}","kind":"use","occurrences":[2121]},{"id":"symbol-f3e334246055738a","tex":"\\rho_{bb}","aliases":"ρ_bb rho","display":"\\rho_{bb}","kind":"use","occurrences":[2157]},{"id":"symbol-d9a79e9beba3ca3f","tex":"\\rho_C","aliases":"ρ_C rho","display":"\\rho_C","kind":"use","occurrences":[3491,3492]},{"id":"symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"use","occurrences":[3590,3593,3595]},{"id":"symbol-6a94deaf3c45a250","tex":"\\rho_{ij}","aliases":"ρ_ij rho","display":"\\rho_{ij}","kind":"use","occurrences":[2167,2168]},{"id":"symbol-c86a36ff5a44dcd8","tex":"\\rho_j","aliases":"ρ_j rho","display":"\\rho_j","kind":"use","occurrences":[2734,2737,2750]},{"id":"symbol-36466f74c3e41259","tex":"\\rho_j^n","aliases":"ρ_j^n rho","display":"\\rho_j^n","kind":"use","occurrences":[2648,2650,2653]},{"id":"symbol-18318974b1f97ee7","tex":"\\rho_j^{n+1}","aliases":"ρ_j^n+1 rho","display":"\\rho_j^{n+1}","kind":"use","occurrences":[2650,2654]},{"id":"symbol-18866b84545a9559","tex":"\\Sigma'","aliases":"Σ' Sigma","display":"\\Sigma'","kind":"use","occurrences":[908,911,918,921]},{"id":"symbol-b8a1bf1bab77ec11","tex":"\\sigma'","aliases":"σ' sigma","display":"\\sigma'","kind":"use","occurrences":[1018,4061,4074,4077,4087,4090,4095]},{"id":"symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"use","occurrences":[4108]},{"id":"symbol-c797b4151d255212","tex":"\\sigma^1","aliases":"σ^1 sigma","display":"\\sigma^1","kind":"use","occurrences":[2811]},{"id":"symbol-0819a6083d4c9117","tex":"\\sigma^3","aliases":"σ^3 sigma","display":"\\sigma^3","kind":"use","occurrences":[2811,3604]},{"id":"symbol-86570cf2b6053ba1","tex":"\\sigma^a","aliases":"σ^a sigma","display":"\\sigma^a","kind":"use","occurrences":[2812,2813]},{"id":"symbol-4cb413620fbda2f1","tex":"\\sigma^b","aliases":"σ^b sigma","display":"\\sigma^b","kind":"use","occurrences":[2813]},{"id":"symbol-59681cd776049191","tex":"\\Sigma_0","aliases":"Σ_0 Sigma","display":"\\Sigma_0","kind":"use","occurrences":[1003]},{"id":"symbol-d0331456e52105eb","tex":"\\sigma_0","aliases":"σ_0 sigma","display":"\\sigma_0","kind":"use","occurrences":[3385]},{"id":"symbol-429412ff1df255be","tex":"\\sigma_A","aliases":"σ_A sigma","display":"\\sigma_A","kind":"use","occurrences":[2142]},{"id":"symbol-782577c277fcf715","tex":"\\Sigma_D","aliases":"Σ_D Sigma","display":"\\Sigma_D","kind":"use","occurrences":[1014]},{"id":"symbol-a702a847774b0ed7","tex":"\\Sigma_h","aliases":"Σ_h Sigma","display":"\\Sigma_h","kind":"use","occurrences":[1001]},{"id":"symbol-e05b751fe47a304b","tex":"\\Sigma_I","aliases":"Σ_I Sigma","display":"\\Sigma_I","kind":"use","occurrences":[3272,3279]},{"id":"symbol-746ee34210cc6942","tex":"\\sigma_I","aliases":"σ_I sigma","display":"\\sigma_I","kind":"use","occurrences":[3341,3342,3345,3346]},{"id":"symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"use","occurrences":[2988,2994,2997,3386]},{"id":"symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","occurrences":[2990,2997]},{"id":"symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","occurrences":[1545,1552,1556,1558]},{"id":"symbol-d8dd014618e9e438","tex":"\\sigma_j^*","aliases":"σ_j^* sigma","display":"\\sigma_j^*","kind":"use","occurrences":[1556]},{"id":"symbol-d11986cbb3c06acc","tex":"\\sigma_K","aliases":"σ_K sigma","display":"\\sigma_K","kind":"use","occurrences":[3346]},{"id":"symbol-fa431dd60a582371","tex":"\\sigma_k","aliases":"σ_k sigma","display":"\\sigma_k","kind":"use","occurrences":[1556]},{"id":"symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","occurrences":[295,679,681]},{"id":"symbol-795b54231d0400ed","tex":"\\Sigma_{p,w}","aliases":"Σ_p,w Sigma","display":"\\Sigma_{p,w}","kind":"use","occurrences":[336,337]},{"id":"symbol-a63fa784be234859","tex":"\\Sigma_r","aliases":"Σ_r Sigma","display":"\\Sigma_r","kind":"use","occurrences":[1002]},{"id":"symbol-b5c9bb4ed7f4bb6f","tex":"\\Sigma_{r+1}","aliases":"Σ_r+1 Sigma","display":"\\Sigma_{r+1}","kind":"use","occurrences":[1002]},{"id":"symbol-7499d2052ab744b7","tex":"\\sigma_x","aliases":"σ_x sigma","display":"\\sigma_x","kind":"use","occurrences":[1010,1546,1554]},{"id":"symbol-ae6135878257b347","tex":"\\sigma_y","aliases":"σ_y sigma","display":"\\sigma_y","kind":"use","occurrences":[1546,1554]},{"id":"symbol-3fa147d5109a56b1","tex":"\\sigma_z","aliases":"σ_z sigma","display":"\\sigma_z","kind":"use","occurrences":[1010,1546,1554,1555]},{"id":"symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","occurrences":[3704,3715]},{"id":"symbol-7224d7199b93ec25","tex":"\\tau^3","aliases":"τ^3 tau","display":"\\tau^3","kind":"use","occurrences":[3703,3705,3708,3710,3713]},{"id":"symbol-35af929f24440334","tex":"\\tau_0","aliases":"τ_0 tau","display":"\\tau_0","kind":"use","occurrences":[3642,3643]},{"id":"symbol-db508c3ce80a5e43","tex":"\\tau_B","aliases":"τ_B tau","display":"\\tau_B","kind":"use","occurrences":[2142]},{"id":"symbol-2ed1bcaa9d2d3dfe","tex":"\\tau_i","aliases":"τ_i tau","display":"\\tau_i","kind":"use","occurrences":[2989,2991]},{"id":"symbol-93f600ceeb9d3b50","tex":"\\tau_N","aliases":"τ_N tau","display":"\\tau_N","kind":"use","occurrences":[3227,3230,3239]},{"id":"symbol-80ad2e368455b323","tex":"\\upsilon","aliases":"υ upsilon","display":"\\upsilon","kind":"use","occurrences":[3302,3304,3305,3349]},{"id":"symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","occurrences":[76,3270,3279,3285,3286,3288,3292,3293,3294,3341,3345,3519,3520,3521,3522,3525,3529,3533,3548,3550,4042,4043,4044,4048,4057,4064,4131,4132]},{"id":"symbol-c89e189652ea7cb2","tex":"\\Phi'","aliases":"Φ' Phi","display":"\\Phi'","kind":"use","occurrences":[3581,3582,3583,3585,4085]},{"id":"symbol-2c08a3f3921a7de9","tex":"\\varphi'","aliases":"φ' varphi","display":"\\varphi'","kind":"use","occurrences":[4044,4048,4064]},{"id":"symbol-c2baa4e22eef86e2","tex":"\\varphi'^2","aliases":"φ'^2 varphi","display":"\\varphi'^2","kind":"use","occurrences":[4044]},{"id":"symbol-aad33d2dbafe3856","tex":"\\Phi^2","aliases":"Φ^2 Phi","display":"\\Phi^2","kind":"use","occurrences":[3609,4059,4075,4077,4078,4105,4161]},{"id":"symbol-607d2c331667a908","tex":"\\varphi^{23}","aliases":"φ^23 varphi","display":"\\varphi^{23}","kind":"use","occurrences":[3292]},{"id":"symbol-2e9bba4479762fb1","tex":"\\varphi^{IJ}","aliases":"φ^IJ varphi","display":"\\varphi^{IJ}","kind":"use","occurrences":[3279,3286]},{"id":"symbol-defceda344190192","tex":"\\varphi^{KL}","aliases":"φ^KL varphi","display":"\\varphi^{KL}","kind":"use","occurrences":[3345]},{"id":"symbol-28b947267f902868","tex":"\\phi_0","aliases":"ϕ_0 phi","display":"\\phi_0","kind":"use","occurrences":[1169,1172]},{"id":"symbol-0dd8e14cc9a785dc","tex":"\\Phi_1","aliases":"Φ_1 Phi","display":"\\Phi_1","kind":"use","occurrences":[3422,3444,3451]},{"id":"symbol-7da0ece20570a181","tex":"\\phi_1","aliases":"ϕ_1 phi","display":"\\phi_1","kind":"use","occurrences":[1187]},{"id":"symbol-d33036383a860124","tex":"\\Phi_2","aliases":"Φ_2 Phi","display":"\\Phi_2","kind":"use","occurrences":[3444,3451]},{"id":"symbol-570f5323737bcd9d","tex":"\\Phi_{2m}","aliases":"Φ_2m Phi","display":"\\Phi_{2m}","kind":"use","occurrences":[3422]},{"id":"symbol-8cc9e6674048bfda","tex":"\\Phi_e'","aliases":"Φ_e' Phi","display":"\\Phi_e'","kind":"use","occurrences":[3897,3899]},{"id":"symbol-43812a1e322e8ca5","tex":"\\varphi_{IJ}","aliases":"φ_IJ varphi","display":"\\varphi_{IJ}","kind":"use","occurrences":[3270,3279,3341,3345]},{"id":"symbol-d916d96a3f8d422a","tex":"\\varphi_j","aliases":"φ_j varphi","display":"\\varphi_j","kind":"use","occurrences":[3519,3520,3522]},{"id":"symbol-9df5dbbaa7b552c8","tex":"\\varphi_{j+1}","aliases":"φ_j+1 varphi","display":"\\varphi_{j+1}","kind":"use","occurrences":[3520]},{"id":"symbol-37d36f4d52c9812c","tex":"\\varphi_j^2","aliases":"φ_j^2 varphi","display":"\\varphi_j^2","kind":"use","occurrences":[3521,3525,3533]},{"id":"symbol-2dc37ac785ef60ce","tex":"\\Phi_j^T","aliases":"Φ_j^T Phi","display":"\\Phi_j^T","kind":"use","occurrences":[3427]},{"id":"symbol-694cee5f94c55798","tex":"\\varphi_{KL}","aliases":"φ_KL varphi","display":"\\varphi_{KL}","kind":"use","occurrences":[3345]},{"id":"symbol-decd81e7725d5b31","tex":"\\phi_m","aliases":"ϕ_m phi","display":"\\phi_m","kind":"use","occurrences":[1170,1185]},{"id":"symbol-0aaab945af7d0743","tex":"\\phi_{m/p}","aliases":"ϕ_m/p phi","display":"\\phi_{m/p}","kind":"use","occurrences":[1170]},{"id":"symbol-72ec78a83e15672c","tex":"\\phi_n","aliases":"ϕ_n phi","display":"\\phi_n","kind":"use","occurrences":[1169,1170,1178,1183,1185]},{"id":"symbol-abd2409cfb8360f5","tex":"\\phi_{pn}","aliases":"ϕ_pn phi","display":"\\phi_{pn}","kind":"use","occurrences":[1170,1183,1185]},{"id":"symbol-e5500c68c310f9c1","tex":"\\Phi_{\\rm emit}","aliases":"Φ_rm emit Phi rm","display":"\\Phi_{\\rm emit}","kind":"use","occurrences":[3609]},{"id":"symbol-531add35ef892f7f","tex":"\\Phi_{\\rm obs}","aliases":"Φ_rm obs Phi rm","display":"\\Phi_{\\rm obs}","kind":"use","occurrences":[3609]},{"id":"symbol-0c73fde37e3f6f1b","tex":"\\phi_{\\rm phys}","aliases":"ϕ_rm phys phi rm","display":"\\phi_{\\rm phys}","kind":"use","occurrences":[4042]},{"id":"symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"use","occurrences":[3554,3558]},{"id":"symbol-f5c9f73ef8b8d6e6","tex":"\\phi_{tt}","aliases":"ϕ_tt phi","display":"\\phi_{tt}","kind":"use","occurrences":[3556,3559,3569]},{"id":"symbol-121c0f11eac3d462","tex":"\\phi_x","aliases":"ϕ_x phi","display":"\\phi_x","kind":"use","occurrences":[3554,3558]},{"id":"symbol-c4279ca52108141d","tex":"\\varphi_x","aliases":"φ_x varphi","display":"\\varphi_x","kind":"use","occurrences":[3550]},{"id":"symbol-418fd32d4d04d557","tex":"\\phi_x^*","aliases":"ϕ_x^* phi","display":"\\phi_x^*","kind":"use","occurrences":[2624]},{"id":"symbol-e9adf0621d8b11bc","tex":"\\phi_{xx}","aliases":"ϕ_xx phi","display":"\\phi_{xx}","kind":"use","occurrences":[3556,3560,3569]},{"id":"symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","occurrences":[769,770,771,1104,1105,2757,2758,2761,3302,3304,3305,3306,3308,3309,3349,3353,3388,3562,3563,3565,3566,3567,3572,3573,3576,4132]},{"id":"symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","occurrences":[3304]},{"id":"symbol-973edae78b5c5539","tex":"\\chi^{IJ}","aliases":"χ^IJ chi","display":"\\chi^{IJ}","kind":"use","occurrences":[3305,3309]},{"id":"symbol-0bbd4413777b414e","tex":"\\chi_A","aliases":"χ_A chi","display":"\\chi_A","kind":"use","occurrences":[3388]},{"id":"symbol-c2171a0d6968e6fe","tex":"\\chi_j","aliases":"χ_j chi","display":"\\chi_j","kind":"use","occurrences":[1104,1105,2757,2758,2761]},{"id":"symbol-04fa9c1343879b99","tex":"\\chi_{j+1}","aliases":"χ_j+1 chi","display":"\\chi_{j+1}","kind":"use","occurrences":[1104,2758]},{"id":"symbol-660c5fbfabd67d33","tex":"\\chi_{j-1}","aliases":"χ_j-1 chi","display":"\\chi_{j-1}","kind":"use","occurrences":[1104]},{"id":"symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","occurrences":[655,656,2144,2150,2913,2929,2931,2974,3390,3391,3392,3393,3402,3403,3405,3407,3408,3409]},{"id":"symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","occurrences":[2783]},{"id":"symbol-86694f2d67d3ff7b","tex":"\\Psi^\\dagger","aliases":"Ψ^† Psi dagger","display":"\\Psi^\\dagger","kind":"use","occurrences":[3391,3392,3393,3402,3403,3405]},{"id":"symbol-674719b5c71f505f","tex":"\\psi^\\dagger","aliases":"ψ^† psi dagger","display":"\\psi^\\dagger","kind":"use","occurrences":[3383,3387,3391,3392]},{"id":"symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"use","occurrences":[474]},{"id":"symbol-0e5ee24e05fc4a76","tex":"\\psi_0^{(h)}","aliases":"ψ_0^(h) psi","display":"\\psi_0^{(h)}","kind":"use","occurrences":[162]},{"id":"symbol-7279e6042a44c457","tex":"\\Psi_A","aliases":"Ψ_A Psi","display":"\\Psi_A","kind":"use","occurrences":[3393]},{"id":"symbol-1d16d5974f691ea4","tex":"\\psi_A","aliases":"ψ_A psi","display":"\\psi_A","kind":"use","occurrences":[1985]},{"id":"symbol-c2b61b1b4b58bf76","tex":"\\psi_B","aliases":"ψ_B psi","display":"\\psi_B","kind":"use","occurrences":[1986]},{"id":"symbol-2d48c69bc1aecc34","tex":"\\psi_b","aliases":"ψ_b psi","display":"\\psi_b","kind":"use","occurrences":[2016,2017,2022,2024,2032]},{"id":"symbol-098b2a09651c388e","tex":"\\Psi_B^\\dagger","aliases":"Ψ_B^† Psi dagger","display":"\\Psi_B^\\dagger","kind":"use","occurrences":[3393]},{"id":"symbol-f2e39e3d75403e53","tex":"\\psi_j","aliases":"ψ_j psi","display":"\\psi_j","kind":"use","occurrences":[2647,2721,2725,2726,2730,2734,2739,2741,2758,2762]},{"id":"symbol-58aa479f4a87b2a0","tex":"\\psi_{j+1}","aliases":"ψ_j+1 psi","display":"\\psi_{j+1}","kind":"use","occurrences":[2725,2730,2731,2882]},{"id":"symbol-d242c66daa18da63","tex":"\\psi_{j-1}","aliases":"ψ_j-1 psi","display":"\\psi_{j-1}","kind":"use","occurrences":[2730]},{"id":"symbol-c796ff8e7977df39","tex":"\\psi_{j-1}^*","aliases":"ψ_j-1^* psi","display":"\\psi_{j-1}^*","kind":"use","occurrences":[2741]},{"id":"symbol-6507e649e586c341","tex":"\\psi_j^*","aliases":"ψ_j^* psi","display":"\\psi_j^*","kind":"use","occurrences":[2725,2726,2730,2731,2734,2738,2739]},{"id":"symbol-2722d43d565a24a4","tex":"\\psi_j^n","aliases":"ψ_j^n psi","display":"\\psi_j^n","kind":"use","occurrences":[2632,2638,2648]},{"id":"symbol-1c9c1e667b6f998c","tex":"\\psi_j^{n+1}","aliases":"ψ_j^n+1 psi","display":"\\psi_j^{n+1}","kind":"use","occurrences":[2638]},{"id":"symbol-a411276b17f3a178","tex":"\\psi_{m,a,q}","aliases":"ψ_m,a,q psi","display":"\\psi_{m,a,q}","kind":"use","occurrences":[468]},{"id":"symbol-d9e406735fd906a4","tex":"\\psi_n","aliases":"ψ_n psi","display":"\\psi_n","kind":"use","occurrences":[1040]},{"id":"symbol-e0b43fdc9ef3ba09","tex":"\\psi_{n+1}","aliases":"ψ_n+1 psi","display":"\\psi_{n+1}","kind":"use","occurrences":[1040]},{"id":"symbol-abfa54f18e6ebc6d","tex":"\\psi_{n-1}","aliases":"ψ_n-1 psi","display":"\\psi_{n-1}","kind":"use","occurrences":[1040]},{"id":"symbol-bd8f16f7432ef794","tex":"\\psi_{pj}","aliases":"ψ_pj psi","display":"\\psi_{pj}","kind":"use","occurrences":[1105]},{"id":"symbol-1d54446a517f972c","tex":"\\psi_r","aliases":"ψ_r psi","display":"\\psi_r","kind":"use","occurrences":[821,822,824,825]},{"id":"symbol-08cf7f88c5994a7c","tex":"\\psi_s","aliases":"ψ_s psi","display":"\\psi_s","kind":"use","occurrences":[819,822,824,825]},{"id":"symbol-bc736e8ef3361342","tex":"\\psi_x","aliases":"ψ_x psi","display":"\\psi_x","kind":"use","occurrences":[2603,2608,2617,2624,2626,2803,2812,2816]},{"id":"symbol-8222a2a1e93111e6","tex":"\\psi_{x+\\widehat\\mu}","aliases":"ψ_x+widehatμ psi widehat mu","display":"\\psi_{x+\\widehat\\mu}","kind":"use","occurrences":[2603,2609,2617,2626,2806]},{"id":"symbol-a5b2d2df696a3716","tex":"\\psi_{x+\\widehat\\mu}^g","aliases":"ψ_x+widehatμ^g psi widehat mu","display":"\\psi_{x+\\widehat\\mu}^g","kind":"use","occurrences":[2609]},{"id":"symbol-247280f36d547b56","tex":"\\psi_{x-\\widehat\\mu}","aliases":"ψ_x-widehatμ psi widehat mu","display":"\\psi_{x-\\widehat\\mu}","kind":"use","occurrences":[2617]},{"id":"symbol-243488011205ccd6","tex":"\\psi_x^*","aliases":"ψ_x^* psi","display":"\\psi_x^*","kind":"use","occurrences":[2626,2806,2812,2816]},{"id":"symbol-1dda4f82a9e3e86f","tex":"\\psi_x^g","aliases":"ψ_x^g psi","display":"\\psi_x^g","kind":"use","occurrences":[2608]},{"id":"symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","occurrences":[3576]},{"id":"symbol-50cc5e20a1c2c14c","tex":"\\Omega^{0i}","aliases":"Ω^0i Omega","display":"\\Omega^{0i}","kind":"use","occurrences":[3372]},{"id":"symbol-768389d483cf5935","tex":"\\omega^I","aliases":"ω^I omega","display":"\\omega^I","kind":"use","occurrences":[3272]},{"id":"symbol-a8ecdd64f9da9bae","tex":"\\Omega^{IJ}","aliases":"Ω^IJ Omega","display":"\\Omega^{IJ}","kind":"use","occurrences":[3337,3342,3374]},{"id":"symbol-021aea4873729e82","tex":"\\omega^{IJ}","aliases":"ω^IJ omega","display":"\\omega^{IJ}","kind":"use","occurrences":[3268,3272,3275,3337]},{"id":"symbol-8e4eeccd528d4197","tex":"\\Omega^{jk}","aliases":"Ω^jk Omega","display":"\\Omega^{jk}","kind":"use","occurrences":[3373]},{"id":"symbol-d0e92a3d0b375b2d","tex":"\\omega^{KJ}","aliases":"ω^KJ omega","display":"\\omega^{KJ}","kind":"use","occurrences":[3272]},{"id":"symbol-0fc4543680ca1902","tex":"\\omega_+","aliases":"ω_+ omega","display":"\\omega_+","kind":"use","occurrences":[3573]},{"id":"symbol-0837ec06cd7132fd","tex":"\\omega_+'","aliases":"ω_+' omega","display":"\\omega_+'","kind":"use","occurrences":[3573]},{"id":"symbol-cfeb9eb96e89714e","tex":"\\Omega_C","aliases":"Ω_C Omega","display":"\\Omega_C","kind":"use","occurrences":[3212,3213]},{"id":"symbol-ff45fb79f7e1cb1e","tex":"\\Omega_F","aliases":"Ω_F Omega","display":"\\Omega_F","kind":"use","occurrences":[3209,3210,3211]},{"id":"symbol-a1bb204421e927bd","tex":"\\Omega_F^{(\\ge3)}","aliases":"Ω_F^(≥3) Omega ge","display":"\\Omega_F^{(\\ge3)}","kind":"use","occurrences":[3209]},{"id":"symbol-f954fc517876491c","tex":"\\omega_h","aliases":"ω_h omega","display":"\\omega_h","kind":"use","occurrences":[3789]},{"id":"symbol-f8cdd4391d2292b5","tex":"\\omega_h^2","aliases":"ω_h^2 omega","display":"\\omega_h^2","kind":"use","occurrences":[3786,3789]},{"id":"symbol-7c5e43c1625ae71d","tex":"\\Omega_{ij}","aliases":"Ω_ij Omega","display":"\\Omega_{ij}","kind":"use","occurrences":[3080]},{"id":"symbol-10ac20847e543481","tex":"\\Omega_{\\mathrm{int}}","aliases":"Ω_int Omega mathrm","display":"\\Omega_{\\mathrm{int}}","kind":"use","occurrences":[3349,3350]},{"id":"symbol-17643fc6afa4dc8b","tex":"\\omega_L","aliases":"ω_L omega","display":"\\omega_L","kind":"use","occurrences":[1205]},{"id":"symbol-9832b5c4fc1bb7e1","tex":"\\omega_L^{(n-j)\\ell}","aliases":"ω_L^(n-j)ℓ omega ell","display":"\\omega_L^{(n-j)\\ell}","kind":"use","occurrences":[1212]},{"id":"symbol-f917b6e617bb2331","tex":"\\omega_L^{-j\\ell}","aliases":"ω_L^-jℓ omega ell","display":"\\omega_L^{-j\\ell}","kind":"use","occurrences":[1208,1243,1249,1324,1330]},{"id":"symbol-c44b43b4aea54580","tex":"\\omega_L^\\ell","aliases":"ω_L^ℓ omega ell","display":"\\omega_L^\\ell","kind":"use","occurrences":[1208,1243,1249,1330,1342]},{"id":"symbol-a102bfba12f83124","tex":"\\omega_p^{(e-d)\\ell}","aliases":"ω_p^(e-d)ℓ omega ell","display":"\\omega_p^{(e-d)\\ell}","kind":"use","occurrences":[211]},{"id":"symbol-c3a30be014515298","tex":"\\omega_p^{(q'-q)d}","aliases":"ω_p^(q'-q)d omega","display":"\\omega_p^{(q'-q)d}","kind":"use","occurrences":[170,472]},{"id":"symbol-db0401080bf91c98","tex":"\\omega_p^{-d\\ell}","aliases":"ω_p^-dℓ omega ell","display":"\\omega_p^{-d\\ell}","kind":"use","occurrences":[209]},{"id":"symbol-61cdd9fbde51e8b1","tex":"\\omega_p^d","aliases":"ω_p^d omega","display":"\\omega_p^d","kind":"use","occurrences":[207]},{"id":"symbol-58cd7d70a3503669","tex":"\\omega_p^{qd}","aliases":"ω_p^qd omega","display":"\\omega_p^{qd}","kind":"use","occurrences":[165,468,469]},{"id":"symbol-59875dc6e7f002db","tex":"\\Omega_{xy}","aliases":"Ω_xy Omega","display":"\\Omega_{xy}","kind":"use","occurrences":[2952,2963,2964,2972,2985,3096,3113]},{"id":"symbol-e2b7ed9de8f0f49b","tex":"\\Omega_{xz}","aliases":"Ω_xz Omega","display":"\\Omega_{xz}","kind":"use","occurrences":[2963,2964,2985]},{"id":"symbol-994125d019c8421f","tex":"\\Omega_{yz}","aliases":"Ω_yz Omega","display":"\\Omega_{yz}","kind":"use","occurrences":[2963,2964,2985]},{"id":"symbol-6aa524e36cbbf167","tex":"\\omega_\\perp","aliases":"ω_⊥ omega perp","display":"\\omega_\\perp","kind":"use","occurrences":[3340]},{"id":"symbol-18f90c283b071b01","tex":"\\omega_\\perp^{IJ}","aliases":"ω_⊥^IJ omega perp","display":"\\omega_\\perp^{IJ}","kind":"use","occurrences":[3337]},{"id":"symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","occurrences":[1349,1351,1353,3383,3387,3391,3392,3393,3402,3403,3405]},{"id":"symbol-2c05d83b370cdcc7","tex":"\\ell_x","aliases":"ℓ_x ell","display":"\\ell_x","kind":"use","occurrences":[2946,2951]},{"id":"symbol-a8e35b84dc016d32","tex":"\\ell_y","aliases":"ℓ_y ell","display":"\\ell_y","kind":"use","occurrences":[2947,2951]},{"id":"symbol-7bbf65f26182fbdc","tex":"\\ell_z","aliases":"ℓ_z ell","display":"\\ell_z","kind":"use","occurrences":[2948]},{"id":"symbol-cb520b9f36cbee35","tex":"\\ell_\\pm","aliases":"ℓ_± ell pm","display":"\\ell_\\pm","kind":"use","occurrences":[4108]},{"id":"symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","occurrences":[223]},{"id":"symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","occurrences":[1513,1565,1566,2652,2693,2727,2728,2729,2732,2753,2756,2778,2801,3076,3077,3079,3083,3085,3100,3102,3107,3112,3116,3118,3221,3230,3240,3281,3282,3304,3366,3405,3406,3408,3482,3489,3528,3538,3548,3558,3613,3625,3631,3637,3654,3725,3796,3797,3850,3955,3977,4063,4079,4080,4087,4108,4131]},{"id":"symbol-d74afdd2e29665d0","tex":"\\partial_+","aliases":"∂_+ partial","display":"\\partial_+","kind":"use","occurrences":[2652]},{"id":"symbol-59508e0d19e81bbe","tex":"\\partial_-","aliases":"∂_- partial","display":"\\partial_-","kind":"use","occurrences":[2652]},{"id":"symbol-3546b807788704d6","tex":"\\partial_a","aliases":"∂_a partial","display":"\\partial_a","kind":"use","occurrences":[3221,3230,3405,3406,3408]},{"id":"symbol-5735f6c055adb0c0","tex":"\\partial_{A_j}","aliases":"∂_A_j partial","display":"\\partial_{A_j}","kind":"use","occurrences":[2732]},{"id":"symbol-e22e9fc616e6e066","tex":"\\partial_b","aliases":"∂_b partial","display":"\\partial_b","kind":"use","occurrences":[3366,3613]},{"id":"symbol-3e7cebb8f331465a","tex":"\\partial_I","aliases":"∂_I partial","display":"\\partial_I","kind":"use","occurrences":[3281,3282]},{"id":"symbol-c97379bf19e9e4d6","tex":"\\partial_i","aliases":"∂_i partial","display":"\\partial_i","kind":"use","occurrences":[3076,3077,3079,3083,3085]},{"id":"symbol-100dccaf595fc888","tex":"\\partial_J","aliases":"∂_J partial","display":"\\partial_J","kind":"use","occurrences":[3282]},{"id":"symbol-1cd2f6e5cfcfeb44","tex":"\\partial_j","aliases":"∂_j partial","display":"\\partial_j","kind":"use","occurrences":[1513,1565,1566,3079]},{"id":"symbol-e9a3eb10d7e3c2e0","tex":"\\partial_p","aliases":"∂_p partial","display":"\\partial_p","kind":"use","occurrences":[3796]},{"id":"symbol-168e83b92c806624","tex":"\\partial_q","aliases":"∂_q partial","display":"\\partial_q","kind":"use","occurrences":[3797]},{"id":"symbol-1a40a8de294e1b51","tex":"\\partial_R","aliases":"∂_R partial","display":"\\partial_R","kind":"use","occurrences":[3955]},{"id":"symbol-f6d8946150f4ab94","tex":"\\partial_r","aliases":"∂_r partial","display":"\\partial_r","kind":"use","occurrences":[4063,4079,4087,4108,4131]},{"id":"symbol-dd94c17e5e087649","tex":"\\partial_t","aliases":"∂_t partial","display":"\\partial_t","kind":"use","occurrences":[3528,3538,3977,4079,4080,4087,4108]},{"id":"symbol-d58378212f1d1d15","tex":"\\partial_x","aliases":"∂_x partial","display":"\\partial_x","kind":"use","occurrences":[2778,2801,3116,3548]},{"id":"symbol-605218e5b93e5d47","tex":"\\partial_{x_1}","aliases":"∂_x_1 partial","display":"\\partial_{x_1}","kind":"use","occurrences":[3482,3489]},{"id":"symbol-85be16485a8fd286","tex":"\\partial_{x_2}","aliases":"∂_x_2 partial","display":"\\partial_{x_2}","kind":"use","occurrences":[3482,3489]},{"id":"symbol-422f05688ea1c04e","tex":"\\partial_\\theta","aliases":"∂_θ partial theta","display":"\\partial_\\theta","kind":"use","occurrences":[3625,3631,3637]},{"id":"symbol-b8be09e8f717ffe2","tex":"\\partial_\\nu","aliases":"∂_ν partial nu","display":"\\partial_\\nu","kind":"use","occurrences":[3304]},{"id":"symbol-a2f34286c4589ec1","tex":"\\partial_\\tau","aliases":"∂_τ partial tau","display":"\\partial_\\tau","kind":"use","occurrences":[3625,3631,3637,3725]},{"id":"symbol-9fb77d52cdb67151","tex":"\\partial_\\tau^2","aliases":"∂_τ^2 partial tau","display":"\\partial_\\tau^2","kind":"use","occurrences":[3654]},{"id":"symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"use","occurrences":[3151,3155,3156,3424,3678,3682,3683,3766,3769]},{"id":"symbol-876cb452f8440f63","tex":"\\nabla_h","aliases":"∇_h nabla","display":"\\nabla_h","kind":"use","occurrences":[3766,3769]},{"id":"symbol-85722266da76cd67","tex":"\\nabla_P","aliases":"∇_P nabla","display":"\\nabla_P","kind":"use","occurrences":[3155,3678]},{"id":"symbol-2f12e7a302f43022","tex":"\\nabla_{P_A}","aliases":"∇_P_A nabla","display":"\\nabla_{P_A}","kind":"use","occurrences":[3151]},{"id":"symbol-a409f6974e1d4356","tex":"\\nabla_Q","aliases":"∇_Q nabla","display":"\\nabla_Q","kind":"use","occurrences":[3678]},{"id":"symbol-be9d7dca88bd31c4","tex":"\\nabla_U","aliases":"∇_U nabla","display":"\\nabla_U","kind":"use","occurrences":[3678]},{"id":"symbol-b1afd481338bf2d7","tex":"\\nabla_V","aliases":"∇_V nabla","display":"\\nabla_V","kind":"use","occurrences":[3678]},{"id":"symbol-346cb4e378e60b5d","tex":"\\nabla_Z","aliases":"∇_Z nabla","display":"\\nabla_Z","kind":"use","occurrences":[3156]},{"id":"symbol-27d5c72219127c6a","tex":"\\nabla_{Z^A}","aliases":"∇_Z^A nabla","display":"\\nabla_{Z^A}","kind":"use","occurrences":[3151]},{"id":"symbol-a75ba8bdaddfa75a","tex":"\\nabla_\\pi","aliases":"∇_π nabla pi","display":"\\nabla_\\pi","kind":"use","occurrences":[3766,3769]},{"id":"symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","occurrences":[248,528,678,748,869,907,1362,2021]},{"id":"symbol-6b6fc7fb58c7b90c","tex":"\\prod_{f\\subset\\partial c}","aliases":"∏_f⊂∂ c prod subset partial","display":"\\prod_{f\\subset\\partial c}","kind":"use","occurrences":[2693]},{"id":"symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","occurrences":[3230]},{"id":"symbol-9335981ae95d8e40","tex":"\\prod_{k=0}^{q-1}","aliases":"∏_k=0^q-1 prod","display":"\\prod_{k=0}^{q-1}","kind":"use","occurrences":[1221]},{"id":"symbol-05c5b21bb8d4781f","tex":"\\prod_{k=1}^{m}","aliases":"∏_k=1^m prod","display":"\\prod_{k=1}^{m}","kind":"use","occurrences":[2567,2581]},{"id":"symbol-8cbaa1215b56d5e0","tex":"\\prod_{p\\in\\mathcal P_{\\mathrm o}}","aliases":"∏_p∈ P_ o prod in mathcal mathrm","display":"\\prod_{p\\in\\mathcal P_{\\mathrm o}}","kind":"use","occurrences":[72,76]},{"id":"symbol-401f2450d734174a","tex":"\\prod_{r=1}^j","aliases":"∏_r=1^j prod","display":"\\prod_{r=1}^j","kind":"use","occurrences":[1308]},{"id":"symbol-b7bce99afe738aa6","tex":"\\prod_{r=1}^{L}","aliases":"∏_r=1^L prod","display":"\\prod_{r=1}^{L}","kind":"use","occurrences":[1221]},{"id":"symbol-cec653bd65c24ad4","tex":"\\sum_{\\mathbf d}","aliases":"∑_ d sum mathbf","display":"\\sum_{\\mathbf d}","kind":"use","occurrences":[300,301,307,312,314]},{"id":"symbol-dfa5b4b45388d395","tex":"\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}","aliases":"∑_ d∈{0,…,p-1}^w sum mathbf in ldots","display":"\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}","kind":"use","occurrences":[192]},{"id":"symbol-62016043dd92b660","tex":"\\sum_{\\|k\\|_\\infty\\le K}","aliases":"∑_ k _∞≤ K sum infty le","display":"\\sum_{\\|k\\|_\\infty\\le K}","kind":"use","occurrences":[3220]},{"id":"symbol-9682ef91352bf906","tex":"\\sum_{\\mathbf r}","aliases":"∑_ r sum mathbf","display":"\\sum_{\\mathbf r}","kind":"use","occurrences":[617,622]},{"id":"symbol-54d79a998f5bb042","tex":"\\sum_A","aliases":"∑_A sum","display":"\\sum_A","kind":"use","occurrences":[2049,3151,3153]},{"id":"symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","occurrences":[1955,2269]},{"id":"symbol-9068784c641ac3f6","tex":"\\sum_{a=0}^{p-1}","aliases":"∑_a=0^p-1 sum","display":"\\sum_{a=0}^{p-1}","kind":"use","occurrences":[1318]},{"id":"symbol-b7aa8d9329a85bf6","tex":"\\sum_{a=0}^{p^{r-1}-1}","aliases":"∑_a=0^p^r-1-1 sum","display":"\\sum_{a=0}^{p^{r-1}-1}","kind":"use","occurrences":[173]},{"id":"symbol-b1314006a37070ed","tex":"\\sum_{a\\in C(A)}","aliases":"∑_a∈ C(A) sum in","display":"\\sum_{a\\in C(A)}","kind":"use","occurrences":[1960,1962]},{"id":"symbol-c8d172fdcd28f72a","tex":"\\sum_{a\\in C_A}","aliases":"∑_a∈ C_A sum in","display":"\\sum_{a\\in C_A}","kind":"use","occurrences":[2045,2048]},{"id":"symbol-02071cad21612a9a","tex":"\\sum_{a\\in C_\\alpha}","aliases":"∑_a∈ C_α sum in alpha","display":"\\sum_{a\\in C_\\alpha}","kind":"use","occurrences":[2349,2350,2353,2531]},{"id":"symbol-10721c0e2add56b1","tex":"\\sum_{a\\notin C}","aliases":"∑_a∉ C sum notin","display":"\\sum_{a\\notin C}","kind":"use","occurrences":[2021]},{"id":"symbol-c7968355940f474b","tex":"\\sum_{A\\ne B}","aliases":"∑_A≠ B sum ne","display":"\\sum_{A\\ne B}","kind":"use","occurrences":[2176]},{"id":"symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","occurrences":[1996,1999,2003,2184,2239,2241,2248,2357,2813]},{"id":"symbol-c43729fd2f2f98cd","tex":"\\sum_{b,a}","aliases":"∑_b,a sum","display":"\\sum_{b,a}","kind":"use","occurrences":[1996,2184,2186,2189,2200,2202]},{"id":"symbol-6c78ad579d275dc7","tex":"\\sum_{b=0}^h","aliases":"∑_b=0^h sum","display":"\\sum_{b=0}^h","kind":"use","occurrences":[997]},{"id":"symbol-f5bff478efdb5579","tex":"\\sum_{b\\in A}","aliases":"∑_b∈ A sum in","display":"\\sum_{b\\in A}","kind":"use","occurrences":[1959]},{"id":"symbol-5a63562b0ab80881","tex":"\\sum_{b\\in C}","aliases":"∑_b∈ C sum in","display":"\\sum_{b\\in C}","kind":"use","occurrences":[2021]},{"id":"symbol-8f2acc418631442b","tex":"\\sum_{b\\in C(B)}","aliases":"∑_b∈ C(B) sum in","display":"\\sum_{b\\in C(B)}","kind":"use","occurrences":[1961,1962]},{"id":"symbol-4ea9c40bbaa4e5c2","tex":"\\sum_{b\\in C_B}","aliases":"∑_b∈ C_B sum in","display":"\\sum_{b\\in C_B}","kind":"use","occurrences":[2048,2174]},{"id":"symbol-b85c57e31dd1fb9d","tex":"\\sum_{b\\in C_\\beta}","aliases":"∑_b∈ C_β sum in beta","display":"\\sum_{b\\in C_\\beta}","kind":"use","occurrences":[2353]},{"id":"symbol-2046ba257a075646","tex":"\\sum_{b\\ne a}","aliases":"∑_b≠ a sum ne","display":"\\sum_{b\\ne a}","kind":"use","occurrences":[2156]},{"id":"symbol-996c298de46e2347","tex":"\\sum_{b\\ne i}","aliases":"∑_b≠ i sum ne","display":"\\sum_{b\\ne i}","kind":"use","occurrences":[2165]},{"id":"symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"use","occurrences":[2523,2524]},{"id":"symbol-581e46df03c6e7a6","tex":"\\sum_d","aliases":"∑_d sum","display":"\\sum_d","kind":"use","occurrences":[186,187,214,469,472,970,978]},{"id":"symbol-ed7bfe0fc25b6ba1","tex":"\\sum_{d=0}^{p-1}","aliases":"∑_d=0^p-1 sum","display":"\\sum_{d=0}^{p-1}","kind":"use","occurrences":[165,170,188,392,468,972]},{"id":"symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","occurrences":[3729,3773,3786,3789]},{"id":"symbol-a9b279e4a6b63e78","tex":"\\sum_{i=0}^{M-1}","aliases":"∑_i=0^M-1 sum","display":"\\sum_{i=0}^{M-1}","kind":"use","occurrences":[3590,3598]},{"id":"symbol-76c8d38e858879d4","tex":"\\sum_{i=1}^{M-1}","aliases":"∑_i=1^M-1 sum","display":"\\sum_{i=1}^{M-1}","kind":"use","occurrences":[3590]},{"id":"symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","occurrences":[1210,1270,1283,1284,1300,1327,1334,1513,1581,1593,1598,1743,1749,2433,2725,2726,2734,2751,2754,2756,3229,3521,3527,3528,3546,3662,3675]},{"id":"symbol-c57a71fec0503f9d","tex":"\\sum_{j=0}^{L-1}","aliases":"∑_j=0^L-1 sum","display":"\\sum_{j=0}^{L-1}","kind":"use","occurrences":[1249]},{"id":"symbol-7d1d47c0c0a47e08","tex":"\\sum_{j=1}^N","aliases":"∑_j=1^N sum","display":"\\sum_{j=1}^N","kind":"use","occurrences":[3041,3043]},{"id":"symbol-b9b31c7c60715e60","tex":"\\sum_{j=a}^{b}","aliases":"∑_j=a^b sum","display":"\\sum_{j=a}^{b}","kind":"use","occurrences":[2750]},{"id":"symbol-c30312fae2625b14","tex":"\\sum_{j=r}^s","aliases":"∑_j=r^s sum","display":"\\sum_{j=r}^s","kind":"use","occurrences":[3541]},{"id":"symbol-c233b15936db2d27","tex":"\\sum_{j=x,y,z}","aliases":"∑_j=x,y,z sum","display":"\\sum_{j=x,y,z}","kind":"use","occurrences":[1565,1701,1702,1865]},{"id":"symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","occurrences":[1604,2139,3740,3777,3778]},{"id":"symbol-a8b3533c8f1752a0","tex":"\\sum_{k=0}^{N}","aliases":"∑_k=0^N sum","display":"\\sum_{k=0}^{N}","kind":"use","occurrences":[1802]},{"id":"symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","occurrences":[1164]},{"id":"symbol-d163c0162318851c","tex":"\\sum_{k=0}^\\infty","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^\\infty","kind":"use","occurrences":[1254]},{"id":"symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","occurrences":[1603]},{"id":"symbol-b5d62fa15f1a3c0e","tex":"\\sum_{k=1}^{\\infty}","aliases":"∑_k=1^∞ sum infty","display":"\\sum_{k=1}^{\\infty}","kind":"use","occurrences":[1772,1874]},{"id":"symbol-baa8f7877711f9a7","tex":"\\sum_{k=n}^{m-1}","aliases":"∑_k=n^m-1 sum","display":"\\sum_{k=n}^{m-1}","kind":"use","occurrences":[3126]},{"id":"symbol-f26710096ba8e9ae","tex":"\\sum_{k=n}^{\\infty}","aliases":"∑_k=n^∞ sum infty","display":"\\sum_{k=n}^{\\infty}","kind":"use","occurrences":[3124]},{"id":"symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","occurrences":[1483,1485]},{"id":"symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","occurrences":[505,531,592]},{"id":"symbol-fe8ca430ca97a131","tex":"\\sum_{m=0}^M","aliases":"∑_m=0^M sum","display":"\\sum_{m=0}^M","kind":"use","occurrences":[453]},{"id":"symbol-668e17fed94fbb22","tex":"\\sum_{m=0}^{M}","aliases":"∑_m=0^M sum","display":"\\sum_{m=0}^{M}","kind":"use","occurrences":[542]},{"id":"symbol-187f645f69d06fb0","tex":"\\sum_{m=0}^{\\infty}","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^{\\infty}","kind":"use","occurrences":[461,3650]},{"id":"symbol-06ee679c4f230fdb","tex":"\\sum_{m>M}","aliases":"∑_m>M sum","display":"\\sum_{m>M}","kind":"use","occurrences":[479,532,552,553]},{"id":"symbol-abaf68bfbf4ecdd4","tex":"\\sum_{m\\leq M}","aliases":"∑_m≤ M sum leq","display":"\\sum_{m\\leq M}","kind":"use","occurrences":[556]},{"id":"symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"use","occurrences":[476,494,526,528,566,571]},{"id":"symbol-97951b3ae8950fca","tex":"\\sum_{m\\geq1}","aliases":"∑_m≥1 sum geq","display":"\\sum_{m\\geq1}","kind":"use","occurrences":[593]},{"id":"symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","occurrences":[1357,1417,1426,1432,1470,1472,1474,1476,1483,1491,1493,1502,2306,2319,2386]},{"id":"symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","occurrences":[1206,1263,1329]},{"id":"symbol-eced359e9f8ad685","tex":"\\sum_{n=Q+1}^\\infty","aliases":"∑_n=Q+1^∞ sum infty","display":"\\sum_{n=Q+1}^\\infty","kind":"use","occurrences":[1254]},{"id":"symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","occurrences":[2317,2407]},{"id":"symbol-683d9edfe17c29d2","tex":"\\sum_{n\\in S}","aliases":"∑_n∈ S sum in","display":"\\sum_{n\\in S}","kind":"use","occurrences":[1530,1531,1670,1684,1691,1694,1696,1706,1707,1766,1780,1900,1902,1915,1925,1933]},{"id":"symbol-89b99765eefcebef","tex":"\\sum_{n\\in\\Z^3}","aliases":"∑_n∈Z^3 sum in Z","display":null,"kind":"use","occurrences":[1356,1501]},{"id":"symbol-9adcdaef75b599bd","tex":"\\sum_{n\\notin S}","aliases":"∑_n∉ S sum notin","display":"\\sum_{n\\notin S}","kind":"use","occurrences":[1362]},{"id":"symbol-f7c05ed299cc8fcf","tex":"\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}","aliases":"∑_n≡ j (rm mod L) sum equiv rm","display":"\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}","kind":"use","occurrences":[1324]},{"id":"symbol-165ec26aa834867f","tex":"\\sum_{n\\leq N}","aliases":"∑_n≤ N sum leq","display":"\\sum_{n\\leq N}","kind":"use","occurrences":[2316]},{"id":"symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","occurrences":[2401]},{"id":"symbol-e5495c300dba1b9b","tex":"\\sum_{q=0}^\\infty","aliases":"∑_q=0^∞ sum infty","display":"\\sum_{q=0}^\\infty","kind":"use","occurrences":[1210,1216]},{"id":"symbol-a21758814a18a18d","tex":"\\sum_{q=0}^{\\infty}","aliases":"∑_q=0^∞ sum infty","display":"\\sum_{q=0}^{\\infty}","kind":"use","occurrences":[3661]},{"id":"symbol-125fff8908feab30","tex":"\\sum_{q=1}^{p-1}","aliases":"∑_q=1^p-1 sum","display":"\\sum_{q=1}^{p-1}","kind":"use","occurrences":[173]},{"id":"symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","occurrences":[799,807,812,823]},{"id":"symbol-096f1afa7adcb4ee","tex":"\\sum_{r(a)=\\beta}","aliases":"∑_r(a)=β sum beta","display":"\\sum_{r(a)=\\beta}","kind":"use","occurrences":[2250,2251]},{"id":"symbol-83dad0d78f66f949","tex":"\\sum_{r,s=0}^{p-1}","aliases":"∑_r,s=0^p-1 sum","display":"\\sum_{r,s=0}^{p-1}","kind":"use","occurrences":[806]},{"id":"symbol-c7b90c949ac04119","tex":"\\sum_{r=0}^{3}","aliases":"∑_r=0^3 sum","display":"\\sum_{r=0}^{3}","kind":"use","occurrences":[1525]},{"id":"symbol-14001847ba6d9b2f","tex":"\\sum_{r=0}^h","aliases":"∑_r=0^h sum","display":"\\sum_{r=0}^h","kind":"use","occurrences":[136,150,257,262,274,720]},{"id":"symbol-0d24bc4aca4d845c","tex":"\\sum_{r=0}^{k}","aliases":"∑_r=0^k sum","display":"\\sum_{r=0}^{k}","kind":"use","occurrences":[1889]},{"id":"symbol-e1f302dc5a0699e7","tex":"\\sum_{r=0}^{k-1}","aliases":"∑_r=0^k-1 sum","display":"\\sum_{r=0}^{k-1}","kind":"use","occurrences":[1768]},{"id":"symbol-4a8ae622c77aac5b","tex":"\\sum_{r=0}^m","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^m","kind":"use","occurrences":[152]},{"id":"symbol-4b5b1fcf31d6f6c3","tex":"\\sum_{r=1}^b","aliases":"∑_r=1^b sum","display":"\\sum_{r=1}^b","kind":"use","occurrences":[989]},{"id":"symbol-9f756cf70c299a69","tex":"\\sum_{r=1}^h","aliases":"∑_r=1^h sum","display":"\\sum_{r=1}^h","kind":"use","occurrences":[276,997]},{"id":"symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","occurrences":[2576]},{"id":"symbol-8615333b8546053a","tex":"\\sum_{r=m+1}^h","aliases":"∑_r=m+1^h sum","display":"\\sum_{r=m+1}^h","kind":"use","occurrences":[153]},{"id":"symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","occurrences":[812,819,822,824]},{"id":"symbol-c0815e59aa813271","tex":"\\sum_{\\substack{b\\in C_\\beta\\\\a\\in C_\\alpha}}","aliases":"∑_substackb∈ C_β a∈ C_α sum substack in beta a in alpha","display":null,"kind":"use","occurrences":[2532]},{"id":"symbol-6ed65771d02f8363","tex":"\\sum_{\\substack{r(b)=\\beta\\\\r(a)=\\alpha}}","aliases":"∑_substackr(b)=β r(a)=α sum substack beta r alpha","display":null,"kind":"use","occurrences":[2251]},{"id":"symbol-1189cbb810e02392","tex":"\\sum_{\\substack{u\\notin A,\\ v\\in A\\\\u\\sim v}}","aliases":"∑_substacku∉ A, v∈ A u∼ v sum substack notin in u sim","display":null,"kind":"use","occurrences":[748]},{"id":"symbol-e30b86f05538dcdd","tex":"\\sum_{\\substack{v\\in A,\\ u\\notin A\\\\u\\sim v}}","aliases":"∑_substackv∈ A, u∉ A u∼ v sum substack in notin u sim","display":null,"kind":"use","occurrences":[748]},{"id":"symbol-e9c7a389e7da74a1","tex":"\\sum_{v\\in A}","aliases":"∑_v∈ A sum in","display":"\\sum_{v\\in A}","kind":"use","occurrences":[748]},{"id":"symbol-6546cb38aa60d81e","tex":"\\sum_{v\\in\\{0,1\\}^3}","aliases":"∑_v∈{0,1}^3 sum in","display":"\\sum_{v\\in\\{0,1\\}^3}","kind":"use","occurrences":[2974]},{"id":"symbol-8c1a906fe53c62b1","tex":"\\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}","aliases":"∑_w∈{0,…,p-1}^s sum in ldots","display":"\\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}","kind":"use","occurrences":[972]},{"id":"symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","occurrences":[2624,2669,2929,3025,3227]},{"id":"symbol-2be6045d20d089d0","tex":"\\sum_{x,a,i}","aliases":"∑_x,a,i sum","display":"\\sum_{x,a,i}","kind":"use","occurrences":[3240]},{"id":"symbol-576fa14b6fc700f4","tex":"\\sum_{x,\\mu}","aliases":"∑_x,μ sum mu","display":"\\sum_{x,\\mu}","kind":"use","occurrences":[2625]},{"id":"symbol-d540aa8cbc4fbf01","tex":"\\sum_{x,\\mu<\\nu}","aliases":"∑_x,μ<ν sum mu nu","display":"\\sum_{x,\\mu<\\nu}","kind":"use","occurrences":[2686]},{"id":"symbol-b9ea8830cdebefeb","tex":"\\sum_{x\\in A}","aliases":"∑_x∈ A sum in","display":"\\sum_{x\\in A}","kind":"use","occurrences":[3773]},{"id":"symbol-3ec3e94007233342","tex":"\\sum_{x\\in S}","aliases":"∑_x∈ S sum in","display":"\\sum_{x\\in S}","kind":"use","occurrences":[3019]},{"id":"symbol-9faa93cb185e1d0d","tex":"\\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}","aliases":"∑_y≡ x (mod p^r) sum equiv mathrm","display":"\\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}","kind":"use","occurrences":[129]},{"id":"symbol-7f9ca7ecb7712f78","tex":"\\sum_{|v|=r}","aliases":"∑_|v|=r sum","display":"\\sum_{|v|=r}","kind":"use","occurrences":[949]},{"id":"symbol-ade7e6465dee6e86","tex":"\\sum_\\lambda","aliases":"∑_λ sum lambda","display":"\\sum_\\lambda","kind":"use","occurrences":[897,898]},{"id":"symbol-f85a9d219ed81c46","tex":"\\sum_\\mu","aliases":"∑_μ sum mu","display":"\\sum_\\mu","kind":"use","occurrences":[816,817,818,2135,2617]},{"id":"symbol-472fcc6e7ca48333","tex":"\\sum_\\Xi","aliases":"∑_Ξ sum Xi","display":"\\sum_\\Xi","kind":"use","occurrences":[3319]},{"id":"symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","occurrences":[211,861,862,1212]},{"id":"symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","occurrences":[1208,1243,1249,1324,1330]},{"id":"symbol-f9da70e93ac834a8","tex":"\\sum_{\\ell=0}^{p-1}","aliases":"∑_ℓ=0^p-1 sum ell","display":"\\sum_{\\ell=0}^{p-1}","kind":"use","occurrences":[209]},{"id":"symbol-14617be2bc27090e","tex":"\\sum_{\\ell=1}^{L-1}","aliases":"∑_ℓ=1^L-1 sum ell","display":"\\sum_{\\ell=1}^{L-1}","kind":"use","occurrences":[1339]},{"id":"symbol-602b230463ce9d93","tex":"\\mp","aliases":"∓ mp","display":"\\mp","kind":"use","occurrences":[3333]},{"id":"symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","occurrences":[874,1223,2686]},{"id":"symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","occurrences":[1167,3478]},{"id":"symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","occurrences":[461,476,494,496,505,512,579,586,592,603,607,616,617,634,928,934,1181,1206,1210,1216,1254,1263,1277,1281,1282,1284,1287,1291,1294,1295,1296,1300,1301,1329,1356,1426,1470,1474,1501,1603,1772,1786,1789,1790,1803,1874,2204,2306,2319,2386,2433,3123,3124,3128,3220,3615,3650,3661,3928,3954,3960,3972,3989,3998,3999,4000,4001]},{"id":"symbol-9e05adb8ff8d013d","tex":"\\infty^2","aliases":"∞^2 infty","display":"\\infty^2","kind":"use","occurrences":[1790,3928]},{"id":"symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","occurrences":[385,641]},{"id":"symbol-a387bbfa91002daa","tex":"\\int_{\\mathcal L_D}","aliases":"∫_ L_D int mathcal","display":"\\int_{\\mathcal L_D}","kind":"use","occurrences":[3505,3506]},{"id":"symbol-ad06d6293cf51313","tex":"\\int_{\\mathbb T^3}","aliases":"∫_ T^3 int mathbb","display":"\\int_{\\mathbb T^3}","kind":"use","occurrences":[1518,1526,1671,3230]},{"id":"symbol-d0667d7e593026bd","tex":"\\int_{-2}^2","aliases":"∫_-2^2 int","display":"\\int_{-2}^2","kind":"use","occurrences":[1190]},{"id":"symbol-ed738c29977ebe55","tex":"\\int_{-L}^L","aliases":"∫_-L^L int","display":"\\int_{-L}^L","kind":"use","occurrences":[3613]},{"id":"symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","occurrences":[3424]},{"id":"symbol-2547054a0e887b21","tex":"\\int_0^{2\\pi}","aliases":"∫_0^2π int pi","display":"\\int_0^{2\\pi}","kind":"use","occurrences":[1164,1174,3621,3634]},{"id":"symbol-fa12689080284226","tex":"\\int_0^{2\\pi p}","aliases":"∫_0^2π p int pi","display":"\\int_0^{2\\pi p}","kind":"use","occurrences":[1174]},{"id":"symbol-ea9bb4c1b114de90","tex":"\\int_0^h","aliases":"∫_0^h int","display":"\\int_0^h","kind":"use","occurrences":[3070]},{"id":"symbol-75f238c1c506c49d","tex":"\\int_0^{h/2}","aliases":"∫_0^h/2 int","display":"\\int_0^{h/2}","kind":"use","occurrences":[3074]},{"id":"symbol-051f75585a774ce8","tex":"\\int_0^r","aliases":"∫_0^r int","display":"\\int_0^r","kind":"use","occurrences":[3582]},{"id":"symbol-6122a3a715c464e3","tex":"\\int_0^{R_0}","aliases":"∫_0^R_0 int","display":"\\int_0^{R_0}","kind":"use","occurrences":[3586]},{"id":"symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","occurrences":[685,925,926,1884,3097]},{"id":"symbol-712f3452ad319777","tex":"\\int_0^u","aliases":"∫_0^u int","display":"\\int_0^u","kind":"use","occurrences":[1710]},{"id":"symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","occurrences":[1714]},{"id":"symbol-4fb5af8a571997bd","tex":"\\int_0^{\\theta}","aliases":"∫_0^θ int theta","display":"\\int_0^{\\theta}","kind":"use","occurrences":[3643]},{"id":"symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","occurrences":[928,934]},{"id":"symbol-6c39689bc04b0a10","tex":"\\int_A","aliases":"∫_A int","display":"\\int_A","kind":"use","occurrences":[427]},{"id":"symbol-2f67664008c33cd2","tex":"\\int_{B_{m,a}}","aliases":"∫_B_m,a int","display":"\\int_{B_{m,a}}","kind":"use","occurrences":[391]},{"id":"symbol-4fb956e840efa313","tex":"\\int_{C_a}","aliases":"∫_C_a int","display":"\\int_{C_a}","kind":"use","occurrences":[3137]},{"id":"symbol-3ee4763e7e30f936","tex":"\\int_e","aliases":"∫_e int","display":"\\int_e","kind":"use","occurrences":[3557,3577]},{"id":"symbol-a3295146e088975d","tex":"\\int_{h/2}^{h}","aliases":"∫_h/2^h int","display":"\\int_{h/2}^{h}","kind":"use","occurrences":[3074]},{"id":"symbol-fe9351791882eb76","tex":"\\int_I","aliases":"∫_I int","display":"\\int_I","kind":"use","occurrences":[3621]},{"id":"symbol-964ee142a04858f9","tex":"\\int_{\\R}","aliases":"∫_R int R","display":null,"kind":"use","occurrences":[1282,1296]},{"id":"symbol-939963fba5a39d6b","tex":"\\int_\\square","aliases":"∫_square int square","display":"\\int_\\square","kind":"use","occurrences":[3558]},{"id":"symbol-fea858387f776702","tex":"\\int_{\\square}","aliases":"∫_square int square","display":"\\int_{\\square}","kind":"use","occurrences":[2698]},{"id":"symbol-d3b0bf5772538b33","tex":"\\int_x^{x+h}","aliases":"∫_x^x+h int","display":"\\int_x^{x+h}","kind":"use","occurrences":[2772]},{"id":"symbol-741b2176a1c35e3c","tex":"\\int_x^{x+s}","aliases":"∫_x^x+s int","display":"\\int_x^{x+s}","kind":"use","occurrences":[2780]},{"id":"symbol-421b279969e85688","tex":"\\int_{x_j}^{x_j+h}","aliases":"∫_x_j^x_j+h int","display":"\\int_{x_j}^{x_j+h}","kind":"use","occurrences":[2775]},{"id":"symbol-d68a718169ef9594","tex":"\\int_\\Sigma","aliases":"∫_Σ int Sigma","display":"\\int_\\Sigma","kind":"use","occurrences":[3342,3374,3380,3387,3393,3403,3405]},{"id":"symbol-8165d9752a014553","tex":"\\int_{\\tau_0}^{\\tau}","aliases":"∫_τ_0^τ int tau tau","display":"\\int_{\\tau_0}^{\\tau}","kind":"use","occurrences":[3643]},{"id":"symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","occurrences":[3924,3925,4149]},{"id":"symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","occurrences":[66,1931]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[888,1022,1028,1061,1067,1083,1290,1549,1570,1576,1668,1677,1718,1740,1979,2154,2156,2164,2165,2176,2190,2452,2481,2489,2498,2514,2851,3005,3084,3085,3389,3446,3857]},{"id":"symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","occurrences":[292,336,654,772,3780,3787]},{"id":"symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","occurrences":[1388,1395,1396,1398,1403,1427,1495,1705,1707,1709,1711,1712,1715,1716,1721,1723,1733,1738,1739,1741,1755,1770,1771,1772,1774,1778,1780,1789,1803,1871,1872,1899,1900,1902,1905,1921,1922,1933,1935,2200,2749,2792,3214,3216,3220,3461,3462,3474,3928,3952,3954,3971,3972,3975,3977,3987,3990,4102,4125]},{"id":"symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","occurrences":[1400,1480,1483,1485,1536,1630,1713,1768,1892,1901,2107,2153,2198,2630,2691,2715,3209,3473,3952,4103]},{"id":"symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","occurrences":[1433,1450,2020]},{"id":"symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","occurrences":[150,461,616,732,843,1531,2256]},{"id":"symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","occurrences":[230,231,864,2007]},{"id":"symbol-d315191ea29af671","tex":"\\bigoplus_{\\mathbf r}","aliases":"⊕_ r bigoplus mathbf","display":"\\bigoplus_{\\mathbf r}","kind":"use","occurrences":[616]},{"id":"symbol-50db95a0004bcd02","tex":"\\bigoplus_a","aliases":"⊕_a bigoplus","display":"\\bigoplus_a","kind":"use","occurrences":[2256]},{"id":"symbol-ff559903fce4d2fd","tex":"\\bigoplus_{j=0}^{M-1}","aliases":"⊕_j=0^M-1 bigoplus","display":"\\bigoplus_{j=0}^{M-1}","kind":"use","occurrences":[843]},{"id":"symbol-f3bc2dd9a6dd9759","tex":"\\bigoplus_{m=0}^{\\infty}","aliases":"⊕_m=0^∞ bigoplus infty","display":"\\bigoplus_{m=0}^{\\infty}","kind":"use","occurrences":[461]},{"id":"symbol-c1e4b6cbf2585e99","tex":"\\bigoplus_{n\\in S}","aliases":"⊕_n∈ S bigoplus in","display":"\\bigoplus_{n\\in S}","kind":"use","occurrences":[1531]},{"id":"symbol-7e8554b16ebfef24","tex":"\\bigoplus_{r=0}^h","aliases":"⊕_r=0^h bigoplus","display":"\\bigoplus_{r=0}^h","kind":"use","occurrences":[150]},{"id":"symbol-6e1bd670f8830b0f","tex":"\\bigoplus_v","aliases":"⊕_v bigoplus","display":"\\bigoplus_v","kind":"use","occurrences":[732]},{"id":"symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"use","occurrences":[74,78,603,604,606,611]},{"id":"symbol-f7130473f45998ab","tex":"\\bigotimes_p","aliases":"⊗_p bigotimes","display":"\\bigotimes_p","kind":"use","occurrences":[604,606]},{"id":"symbol-06a1b3c4dffd21b3","tex":"\\bigotimes_{p\\in\\mathcal P}","aliases":"⊗_p∈ P bigotimes in mathcal","display":"\\bigotimes_{p\\in\\mathcal P}","kind":"use","occurrences":[603,611]},{"id":"symbol-658c3994137bf7f4","tex":"\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}","aliases":"⊗_p∈ P_ o bigotimes in mathcal mathrm","display":"\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}","kind":"use","occurrences":[74,78]},{"id":"symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","occurrences":[544,641,965,3337,3340]},{"id":"symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","occurrences":[1284]},{"id":"symbol-d42ab8c0624e1096","tex":"\\bigcap_j","aliases":"⋂_j bigcap","display":"\\bigcap_j","kind":"use","occurrences":[1284]},{"id":"symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","occurrences":[432,3223]},{"id":"symbol-4d5ed9808720c1a8","tex":"\\bigcup_K","aliases":"⋃_K bigcup","display":"\\bigcup_K","kind":"use","occurrences":[3223]},{"id":"symbol-e8829e61e00377d3","tex":"\\bigcup_m","aliases":"⋃_m bigcup","display":"\\bigcup_m","kind":"use","occurrences":[432]},{"id":"symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","occurrences":[3287,3289,3291,3293,3332,3333]},{"id":"symbol-d300cc10a142bb79","tex":"\\star^2","aliases":"⋆^2 star","display":"\\star^2","kind":"use","occurrences":[3332]},{"id":"symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","occurrences":[192,203,206,443,637,1218,3041,3043,3205]},{"id":"symbol-6d63ab7a8797f0ed","tex":"\\rangle_h","aliases":"⟩_h rangle","display":"\\rangle_h","kind":"use","occurrences":[3751,3753,3754,3766,3778]},{"id":"symbol-92fa129423d0a85c","tex":"\\rangle_{L^2}","aliases":"⟩_L^2 rangle","display":"\\rangle_{L^2}","kind":"use","occurrences":[1526,1668,1671,1696]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[12]}],"contexts":["Adic refinement first changes the address of a state. Its amplitudes remain\ncomplex numbers, and its measurement probabilities remain ordinary Born\nprobabilities. For a finite address set $X$ we use the counting inner product\non $\\ell^2(X;\\C)$, so the address vectors $|x\\rangle$ are orthonormal.\nA bijection of address sets therefore gives a permutation unitary without\nchanging the quantum probability rule. Arithmetic determines the address\nfactorization. A Hamiltonian and a measurement prescription supply the\nadditional physical choices.","The first coordinate issue is specific to units. For an odd prime $p$ and\nan integer $h\\geq0$, set\n\\[\n U_{p,h}=(\\Z/p^{h+1}\\Z)^\\times.\n\\]\nFor $p=2$ the useful indexing starts one digit later:\n\\[\n U_{2,h}=(\\Z/2^{h+2}\\Z)^\\times.\n\\]\nThe shifted dyadic index retains the sign even when the principal phase\nregister has dimension one. It prevents the two units modulo four from\nbeing collapsed into one phase coordinate.","\\begin{proposition}[Finite unit coordinates]\n\\label{a-finite-unit-coordinates}\nFor odd $p$, choose a unit $\\zeta_{p,h}$ of order $p-1$ whose reduction\ngenerates $(\\Z/p\\Z)^\\times$. The maps\n\\begin{align}\n (t,a)&\\longmapsto\\zeta_{p,h}^{\\,t}(1+p)^a\n       \\pmod{p^{h+1}},\n & (t,a)&\\in\\Z/(p-1)\\Z\\times\\Z/p^h\\Z,\n \\label{a-finite-odd-coordinate}","\\begin{proposition}[Finite unit coordinates]\n\\label{a-finite-unit-coordinates}\nFor odd $p$, choose a unit $\\zeta_{p,h}$ of order $p-1$ whose reduction\ngenerates $(\\Z/p\\Z)^\\times$. The maps\n\\begin{align}\n (t,a)&\\longmapsto\\zeta_{p,h}^{\\,t}(1+p)^a\n       \\pmod{p^{h+1}},\n & (t,a)&\\in\\Z/(p-1)\\Z\\times\\Z/p^h\\Z,\n \\label{a-finite-odd-coordinate}\\\\\n (s,a)&\\longmapsto(-1)^s5^a\\pmod{2^{h+2}},\n & (s,a)&\\in\\Z/2\\Z\\times\\Z/2^h\\Z\n \\label{a-finite-dyadic-coordinate}\n\\end{align}\nare group isomorphisms onto $U_{p,h}$ and $U_{2,h}$ respectively.\n\\end{proposition}","(s,a)&\\longmapsto(-1)^s5^a\\pmod{2^{h+2}},\n & (s,a)&\\in\\Z/2\\Z\\times\\Z/2^h\\Z\n \\label{a-finite-dyadic-coordinate}\n\\end{align}\nare group isomorphisms onto $U_{p,h}$ and $U_{2,h}$ respectively.\n\\end{proposition}","\\begin{proof}\nFor odd $p$ the valuation identity\n\\[\n v_p\\bigl((1+p)^{p^k}-1\\bigr)=k+1\n \\qquad(k\\geq0)\n\\]\nshows that $1+p$ has order $p^h$ modulo $p^{h+1}$. To check the\nidentity, the case $k=0$ is immediate. If $u=1+c p^{k+1}$ with\n$p\\nmid c$, the binomial expansion of $u^p-1$ has first term\n$c p^{k+2}$. Every remaining term has strictly larger $p$-valuation\nwhen $p$ is odd. This proves the induction. The subgroup generated by\n$1+p$ is contained in the kernel of reduction modulo $p$ and has the\nsame cardinality $p^h$, so it is that entire kernel.","luation identity\n\\[\n v_p\\bigl((1+p)^{p^k}-1\\bigr)=k+1\n \\qquad(k\\geq0)\n\\]\nshows that $1+p$ has order $p^h$ modulo $p^{h+1}$. To check the\nidentity, the case $k=0$ is immediate. If $u=1+c p^{k+1}$ with\n$p\\nmid c$, the binomial expansion of $u^p-1$ has first term\n$c p^{k+2}$. Every remaining term has strictly larger $p$-valuation\nwhen $p$ is odd. This proves the induction. The subgroup generated by\n$1+p$ is contained in the kernel of reduction modulo $p$ and has the\nsame cardinality $p^h$, so it is that entire kernel.","\\qquad(k\\geq0)\n\\]\nshows that $1+p$ has order $p^h$ modulo $p^{h+1}$. To check the\nidentity, the case $k=0$ is immediate. If $u=1+c p^{k+1}$ with\n$p\\nmid c$, the binomial expansion of $u^p-1$ has first term\n$c p^{k+2}$. Every remaining term has strictly larger $p$-valuation\nwhen $p$ is odd. This proves the induction. The subgroup generated by\n$1+p$ is contained in the kernel of reduction modulo $p$ and has the\nsame cardinality $p^h$, so it is that entire kernel.","+p$ has order $p^h$ modulo $p^{h+1}$. To check the\nidentity, the case $k=0$ is immediate. If $u=1+c p^{k+1}$ with\n$p\\nmid c$, the binomial expansion of $u^p-1$ has first term\n$c p^{k+2}$. Every remaining term has strictly larger $p$-valuation\nwhen $p$ is odd. This proves the induction. The subgroup generated by\n$1+p$ is contained in the kernel of reduction modulo $p$ and has the\nsame cardinality $p^h$, so it is that entire kernel.","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime ord","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Conse","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, a","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem give","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bou","For completeness, an order-$p-1$ lift of any nonzero residue can be chosen\ncoherently. The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.",". The root equation $x^{p-1}=1$ has nonzero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","ero derivative\n$(p-1)x^{p-2}$ modulo $p$. If $x_n$ is a root modulo $p^n$, substituting\n$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","$x_n+c p^n$ modulo $p^{n+1}$ determines one and only one residue $c$\nthat cancels the error. Starting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","rting with a generator of\n$(\\Z/p\\Z)^\\times$ produces $\\zeta_{p,h}$ with the required order.\nThe cyclicity of this residue group follows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","lows from the elementary fact that\na degree-$d$ polynomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","nomial over a field has at most $d$ roots: in a finite\nsubgroup of the multiplicative group, the least common multiple $m$ of\nthe element orders is realized by an element. To see this, take for each\nprime divisor of $m$ a component having the maximal prime-power order\nand multiply those commuting components of coprime orders. All elements solve\n$x^m=1$. Consequently the subgroup has size at most $m$, and Lagrange's\ntheorem gives the reverse bound. The subgroup is cyclic.","If $\\zeta_{p,h}^{\\,t}(1+p)^a=1$, reduction modulo $p$ gives\n$t=0$ in $\\Z/(p-1)\\Z$. The principal-unit order then gives\n$a=0$ in $\\Z/p^h\\Z$. Thus the displayed map is injective, and its\ndomain has size $(p-1)p^h=|U_{p,h}|$, which proves surjectivity.","For the dyadic case, successive squaring gives\n\\[\n v_2\\bigl(5^{2^k}-1\\bigr)=k+2.\n\\]\nIndeed $5-1$ has valuation two, and for every such power the factor\n$5^{2^k}+1$ has valuation one. Thus $5$ has order $2^h$ modulo\n$2^{h+2}$. Its powers are exactly the $2^h$ units congruent to one\nmodulo four. Multiplication by $-1$ supplies the units congruent to\nthree modulo four. These two classes are disjoint, giving the claimed\nunique coordinates.\n\\end{proof}","In the odd-prime register $t$ is a torsion coordinate and $a$ is a\nprincipal phase coordinate. In the dyadic register $s$ is a sign bit and\n$a$ is again the principal phase. The Hilbert-space factorizations are\ntherefore\n\\begin{align}\n \\ell^2(U_{p,h})&\\cong\n \\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^h\\Z),\n &&p\\text{ odd},\\nonumber","In the odd-prime register $t$ is a torsion coordinate and $a$ is a\nprincipal phase coordinate. In the dyadic register $s$ is a sign bit and\n$a$ is again the principal phase. The Hilbert-space factorizations are\ntherefore\n\\begin{align}\n \\ell^2(U_{p,h})&\\cong\n \\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^h\\Z),\n &&p\\text{ odd},\\nonumber\\\\\n \\ell^2(U_{2,h})&\\cong\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^h\\Z).\n \\label{a-finite-unit-hilbert}\n\\end{align}\nThese are unitary changes of basis. They do not assert that a generic\nstate is a product state, or that a generic Hamiltonian is a sum of\nindependent register Hamiltonians. Entanglement and interactions are\npreserved under the change of coordinates.","The lifting construction in the proof permits the coherent choice\n$\\zeta_{p,h+1}\\equiv\\zeta_{p,h}\\pmod{p^{h+1}}$. With that choice,\nreduction of unit addresses retains $t$ and reduces $a$ modulo $p^h$.\nIf the torsion lifts are instead chosen independently at each depth, the\nfinite factorizations remain exact, but a comparison between levels must\ninclude the resulting coordinate changes. A levelwise permutation is\nnot by itself a specified refinement tower.","Let $\\mathcal P_{\\mathrm o}$ be a finite set of odd primes and put\n\\[\n M=2^{h_2+2}\\prod_{p\\in\\mathcal P_{\\mathrm o}}p^{h_p+1}.\n\\]\nThe Chinese remainder theorem gives a bijection from the unit addresses\nmodulo $M$ to their local unit addresses. Combining it with\nProposition~\\ref{a-finite-unit-coordinates} gives a permutation unitary\n\\begin{equation}\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique","Let $\\mathcal P_{\\mathrm o}$ be a finite set of odd primes and put\n\\[\n M=2^{h_2+2}\\prod_{p\\in\\mathcal P_{\\mathrm o}}p^{h_p+1}.\n\\]\nThe Chinese remainder theorem gives a bijection from the unit addresses\nmodulo $M$ to their local unit addresses. Combining it with\nProposition~\\ref{a-finite-unit-coordinates} gives a permutation unitary\n\\begin{equation}\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique tuple of coordinates. Distinct units have distinct\ntuples, so inner products are preserved. The dim","Let $\\mathcal P_{\\mathrm o}$ be a finite set of odd primes and put\n\\[\n M=2^{h_2+2}\\prod_{p\\in\\mathcal P_{\\mathrm o}}p^{h_p+1}.\n\\]\nThe Chinese remainder theorem gives a bijection from the unit addresses\nmodulo $M$ to their local unit addresses. Combining it with\nProposition~\\ref{a-finite-unit-coordinates} gives a permutation unitary\n\\begin{equation}\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique tuple of coordinates. Distinct units have distinct\ntuples, so inner products are preserved. The dimensions also agree:\n\\[\n \\varphi(M)=2^{h_2+1}\\prod_{p\\in\\mathcal P_{\\mathrm o}}(p-1)p","Let $\\mathcal P_{\\mathrm o}$ be a finite set of odd primes and put\n\\[\n M=2^{h_2+2}\\prod_{p\\in\\mathcal P_{\\mathrm o}}p^{h_p+1}.\n\\]\nThe Chinese remainder theorem gives a bijection from the unit addresses\nmodulo $M$ to their local unit addresses. Combining it with\nProposition~\\ref{a-finite-unit-coordinates} gives a permutation unitary\n\\begin{equation}\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique tuple of coordinates. Distinct units have distinct\ntuples, so inner products are preserved. The dimensions also agree:\n\\[\n \\varphi(M)=2^{h_2+1}\\prod_{p\\in\\mathcal P_{\\mathrm o}}(p-1)p^{h_p}.\n\\]\nMultiplication of units becomes addition in each cyclic coordinate.","with\nProposition~\\ref{a-finite-unit-coordinates} gives a permutation unitary\n\\begin{equation}\n \\mathcal U_{\\mathrm{CRT}}:\n \\ell^2\\bigl((\\Z/M\\Z)^\\times\\bigr)\n \\longrightarrow\n \\C^2_{\\mathrm{sign}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique tuple of coordinates. Distinct units have distinct\ntuples, so inner products are preserved. The dimensions also agree:\n\\[\n \\varphi(M)=2^{h_2+1}\\prod_{p\\in\\mathcal P_{\\mathrm o}}(p-1)p^{h_p}.\n\\]\nMultiplication of units becomes addition in each cyclic coordinate.","n}}\\otimes\\ell^2(\\Z/2^{h_2}\\Z)\n \\otimes\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}\n \\left(\\C^{p-1}_{\\mathrm{tors}}\\otimes\\ell^2(\\Z/p^{h_p}\\Z)\\right).\n \\label{a-finite-crt}\n\\end{equation}\nFor each basis vector, $\\mathcal U_{\\mathrm{CRT}}|u\\rangle$ is the basis\nvector with its unique tuple of coordinates. Distinct units have distinct\ntuples, so inner products are preserved. The dimensions also agree:\n\\[\n \\varphi(M)=2^{h_2+1}\\prod_{p\\in\\mathcal P_{\\mathrm o}}(p-1)p^{h_p}.\n\\]\nMultiplication of units becomes addition in each cyclic coordinate.","We use the positive-phase Fourier convention\n\\begin{equation}\n (F_n)_{k x}=n^{-1/2}\\exp(2\\pi\\mathrm{i}kx/n).\n \\label{a-finite-fourier-convention}\n\\end{equation}\nThe character transform of the unit group consequently factors as\n\\begin{equation}\n F_{U_M}=\\mathcal U_{\\mathrm{CRT}}^*\n \\left[F_2\\otimes F_{2^{h_2}}\\otimes\n \\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}(F_{p-1}\\otimes F_{p^{h_p}})\n \\right]\\mathcal U_{\\mathrm{CRT}}.\n \\label{a-finite-unit-fourier}\n\\end{equation}\nHere the unit-group character labels have been identified with the same\ncyclic coordinate sets. This is a transform on the multiplicative unit\ngroup, rather than the additive Fourier transform on all residues\nmodulo $M$.","ft[F_2\\otimes F_{2^{h_2}}\\otimes\n \\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}(F_{p-1}\\otimes F_{p^{h_p}})\n \\right]\\mathcal U_{\\mathrm{CRT}}.\n \\label{a-finite-unit-fourier}\n\\end{equation}\nHere the unit-group character labels have been identified with the same\ncyclic coordinate sets. This is a transform on the multiplicative unit\ngroup, rather than the additive Fourier transform on all residues\nmodulo $M$.","The additive transform has a related factorization, with different\npermutations on its input and its dual output. Let $a,b$ be positive\ncoprime integers and $N=ab$. Define\n\\begin{align*}\n \\mathcal P_{\\mathrm{in}}|n\\rangle\n &=|n\\bmod a\\rangle\\otimes|n\\bmod b\\rangle,","The additive transform has a related factorization, with different\npermutations on its input and its dual output. Let $a,b$ be positive\ncoprime integers and $N=ab$. Define\n\\begin{align*}\n \\mathcal P_{\\mathrm{in}}|n\\rangle\n &=|n\\bmod a\\rangle\\otimes|n\\bmod b\\rangle,\\\\\n \\mathcal P_{\\mathrm{out}}|k\\rangle\n &=|k b^{-1}\\bmod a\\rangle\\otimes|k a^{-1}\\bmod b\\rangle,\n\\end{align*}\nwhere the inverses are taken in the indicated residue rings. Then\n\\begin{equation}\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output","\\mathcal P_{\\mathrm{out}}|k\\rangle\n &=|k b^{-1}\\bmod a\\rangle\\otimes|k a^{-1}\\bmod b\\rangle,\n\\end{align*}\nwhere the inverses are taken in the indicated residue rings. Then\n\\begin{equation}\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","al P_{\\mathrm{out}}|k\\rangle\n &=|k b^{-1}\\bmod a\\rangle\\otimes|k a^{-1}\\bmod b\\rangle,\n\\end{align*}\nwhere the inverses are taken in the indicated residue rings. Then\n\\begin{equation}\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","gle\n &=|k b^{-1}\\bmod a\\rangle\\otimes|k a^{-1}\\bmod b\\rangle,\n\\end{align*}\nwhere the inverses are taken in the indicated residue rings. Then\n\\begin{equation}\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","}\\bmod b\\rangle,\n\\end{align*}\nwhere the inverses are taken in the indicated residue rings. Then\n\\begin{equation}\n F_N=\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","\\mathcal P_{\\mathrm{out}}^*(F_a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","a\\otimes F_b)\n       \\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","\\mathcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","thcal P_{\\mathrm{in}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","n}}.\n \\label{a-finite-additive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","itive-fourier}\n\\end{equation}\nTo verify the phase, let $n_a=n\\bmod a$ and $n_b=n\\bmod b$. At the\ndual coordinates $k_a=kb^{-1}\\bmod a$ and $k_b=ka^{-1}\\bmod b$, the\nproduct kernel is\n\\[\n N^{-1/2}\\exp\\!\\left[2\\pi\\mathrm{i}\n      \\left(\\frac{k_a n_a}{a}+\\frac{k_b n_b}{b}\\right)\\right].\n\\]\nThe numerator $b k_a+a k_b$ is congruent to $k$ modulo both $a$ and\n$b$, hence modulo $ab$. This kernel equals\n$N^{-1/2}\\exp(2\\pi\\mathrm{i}kn/N)$, proving\n\\eqref{a-finite-additive-fourier}. Omitting the dual-output permutation\nwould generally give a different phase labeling.","The linear address-forgetting map\n\\begin{equation}\n C_{\\mathrm{phase}}|s,a\\rangle=|a\\rangle\n \\label{a-finite-forgetting}\n\\end{equation}\nmaps a $2^{h+1}$-dimensional dyadic unit register onto a\n$2^h$-dimensional principal phase register. Its kernel is the span of\nthe $2^h$ vectors $|0,a\\rangle-|1,a\\rangle$. Thus\n\\[\n \\operatorname{rank}C_{\\mathrm{phase}}=2^h,\n \\qquad\\dim\\ker C_{\\mathrm{phase}}=2^h.\n\\]\nThis is a rank-deficient linear sum of amplitudes. It is not an\nisometry, and it is not the quantum channel that discards a subsystem.\nFor example it sends the normalized state\n$(|0,a\\rangle+|1,a\\rangle)/\\sqrt2$ to $\\sqrt2|a\\rangle$.\nIts adjoint product therefore does not equal the identity.","ster onto a\n$2^h$-dimensional principal phase register. Its kernel is the span of\nthe $2^h$ vectors $|0,a\\rangle-|1,a\\rangle$. Thus\n\\[\n \\operatorname{rank}C_{\\mathrm{phase}}=2^h,\n \\qquad\\dim\\ker C_{\\mathrm{phase}}=2^h.\n\\]\nThis is a rank-deficient linear sum of amplitudes. It is not an\nisometry, and it is not the quantum channel that discards a subsystem.\nFor example it sends the normalized state\n$(|0,a\\rangle+|1,a\\rangle)/\\sqrt2$ to $\\sqrt2|a\\rangle$.\nIts adjoint product therefore does not equal the identity.","l phase register. Its kernel is the span of\nthe $2^h$ vectors $|0,a\\rangle-|1,a\\rangle$. Thus\n\\[\n \\operatorname{rank}C_{\\mathrm{phase}}=2^h,\n \\qquad\\dim\\ker C_{\\mathrm{phase}}=2^h.\n\\]\nThis is a rank-deficient linear sum of amplitudes. It is not an\nisometry, and it is not the quantum channel that discards a subsystem.\nFor example it sends the normalized state\n$(|0,a\\rangle+|1,a\\rangle)/\\sqrt2$ to $\\sqrt2|a\\rangle$.\nIts adjoint product therefore does not equal the identity.","Discarding the sign as a quantum operation instead takes the partial\ntrace $\\rho\\mapsto\\operatorname{Tr}_{\\mathrm{sign}}\\rho$. Both pure\nstates $|0,a\\rangle$ and $|1,a\\rangle$ then produce the same reduced\nphase state $|a\\rangle\\langle a|$. Yet the sign observable\n\\[\n Z_{\\mathrm{sign}}=(|0\\rangle\\langle0|-|1\\rangle\\langle1|)\\otimes I\n\\]\nhas expectations $+1$ and $-1$ on those states. No measurement on the\nretained phase state can reconstruct this difference. The obstruction is\ninformation loss in a stated map, not an alternative probability rule.","A separate arithmetic example shows why integral residue selectors and\nrational spectral idempotents should be distinguished. Let\n\\[\n A=\\begin{pmatrix}1&1\\\\0&-1\\end{pmatrix},\\qquad A^2=I,\n \\qquad E_\\pm=\\frac{I\\pm A}{2}.\n\\]\nThe identities $E_\\pm^2=E_\\pm$, $E_+E_-=0$, and $E_++E_-=I$\nfollow directly from $A^2=I$. Both idempotents contain a denominator\ntwo, and the resultant of $x-1$ and $x+1$ has dyadic valuation one.\nModulo two the two roots agree and $(A-I)^2=0$, while $A-I$ is\nnonzero. Thus the residual matrix can retain a nilpotent extension even\nthough it no longer distinguishes the two rational eigenvalues.\nThese $E_\\pm$ are complementary algebraic spectral idempotents. Since\n$A$ is not self-adjoint in the standard counting inner product, they\nare not orthogonal quantum measurement projectors in that inner product.\nThe unit-address sign projector, by contrast, is an ordinary diagonal\northogonal projecto","0&-1\\end{pmatrix},\\qquad A^2=I,\n \\qquad E_\\pm=\\frac{I\\pm A}{2}.\n\\]\nThe identities $E_\\pm^2=E_\\pm$, $E_+E_-=0$, and $E_++E_-=I$\nfollow directly from $A^2=I$. Both idempotents contain a denominator\ntwo, and the resultant of $x-1$ and $x+1$ has dyadic valuation one.\nModulo two the two roots agree and $(A-I)^2=0$, while $A-I$ is\nnonzero. Thus the residual matrix can retain a nilpotent extension even\nthough it no longer distinguishes the two rational eigenvalues.\nThese $E_\\pm$ are complementary algebraic spectral idempotents. Since\n$A$ is not self-adjoint in the standard counting inner product, they\nare not orthogonal quantum measurement projectors in that inner product.\nThe unit-address sign projector, by contrast, is an ordinary diagonal\northogonal projector.","identities $E_\\pm^2=E_\\pm$, $E_+E_-=0$, and $E_++E_-=I$\nfollow directly from $A^2=I$. Both idempotents contain a denominator\ntwo, and the resultant of $x-1$ and $x+1$ has dyadic valuation one.\nModulo two the two roots agree and $(A-I)^2=0$, while $A-I$ is\nnonzero. Thus the residual matrix can retain a nilpotent extension even\nthough it no longer distinguishes the two rational eigenvalues.\nThese $E_\\pm$ are complementary algebraic spectral idempotents. Since\n$A$ is not self-adjoint in the standard counting inner product, they\nare not orthogonal quantum measurement projectors in that inner product.\nThe unit-address sign projector, by contrast, is an ordinary diagonal\northogonal projector.","directly from $A^2=I$. Both idempotents contain a denominator\ntwo, and the resultant of $x-1$ and $x+1$ has dyadic valuation one.\nModulo two the two roots agree and $(A-I)^2=0$, while $A-I$ is\nnonzero. Thus the residual matrix can retain a nilpotent extension even\nthough it no longer distinguishes the two rational eigenvalues.\nThese $E_\\pm$ are complementary algebraic spectral idempotents. Since\n$A$ is not self-adjoint in the standard counting inner product, they\nare not orthogonal quantum measurement projectors in that inner product.\nThe unit-address sign projector, by contrast, is an ordinary diagonal\northogonal projector.","The principal phase register has a canonical finite filtration. For a\nprime $p$ and depth $h\\geq0$, set\n\\[\n G_{p,h}=\\Z/p^h\\Z,\\qquad\\mathcal H_{p,h}=\\ell^2(G_{p,h};\\C).\n\\]\nAt depth zero this is a one-dimensional space. A level-$r$ cylinder\ncontains all residues with a prescribed prefix modulo $p^r$, and has\n$p^{h-r}$ members. Throughout this section depth refers to the number of\nretained least significant base-$p$ digits.","phase register has a canonical finite filtration. For a\nprime $p$ and depth $h\\geq0$, set\n\\[\n G_{p,h}=\\Z/p^h\\Z,\\qquad\\mathcal H_{p,h}=\\ell^2(G_{p,h};\\C).\n\\]\nAt depth zero this is a one-dimensional space. A level-$r$ cylinder\ncontains all residues with a prescribed prefix modulo $p^r$, and has\n$p^{h-r}$ members. Throughout this section depth refers to the number of\nretained least significant base-$p$ digits.","For $0\\leq r\\leq h$, define the cylinder average\n\\begin{equation}\n (E_r^{(h)}f)(x)=p^{-(h-r)}\n       \\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}f(y).\n \\label{a-finite-cylinder-average}\n\\end{equation}\nIts matrix is real and symmetric. Averaging twice over the same cylinder\nhas the same effect as averaging once. Thus $E_r^{(h)}$ is the\northogonal projector onto functions constant on level-$r$ cylinders.\nThe range has dimension $p^r$.","ne the cylinder average\n\\begin{equation}\n (E_r^{(h)}f)(x)=p^{-(h-r)}\n       \\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}f(y).\n \\label{a-finite-cylinder-average}\n\\end{equation}\nIts matrix is real and symmetric. Averaging twice over the same cylinder\nhas the same effect as averaging once. Thus $E_r^{(h)}$ is the\northogonal projector onto functions constant on level-$r$ cylinders.\nThe range has dimension $p^r$.","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Finite Haar resolution]\n\\label{a-finite-haar-resolution}\nThe cylinder projectors satisfy\n\\[\n E_r^{(h)}E_s^{(h)}=E_{\\min(r,s)}^{(h)},\\qquad E_h^{(h)}=I.\n\\]\nDefine\n\\[\n D_0^{(h)}=E_0^{(h)},\\qquad\n D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\\quad(1\\leq r\\leq h).\n\\]\nThen the $D_r^{(h)}$ are mutually orthogonal projectors and\n\\begin{align}\n \\sum_{r=0}^hD_r^{(h)}&=I,\n &D_r^{(h)}D_s^{(h)}&=\\delta_{rs}D_r^{(h)},\\nonumber","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Finite Haar resolution]\n\\label{a-finite-haar-resolution}\nThe cylinder projectors satisfy\n\\[\n E_r^{(h)}E_s^{(h)}=E_{\\min(r,s)}^{(h)},\\qquad E_h^{(h)}=I.\n\\]\nDefine\n\\[\n D_0^{(h)}=E_0^{(h)},\\qquad\n D_r^{(h)}=E_r^{(h)}-E_{r-1}^{(h)}\\quad(1\\leq r\\leq h).\n\\]\nThen the $D_r^{(h)}$ are mutually orthogonal projectors and\n\\begin{align}\n \\sum_{r=0}^hD_r^{(h)}&=I,\n &D_r^{(h)}D_s^{(h)}&=\\delta_{rs}D_r^{(h)},\\nonumber\\\\\n \\operatorname{rank}D_0^{(h)}&=1,\n &\\operatorname{rank}D_r^{(h)}&=(p-1)p^{r-1}\\quad(r\\geq1).\n \\label{a-finite-detail-ranks}\n\\end{align}\n\\end{proposition}","\\begin{proof}\nIf $r\\leq s$, the level-$s$ cylinders partition each level-$r$\ncylinder into equally sized groups. Averaging over the smaller groups\nand then over the whole group returns the full average, giving\n$E_rE_s=E_r$. The self-adjointness of both sides also gives\n$E_sE_r=E_r$. This proves the minimum rule.","For $r\\geq1$,\n$(E_r-E_{r-1})^2=E_r-2E_{r-1}+E_{r-1}=D_r$.\nFor $1\\leq r<s$, expansion of\n$(E_r-E_{r-1})(E_s-E_{s-1})$ gives zero by the same minimum\nrule. The products with $D_0$ vanish as well. Telescoping gives the\nresolution of the identity. Finally $\\Ran E_{r-1}\\subset\\Ran E_r$,\nso the detail rank is $p^r-p^{r-1}$, while $E_0$ has rank one.\n\\end{proof}","The decomposition is exact at each finite depth:\n\\begin{equation}\n \\mathcal H_{p,h}=\\bigoplus_{r=0}^h\\Ran D_r^{(h)},\\qquad\n \\norm f^2=\\sum_{r=0}^h\\norm{D_r^{(h)}f}^2.\n \\label{a-finite-parseval}\n\\end{equation}\nA depth cutoff $m\\leq h$ retains $E_m^{(h)}=\\sum_{r=0}^mD_r^{(h)}$.\nIts discarded norm is exactly\n\\[\n \\norm{(I-E_m^{(h)})f}^2=\n \\sum_{r=m+1}^h\\norm{D_r^{(h)}f}^2.\n\\]\nNo smoothness assumption is needed for this finite Parseval identity.\nA rate of decrease with $m$ requires an additional bound on the detail\nweights of the state.","thcal H_{p,h}=\\bigoplus_{r=0}^h\\Ran D_r^{(h)},\\qquad\n \\norm f^2=\\sum_{r=0}^h\\norm{D_r^{(h)}f}^2.\n \\label{a-finite-parseval}\n\\end{equation}\nA depth cutoff $m\\leq h$ retains $E_m^{(h)}=\\sum_{r=0}^mD_r^{(h)}$.\nIts discarded norm is exactly\n\\[\n \\norm{(I-E_m^{(h)})f}^2=\n \\sum_{r=m+1}^h\\norm{D_r^{(h)}f}^2.\n\\]\nNo smoothness assumption is needed for this finite Parseval identity.\nA rate of decrease with $m$ requires an additional bound on the detail\nweights of the state.","Write $\\omega_p=\\exp(2\\pi\\mathrm{i}/p)$. For $1\\leq r\\leq h$,\nparent prefix $a\\in\\{0,\\ldots,p^{r-1}-1\\}$, and character\n$q\\in\\{1,\\ldots,p-1\\}$, define\n\\begin{equation}\n \\psi_{r,a,q}^{(h)}(x)=\n \\begin{cases}\n p^{-(h-r+1)/2}\\omega_p^{q d_r(x)},\n       &x\\equiv a\\pmod{p^{r-1}},","Write $\\omega_p=\\exp(2\\pi\\mathrm{i}/p)$. For $1\\leq r\\leq h$,\nparent prefix $a\\in\\{0,\\ldots,p^{r-1}-1\\}$, and character\n$q\\in\\{1,\\ldots,p-1\\}$, define\n\\begin{equation}\n \\psi_{r,a,q}^{(h)}(x)=\n \\begin{cases}\n p^{-(h-r+1)/2}\\omega_p^{q d_r(x)},\n       &x\\equiv a\\pmod{p^{r-1}},\\\\\n 0,&\\text{otherwise},\n \\end{cases}\n \\label{a-finite-wavelet}\n\\end{equation}\nwhere $d_r(x)$ is the digit multiplying $p^{r-1}$. Include the\nconstant scaling vector $\\psi_0^{(h)}(x)=p^{-h/2}$.","0,&\\text{otherwise},\n \\end{cases}\n \\label{a-finite-wavelet}\n\\end{equation}\nwhere $d_r(x)$ is the digit multiplying $p^{r-1}$. Include the\nconstant scaling vector $\\psi_0^{(h)}(x)=p^{-h/2}$.","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted char","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average.","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average an","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\e","Each wavelet is constant on level-$r$ cylinders and has zero average\non its parent level-$r-1$ cylinder, because\n$\\sum_{d=0}^{p-1}\\omega_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent…","_p^{qd}=0$ for nonzero $q$. Thus it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","it lies in\n$\\Ran D_r^{(h)}$. Its support contains $p^{h-r+1}$ points, giving\nunit norm. Distinct parents have disjoint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","joint supports. For a fixed parent\nthe inner product of the characters $q,q'$ is\n\\[\n p^{-1}\\sum_{d=0}^{p-1}\\omega_p^{(q'-q)d}=\\delta_{qq'}.\n\\]\nWavelets at different levels are orthogonal because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","because their detail\nprojectors are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","are orthogonal. There are exactly\n$(p-1)p^{r-1}$ wavelets at level $r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","$r$, equal to the rank in\n\\eqref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","qref{a-finite-detail-ranks}, so they span that detail subspace. Hence\n\\begin{equation}\n D_r^{(h)}=\\sum_{a=0}^{p^{r-1}-1}\\sum_{q=1}^{p-1}\n       |\\psi_{r,a,q}^{(h)}\\rangle\\langle\\psi_{r,a,q}^{(h)}|.\n \\label{a-finite-wavelet-projector}\n\\end{equation}\nThe omitted character $q=0$ is the parent average. For $p=2$ there is\none average and one difference per parent. For odd $p$ there is one\naverage and $p-1$ nontrivial child characters. The dyadic sign factor of\n\\eqref{a-finite-unit-hilbert} is independent of this average/detail\nsplit in the principal phase register.","\\adicfigure{02_haar_channels}{For $p=3$, the normalized child characters\ncontain one average and two orthogonal complex details. Solid curves\nshow real parts and dashed curves show imaginary parts. The level\nrank $(p-1)p^{r-1}$ grows by the branching factor. The atlas uses\n$\\mathsf F_r$ for the corresponding infinite cylinder detail; its\nfinite restriction is $D_r^{(h)}$ for $r\\leq h$.}","One more digit supplies $p$ children for every address. The uniform\nrefinement map is\n\\begin{equation}\n (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),\n \\quad 0\\leq a<p^h,\\quad0\\leq d<p.\n \\label{a-finite-refinement}\n\\end{equation}\nIn tensor coordinates it is $f\\mapsto|+_p\\rangle\\otimes f$,\nwhere $|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle$ and the child digit is\nlisted first. The norm calculation is\n\\[\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n\\]\nThe adjoint is the normalized amplitude sum\n\\begin{equation}\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n\\end{equation}\nConsequently\n\\begin{equation}\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=","One more digit supplies $p$ children for every address. The uniform\nrefinement map is\n\\begin{equation}\n (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),\n \\quad 0\\leq a<p^h,\\quad0\\leq d<p.\n \\label{a-finite-refinement}\n\\end{equation}\nIn tensor coordinates it is $f\\mapsto|+_p\\rangle\\otimes f$,\nwhere $|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle$ and the child digit is\nlisted first. The norm calculation is\n\\[\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n\\]\nThe adjoint is the normalized amplitude sum\n\\begin{equation}\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n\\end{equation}\nConsequently\n\\begin{equation}\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.\n \\label{a-finite-refinement-projectors}\n\\end{equation}\nThe first composition is an identity on the coarse space. The second\nis a projector on the fine space, n","One more digit supplies $p$ children for every address. The uniform\nrefinement map is\n\\begin{equation}\n (V_{p,h}f)(a+d p^h)=p^{-1/2}f(a),\n \\quad 0\\leq a<p^h,\\quad0\\leq d<p.\n \\label{a-finite-refinement}\n\\end{equation}\nIn tensor coordinates it is $f\\mapsto|+_p\\rangle\\otimes f$,\nwhere $|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle$ and the child digit is\nlisted first. The norm calculation is\n\\[\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n\\]\nThe adjoint is the normalized amplitude sum\n\\begin{equation}\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n\\end{equation}\nConsequently\n\\begin{equation}\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.\n \\label{a-finite-refinement-projectors}\n\\end{equation}\nThe first composition is an identity on the coarse space. The second\nis a projector on the fine space, not a fine-space identity.","_{p,h}f)(a+d p^h)=p^{-1/2}f(a),\n \\quad 0\\leq a<p^h,\\quad0\\leq d<p.\n \\label{a-finite-refinement}\n\\end{equation}\nIn tensor coordinates it is $f\\mapsto|+_p\\rangle\\otimes f$,\nwhere $|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle$ and the child digit is\nlisted first. The norm calculation is\n\\[\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n\\]\nThe adjoint is the normalized amplitude sum\n\\begin{equation}\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n\\end{equation}\nConsequently\n\\begin{equation}\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.\n \\label{a-finite-refinement-projectors}\n\\end{equation}\nThe first composition is an identity on the coarse space. The second\nis a projector on the fine space, not a fine-space identity.","is $f\\mapsto|+_p\\rangle\\otimes f$,\nwhere $|+_p\\rangle=p^{-1/2}\\sum_d|d\\rangle$ and the child digit is\nlisted first. The norm calculation is\n\\[\n \\norm{V_{p,h}f}^2=\\sum_a\\sum_d p^{-1}|f(a)|^2=\\norm f^2.\n\\]\nThe adjoint is the normalized amplitude sum\n\\begin{equation}\n (V_{p,h}^*g)(a)=p^{-1/2}\\sum_{d=0}^{p-1}g(a+d p^h).\n \\label{a-finite-refinement-adjoint}\n\\end{equation}\nConsequently\n\\begin{equation}\n V_{p,h}^*V_{p,h}=I,\\qquad\n V_{p,h}V_{p,h}^*=E_h^{(h+1)},\\qquad\n I-V_{p,h}V_{p,h}^*=D_{h+1}^{(h+1)}.\n \\label{a-finite-refinement-projectors}\n\\end{equation}\nThe first composition is an identity on the coarse space. The second\nis a projector on the fine space, not a fine-space identity.","Replicated vectors are constant in the new digit. All old cylinder\naverages commute with replication, giving\n\\begin{align}\n E_r^{(h+1)}V_{p,h}&=V_{p,h}E_r^{(h)}\\quad(0\\leq r\\leq h),\n \\nonumber\\\\\n D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\\quad(0\\leq r\\leq h),\n &D_{h+1}^{(h+1)}V_{p,h}&=0.\n \\label{a-finite-detail-intertwining}\n\\end{align}\nRepeated refinements compose as isometries. For $w\\geq0$ and one\ncoarse address,\n\\begin{equation}\n V_{p,h+w-1}\\cdots V_{p,h}|a\\rangle\n =p^{-w/2}\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}|a,\\mathbf d\\rangle,\n \\label{a-finite-nested-refinement}\n\\end{equation}\nwith the empty product interpreted as the identity. The factor\n$p^{-w/2}$ is forced by the isometry and orthonormality of the\ndescendant addresses.","D_r^{(h+1)}V_{p,h}&=V_{p,h}D_r^{(h)}\\quad(0\\leq r\\leq h),\n &D_{h+1}^{(h+1)}V_{p,h}&=0.\n \\label{a-finite-detail-intertwining}\n\\end{align}\nRepeated refinements compose as isometries. For $w\\geq0$ and one\ncoarse address,\n\\begin{equation}\n V_{p,h+w-1}\\cdots V_{p,h}|a\\rangle\n =p^{-w/2}\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}|a,\\mathbf d\\rangle,\n \\label{a-finite-nested-refinement}\n\\end{equation}\nwith the empty product interpreted as the identity. The factor\n$p^{-w/2}$ is forced by the isometry and orthonormality of the\ndescendant addresses.","h),\n &D_{h+1}^{(h+1)}V_{p,h}&=0.\n \\label{a-finite-detail-intertwining}\n\\end{align}\nRepeated refinements compose as isometries. For $w\\geq0$ and one\ncoarse address,\n\\begin{equation}\n V_{p,h+w-1}\\cdots V_{p,h}|a\\rangle\n =p^{-w/2}\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}|a,\\mathbf d\\rangle,\n \\label{a-finite-nested-refinement}\n\\end{equation}\nwith the empty product interpreted as the identity. The factor\n$p^{-w/2}$ is forced by the isometry and orthonormality of the\ndescendant addresses.","For $h\\geq1$, put $N=p^h$ and $m=p^{h-1}$. Two permutation\nunitaries separate different digits:\n\\begin{align}\n \\mathcal A^{\\mathrm{std}}_{p,h}|pk+r\\rangle\n &=|r\\rangle\\otimes|k\\rangle,\n &0\\leq r<p,\\quad0\\leq k<m,\\nonumber","For $h\\geq1$, put $N=p^h$ and $m=p^{h-1}$. Two permutation\nunitaries separate different digits:\n\\begin{align}\n \\mathcal A^{\\mathrm{std}}_{p,h}|pk+r\\rangle\n &=|r\\rangle\\otimes|k\\rangle,\n &0\\leq r<p,\\quad0\\leq k<m,\\nonumber\\\\\n \\mathcal A^{\\mathrm{ad}}_{p,h}|a+d p^{h-1}\\rangle\n &=|d\\rangle\\otimes|a\\rangle,\n &0\\leq d<p,\\quad0\\leq a<m.\n \\label{a-finite-digit-splits}\n\\end{align}\nThe standard polyphase split extracts the least significant residue.\nThe adic split extracts the newest, most significant digit at the\ncurrent depth. Let $\\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,\nwith $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the","\\mathcal A^{\\mathrm{ad}}_{p,h}|a+d p^{h-1}\\rangle\n &=|d\\rangle\\otimes|a\\rangle,\n &0\\leq d<p,\\quad0\\leq a<m.\n \\label{a-finite-digit-splits}\n\\end{align}\nThe standard polyphase split extracts the least significant residue.\nThe adic split extracts the newest, most significant digit at the\ncurrent depth. Let $\\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,\nwith $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","p^{h-1}\\rangle\n &=|d\\rangle\\otimes|a\\rangle,\n &0\\leq d<p,\\quad0\\leq a<m.\n \\label{a-finite-digit-splits}\n\\end{align}\nThe standard polyphase split extracts the least significant residue.\nThe adic split extracts the newest, most significant digit at the\ncurrent depth. Let $\\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,\nwith $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","ue.\nThe adic split extracts the newest, most significant digit at the\ncurrent depth. Let $\\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,\nwith $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","most significant digit at the\ncurrent depth. Let $\\mathcal R_{p,h}$ reverse the $h$ base-$p$ digits,\nwith $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","h $\\mathcal R_{p,0}=I$ on the one-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","-dimensional space. Then\n\\begin{equation}\n \\mathcal A^{\\mathrm{ad}}_{p,h}=\n (I_p\\otimes\\mathcal R_{p,h-1})\n \\mathcal A^{\\mathrm{std}}_{p,h}\\mathcal R_{p,h}.\n \\label{a-finite-digit-reversal}\n\\end{equation}\nIndeed reversing $d_0+d_1p+\\cdots+d_{h-1}p^{h-1}$ makes\n$d_{h-1}$ the least significant digit. Standard analysis removes that\ndigit, and reversing the remaining $h-1$ digits restores the prefix\n$a=d_0+\\cdots+d_{h-2}p^{h-2}$. This proves the identity on an\northonormal basis.","Let the new-digit modulation be\n$M_{p,h}|a+d p^{h-1}\\rangle=\\omega_p^d|a+d p^{h-1}\\rangle$.\nThe selector of child digit $d$ is the exact roots-of-unity filter\n\\begin{equation}\n R_d=\\frac1p\\sum_{\\ell=0}^{p-1}\\omega_p^{-d\\ell}M_{p,h}^{\\ell}.\n \\label{a-finite-root-selector}\n\\end{equation}\nOn a child with digit $e$, the coefficient is\n$p^{-1}\\sum_\\ell\\omega_p^{(e-d)\\ell}=\\delta_{de}$. Therefore\n$R_dR_e=\\delta_{de}R_d$, $R_d^*=R_d$, and $\\sum_dR_d=I$.\nThese projectors select address digits. They are different from the\nprojectors onto Fourier characters of a child register.","dulation be\n$M_{p,h}|a+d p^{h-1}\\rangle=\\omega_p^d|a+d p^{h-1}\\rangle$.\nThe selector of child digit $d$ is the exact roots-of-unity filter\n\\begin{equation}\n R_d=\\frac1p\\sum_{\\ell=0}^{p-1}\\omega_p^{-d\\ell}M_{p,h}^{\\ell}.\n \\label{a-finite-root-selector}\n\\end{equation}\nOn a child with digit $e$, the coefficient is\n$p^{-1}\\sum_\\ell\\omega_p^{(e-d)\\ell}=\\delta_{de}$. Therefore\n$R_dR_e=\\delta_{de}R_d$, $R_d^*=R_d$, and $\\sum_dR_d=I$.\nThese projectors select address digits. They are different from the\nprojectors onto Fourier characters of a child register.",",h}|a+d p^{h-1}\\rangle=\\omega_p^d|a+d p^{h-1}\\rangle$.\nThe selector of child digit $d$ is the exact roots-of-unity filter\n\\begin{equation}\n R_d=\\frac1p\\sum_{\\ell=0}^{p-1}\\omega_p^{-d\\ell}M_{p,h}^{\\ell}.\n \\label{a-finite-root-selector}\n\\end{equation}\nOn a child with digit $e$, the coefficient is\n$p^{-1}\\sum_\\ell\\omega_p^{(e-d)\\ell}=\\delta_{de}$. Therefore\n$R_dR_e=\\delta_{de}R_d$, $R_d^*=R_d$, and $\\sum_dR_d=I$.\nThese projectors select address digits. They are different from the\nprojectors onto Fourier characters of a child register.","Apply the child Fourier transform after adic analysis:\n\\begin{equation}\n \\mathcal W_{p,h}=(F_p\\otimes I_m)\\mathcal A^{\\mathrm{ad}}_{p,h}.\n \\label{a-finite-child-fourier}\n\\end{equation}\nThe zero child-frequency row of $F_p$ is $p^{-1/2}(1,\\ldots,1)$.\nHence\n\\begin{equation}\n\\begin{aligned}\n (\\langle0|\\otimes I_m)\\mathcal W_{p,h}&=V_{p,h-1}^*,","Apply the child Fourier transform after adic analysis:\n\\begin{equation}\n \\mathcal W_{p,h}=(F_p\\otimes I_m)\\mathcal A^{\\mathrm{ad}}_{p,h}.\n \\label{a-finite-child-fourier}\n\\end{equation}\nThe zero child-frequency row of $F_p$ is $p^{-1/2}(1,\\ldots,1)$.\nHence\n\\begin{equation}\n\\begin{aligned}\n (\\langle0|\\otimes I_m)\\mathcal W_{p,h}&=V_{p,h-1}^*,\\\\\n \\mathcal W_{p,h}^*\n \\bigl[(I_p-|0\\rangle\\langle0|)\\otimes I_m\\bigr]\n \\mathcal W_{p,h}&=D_h^{(h)}.\n\\end{aligned}\n\\label{a-finite-child-detail}\n\\end{equation}\nThe second identity subtracts the coarse zero-frequency projector from\nthe identity using \\eqref{a-finite-refinement-projectors}. Recursive\napplication to the zero-frequency channel alone constructs a unitary\nHaar analysis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonz…","\\mathcal W_{p,h}^*\n \\bigl[(I_p-|0\\rangle\\langle0|)\\otimes I_m\\bigr]\n \\mathcal W_{p,h}&=D_h^{(h)}.\n\\end{aligned}\n\\label{a-finite-child-detail}\n\\end{equation}\nThe second identity subtracts the coarse zero-frequency projector from\nthe identity using \\eqref{a-finite-refinement-projectors}. Recursive\napplication to the zero-frequency channel alone constructs a unitary\nHaar analysis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonzero character labels $q\\leftrightarrow p-q$ and leaves the\ndetail projectors unchanged.","p-|0\\rangle\\langle0|)\\otimes I_m\\bigr]\n \\mathcal W_{p,h}&=D_h^{(h)}.\n\\end{aligned}\n\\label{a-finite-child-detail}\n\\end{equation}\nThe second identity subtracts the coarse zero-frequency projector from\nthe identity using \\eqref{a-finite-refinement-projectors}. Recursive\napplication to the zero-frequency channel alone constructs a unitary\nHaar analysis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonzero character labels $q\\leftrightarrow p-q$ and leaves the\ndetail projectors unchanged.","d}\n\\label{a-finite-child-detail}\n\\end{equation}\nThe second identity subtracts the coarse zero-frequency projector from\nthe identity using \\eqref{a-finite-refinement-projectors}. Recursive\napplication to the zero-frequency channel alone constructs a unitary\nHaar analysis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonzero character labels $q\\leftrightarrow p-q$ and leaves the\ndetail projectors unchanged.","t-projectors}. Recursive\napplication to the zero-frequency channel alone constructs a unitary\nHaar analysis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonzero character labels $q\\leftrightarrow p-q$ and leaves the\ndetail projectors unchanged.","sis map $\\mathcal T_{p,h}$ whose channel projectors $\\Pi_r$\nhave the ranks in \\eqref{a-finite-detail-ranks} and obey\n\\begin{equation}\n \\mathcal T_{p,h}^*\\Pi_r\\mathcal T_{p,h}=D_r^{(h)}.\n \\label{a-finite-haar-analysis}\n\\end{equation}\nWith our positive Fourier sign, the analysis row of character $q$\npairs with the complex-conjugate wavelet character. This only permutes\nthe nonzero character labels $q\\leftrightarrow p-q$ and leaves the\ndetail projectors unchanged.","The zero child-frequency channel retains a uniform superposition over\nthe new digit. Retaining standard residue zero instead keeps one\naddress out of each $p$ addresses. Both spaces have the same dimension,\nbut they are distinct subspaces. Their compressed dynamics and Schur\ncomplements consequently need not agree. Digit reversal and child\nFourier analysis specify the transformations needed to compare them.","\\begin{theorem}[Finite refinement classification]\n\\label{a-finite-hamiltonian-classification}\nLet $V:\\mathcal H_c\\to\\mathcal H_f$ be an isometry between finite\nHilbert spaces. Set $P=VV^*$ and $Q=I-P$. For self-adjoint operators\n$H_c,H_f$, the following conditions are equivalent:\n\\begin{align}\n H_fV&=VH_c,\\label{a-finite-hamiltonian-intertwine}","\\begin{theorem}[Finite refinement classification]\n\\label{a-finite-hamiltonian-classification}\nLet $V:\\mathcal H_c\\to\\mathcal H_f$ be an isometry between finite\nHilbert spaces. Set $P=VV^*$ and $Q=I-P$. For self-adjoint operators\n$H_c,H_f$, the following conditions are equivalent:\n\\begin{align}\n H_fV&=VH_c,\\label{a-finite-hamiltonian-intertwine}\\\\\n QH_fP&=0,\\qquad PH_fP=VH_cV^*.\n \\label{a-finite-hamiltonian-blocks}\n\\end{align}\nThey hold exactly when, on $\\Ran P\\oplus\\Ran Q$,\n\\begin{equation}\n H_f=(VH_cV^*)|_{\\Ran P}\\ \\oplus\\ H_Q,\n \\qquad H_Q=(QH_fQ)|_{\\Ran Q}.\n \\label{a-finite-hamiltonian-direct-sum}\n\\end{equation}\nThe detail operator $H_Q$ may be any self-adjoint operator on\n$\\Ran Q$.\n\\end{theorem}","QH_fP&=0,\\qquad PH_fP=VH_cV^*.\n \\label{a-finite-hamiltonian-blocks}\n\\end{align}\nThey hold exactly when, on $\\Ran P\\oplus\\Ran Q$,\n\\begin{equation}\n H_f=(VH_cV^*)|_{\\Ran P}\\ \\oplus\\ H_Q,\n \\qquad H_Q=(QH_fQ)|_{\\Ran Q}.\n \\label{a-finite-hamiltonian-direct-sum}\n\\end{equation}\nThe detail operator $H_Q$ may be any self-adjoint operator on\n$\\Ran Q$.\n\\end{theorem}","\\begin{proof}\nRight multiplication of \\eqref{a-finite-hamiltonian-intertwine} by\n$V^*$ gives $H_fP=VH_cV^*$. Left multiplication by $Q$ and $P$\ngives \\eqref{a-finite-hamiltonian-blocks}. Conversely $PV=V$ and\n$V^*V=I$ imply\n\\[\n H_fV=(PH_fP+QH_fP)V=VH_cV^*V=VH_c.\n\\]\nSelf-adjointness gives $PH_fQ=(QH_fP)^*=0$. Thus both off-diagonal\nblocks vanish, and the remaining blocks are exactly those in\n\\eqref{a-finite-hamiltonian-direct-sum}. Conversely that direct sum\nhas the two block properties. There is no further condition on the\nself-adjoint detail block.\n\\end{proof}","The distinction between invariance and compression is essential.\nDefining $H_c=V^*H_fV$ always matches the coarse block, but leaves\n$QH_fV$ unconstrained. Thus compression alone does not establish\n\\eqref{a-finite-hamiltonian-intertwine}. When it does hold, the coarse\nsubspace is reducing, and fine evolution introduces no new detail\namplitude from a coarse initial state.","\\begin{corollary}[Functional-calculus intertwining]\n\\label{a-finite-functional-calculus}\nUnder Theorem~\\ref{a-finite-hamiltonian-classification}, for every\ncomplex function $f$ defined on the finite union of the spectra,\n\\[\n f(H_f)V=Vf(H_c).\n\\]\nIn particular, for real $t$, real $\\beta$, and\n$z\\notin\\sigma(H_f)$,\n\\begin{align}\n e^{-\\mathrm{i}tH_f}V&=Ve^{-\\mathrm{i}tH_c},\\nonumber","\\begin{corollary}[Functional-calculus intertwining]\n\\label{a-finite-functional-calculus}\nUnder Theorem~\\ref{a-finite-hamiltonian-classification}, for every\ncomplex function $f$ defined on the finite union of the spectra,\n\\[\n f(H_f)V=Vf(H_c).\n\\]\nIn particular, for real $t$, real $\\beta$, and\n$z\\notin\\sigma(H_f)$,\n\\begin{align}\n e^{-\\mathrm{i}tH_f}V&=Ve^{-\\mathrm{i}tH_c},\\nonumber\\\\\n e^{-\\beta H_f}V&=Ve^{-\\beta H_c},\\nonumber\\\\\n (z-H_f)^{-1}V&=V(z-H_c)^{-1}.\n \\label{a-finite-evolution-intertwining}\n\\end{align}\n\\end{corollary}","\\begin{proof}\nOn $\\Ran P$, $V$ is a unitary identification of $H_c$ with the\nfirst block of $H_f$. The finite spectral decomposition therefore\nmaps each coarse spectral subspace to the corresponding first-block\nspectral subspace. Applying $f$ to the eigenvalues gives the identity.\nThe three displayed functions are defined on the relevant spectra.\n\\end{proof}","The Euclidean operator $e^{-\\beta H}$ in this corollary is a spectral\nmultiplier. It is not generally unitary, and a normalized state after\nEuclidean evolution requires its own normalization. The identity does\nnot compare normalized Gibbs states of the entire fine and coarse\nsystems, since the fine partition function also counts the detail\nblock. For a compatible finite tower, the coarse propagator and its\nembedded fine propagator agree exactly. This does not make an arbitrary\ninteracting fine Hamiltonian compatible.","Choose a real sequence of level energies $\\epsilon_{p,r}$ and define\n\\begin{equation}\n H_{p,h}^{\\mathrm{Haar}}=\\sum_{r=0}^h\n          \\epsilon_{p,r}D_r^{(h)}.\n \\label{a-finite-hierarchical-hamiltonian}\n\\end{equation}\nEvery wavelet in $\\Ran D_r^{(h)}$ has energy $\\epsilon_{p,r}$.\nThe projectors in \\eqref{a-finite-detail-intertwining} show directly\nthat $H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}$. The new detail energy\n$\\epsilon_{p,h+1}$ does not alter the embedded older levels.\nMoreover the recursive Haar transform diagonalizes the level blocks:\n\\[\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n\\]\nThis is exact for each finite $h$ and every finite real choice of the\nenergies. Th","Choose a real sequence of level energies $\\epsilon_{p,r}$ and define\n\\begin{equation}\n H_{p,h}^{\\mathrm{Haar}}=\\sum_{r=0}^h\n          \\epsilon_{p,r}D_r^{(h)}.\n \\label{a-finite-hierarchical-hamiltonian}\n\\end{equation}\nEvery wavelet in $\\Ran D_r^{(h)}$ has energy $\\epsilon_{p,r}$.\nThe projectors in \\eqref{a-finite-detail-intertwining} show directly\nthat $H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}$. The new detail energy\n$\\epsilon_{p,h+1}$ does not alter the embedded older levels.\nMoreover the recursive Haar transform diagonalizes the level blocks:\n\\[\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n\\]\nThis is exact for each finite $h$ and every finite real choice of the\nenergies. Their growth becomes a domain question only when an\ninfinite-depth operator is constructed.","silon_{p,r}$ and define\n\\begin{equation}\n H_{p,h}^{\\mathrm{Haar}}=\\sum_{r=0}^h\n          \\epsilon_{p,r}D_r^{(h)}.\n \\label{a-finite-hierarchical-hamiltonian}\n\\end{equation}\nEvery wavelet in $\\Ran D_r^{(h)}$ has energy $\\epsilon_{p,r}$.\nThe projectors in \\eqref{a-finite-detail-intertwining} show directly\nthat $H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}$. The new detail energy\n$\\epsilon_{p,h+1}$ does not alter the embedded older levels.\nMoreover the recursive Haar transform diagonalizes the level blocks:\n\\[\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n\\]\nThis is exact for each finite $h$ and every finite real choice of the\nenergies. Their growth becomes a domain question only when an\ninfinite-depth operator is constructed.","hierarchical-hamiltonian}\n\\end{equation}\nEvery wavelet in $\\Ran D_r^{(h)}$ has energy $\\epsilon_{p,r}$.\nThe projectors in \\eqref{a-finite-detail-intertwining} show directly\nthat $H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}$. The new detail energy\n$\\epsilon_{p,h+1}$ does not alter the embedded older levels.\nMoreover the recursive Haar transform diagonalizes the level blocks:\n\\[\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n\\]\nThis is exact for each finite $h$ and every finite real choice of the\nenergies. Their growth becomes a domain question only when an\ninfinite-depth operator is constructed.","a-finite-detail-intertwining} show directly\nthat $H_{p,h+1}^{\\mathrm{Haar}}V_{p,h}\n=V_{p,h}H_{p,h}^{\\mathrm{Haar}}$. The new detail energy\n$\\epsilon_{p,h+1}$ does not alter the embedded older levels.\nMoreover the recursive Haar transform diagonalizes the level blocks:\n\\[\n \\mathcal T_{p,h}H_{p,h}^{\\mathrm{Haar}}\\mathcal T_{p,h}^*\n =\\sum_{r=0}^h\\epsilon_{p,r}\\Pi_r.\n\\]\nThis is exact for each finite $h$ and every finite real choice of the\nenergies. Their growth becomes a domain question only when an\ninfinite-depth operator is constructed.","Let the cyclic translation be $(S_{p,h}f)(x)=f(x-1)$, with\nsubtraction modulo $p^h$. Since a fine replicated function depends only\non the residue modulo $p^h$,\n\\[\n S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.\n\\]\nThe adjoint translation obeys the same identity. Therefore the cyclic\nLaplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real\n$\\kappa_p$ the family\n\\begin{equation}\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n\\end{equation}\nis self-adjoint and refinement compatible. Translation permutes the\nlevel-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and\n$D_r^{(h)}$. The Laplacian thus preserves every detail subspace,\nalthough a chosen spatially localized Haar wavelet need not be its\neigenvect","Let the cyclic translation be $(S_{p,h}f)(x)=f(x-1)$, with\nsubtraction modulo $p^h$. Since a fine replicated function depends only\non the residue modulo $p^h$,\n\\[\n S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.\n\\]\nThe adjoint translation obeys the same identity. Therefore the cyclic\nLaplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real\n$\\kappa_p$ the family\n\\begin{equation}\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n\\end{equation}\nis self-adjoint and refinement compatible. Translation permutes the\nlevel-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and\n$D_r^{(h)}$. The Laplacian thus preserves every detail subspace,\nalthough a chosen spatially localized Haar wavelet need not be its\neigenvector.","^h$,\n\\[\n S_{p,h+1}V_{p,h}=V_{p,h}S_{p,h}.\n\\]\nThe adjoint translation obeys the same identity. Therefore the cyclic\nLaplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real\n$\\kappa_p$ the family\n\\begin{equation}\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n\\end{equation}\nis self-adjoint and refinement compatible. Translation permutes the\nlevel-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and\n$D_r^{(h)}$. The Laplacian thus preserves every detail subspace,\nalthough a chosen spatially localized Haar wavelet need not be its\neigenvector.",".\n\\]\nThe adjoint translation obeys the same identity. Therefore the cyclic\nLaplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real\n$\\kappa_p$ the family\n\\begin{equation}\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n\\end{equation}\nis self-adjoint and refinement compatible. Translation permutes the\nlevel-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and\n$D_r^{(h)}$. The Laplacian thus preserves every detail subspace,\nalthough a chosen spatially localized Haar wavelet need not be its\neigenvector.","translation obeys the same identity. Therefore the cyclic\nLaplacian $L_{p,h}=2I-S_{p,h}-S_{p,h}^*$ intertwines, and for real\n$\\kappa_p$ the family\n\\begin{equation}\n H_{p,h}=H_{p,h}^{\\mathrm{Haar}}+\\kappa_p L_{p,h}\n \\label{a-finite-compatible-laplacian}\n\\end{equation}\nis self-adjoint and refinement compatible. Translation permutes the\nlevel-$r$ cylinders, so it commutes with each $E_r^{(h)}$ and\n$D_r^{(h)}$. The Laplacian thus preserves every detail subspace,\nalthough a chosen spatially localized Haar wavelet need not be its\neigenvector.","For the pure hierarchical Hamiltonian, the finite spectral theorem gives\n\\begin{equation}\n U_{p,h}(t)=e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}\n =\\sum_{r=0}^he^{-\\mathrm{i}t\\epsilon_{p,r}}D_r^{(h)}.\n \\label{a-finite-shell-propagator}\n\\end{equation}\nThe cylinder kernel is\n\\[\n E_r^{(h)}(x,y)=p^{-(h-r)}\n              \\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}.\n\\]\nConsequently\n\\begin{align}\n U_{p,h}(t;x,y)\n &=p^{-h}e^{-\\mathrm{i}t\\epsilon_{p,0}}\\nonumber","For the pure hierarchical Hamiltonian, the finite spectral theorem gives\n\\begin{equation}\n U_{p,h}(t)=e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}\n =\\sum_{r=0}^he^{-\\mathrm{i}t\\epsilon_{p,r}}D_r^{(h)}.\n \\label{a-finite-shell-propagator}\n\\end{equation}\nThe cylinder kernel is\n\\[\n E_r^{(h)}(x,y)=p^{-(h-r)}\n              \\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}.\n\\]\nConsequently\n\\begin{align}\n U_{p,h}(t;x,y)\n &=p^{-h}e^{-\\mathrm{i}t\\epsilon_{p,0}}\\nonumber\\\\\n &\\quad+\\sum_{r=1}^he^{-\\mathrm{i}t\\epsilon_{p,r}}\n \\left[p^{-(h-r)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}\n       -p^{-(h-r+1)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n\\end{align}\nEvery indicator is determined by the truncated valuation\n$\\min\\{v_p(x-y),h\\}$, with the diagonal assigned value $h$.\nThe transition amplitude, and therefore the individual-address Born\nprobability, is constant on each such valuation shell. At a fixed\ninitial address $y$, the shell of valuation $k<h$ contains\n$(p-1)p^{h-k-1}$ final addresses. Its total probability includes\nthis multiplicity. The diagonal has o…","&\\quad+\\sum_{r=1}^he^{-\\mathrm{i}t\\epsilon_{p,r}}\n \\left[p^{-(h-r)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}\n       -p^{-(h-r+1)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n\\end{align}\nEvery indicator is determined by the truncated valuation\n$\\min\\{v_p(x-y),h\\}$, with the diagonal assigned value $h$.\nThe transition amplitude, and therefore the individual-address Born\nprobability, is constant on each such valuation shell. At a fixed\ninitial address $y$, the shell of valuation $k<h$ contains\n$(p-1)p^{h-k-1}$ final addresses. Its total probability includes\nthis multiplicity. The diagonal has one address. Uniform probability\nper address on a shell does not mean equal total probabilities of\ndifferent shells.","{mod}\\,p^r)}\n       -p^{-(h-r+1)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n\\end{align}\nEvery indicator is determined by the truncated valuation\n$\\min\\{v_p(x-y),h\\}$, with the diagonal assigned value $h$.\nThe transition amplitude, and therefore the individual-address Born\nprobability, is constant on each such valuation shell. At a fixed\ninitial address $y$, the shell of valuation $k<h$ contains\n$(p-1)p^{h-k-1}$ final addresses. Its total probability includes\nthis multiplicity. The diagonal has one address. Uniform probability\nper address on a shell does not mean equal total probabilities of\ndifferent shells.","r+1)}\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n\\end{align}\nEvery indicator is determined by the truncated valuation\n$\\min\\{v_p(x-y),h\\}$, with the diagonal assigned value $h$.\nThe transition amplitude, and therefore the individual-address Born\nprobability, is constant on each such valuation shell. At a fixed\ninitial address $y$, the shell of valuation $k<h$ contains\n$(p-1)p^{h-k-1}$ final addresses. Its total probability includes\nthis multiplicity. The diagonal has one address. Uniform probability\nper address on a shell does not mean equal total probabilities of\ndifferent shells.","x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}\n \\right].\n \\label{a-finite-shell-kernel}\n\\end{align}\nEvery indicator is determined by the truncated valuation\n$\\min\\{v_p(x-y),h\\}$, with the diagonal assigned value $h$.\nThe transition amplitude, and therefore the individual-address Born\nprobability, is constant on each such valuation shell. At a fixed\ninitial address $y$, the shell of valuation $k<h$ contains\n$(p-1)p^{h-k-1}$ final addresses. Its total probability includes\nthis multiplicity. The diagonal has one address. Uniform probability\nper address on a shell does not mean equal total probabilities of\ndifferent shells.","For a self-adjoint fine Hamiltonian with coarse projector $P$, set\n$B=QH_fP$. If a unit vector $\\psi$ lies in $\\Ran P$, its initial\ndetail amplitude is\n\\[\n Qe^{-\\mathrm{i}tH_f}\\psi=-\\mathrm{i}tB\\psi+O(t^2),\n \\qquad\n \\norm{Qe^{-\\mathrm{i}tH_f}\\psi}^2\n =t^2\\norm{B\\psi}^2+O(|t|^3).\n\\]\nThese are finite Taylor expansions for fixed $H_f$ and $\\psi$.\nThus a nonzero $B$ supplies a state with nonzero short-time leakage.\nThe two-dimensional example\n\\[\n H_f=\\begin{pmatrix}0&\\overline\\gamma","se projector $P$, set\n$B=QH_fP$. If a unit vector $\\psi$ lies in $\\Ran P$, its initial\ndetail amplitude is\n\\[\n Qe^{-\\mathrm{i}tH_f}\\psi=-\\mathrm{i}tB\\psi+O(t^2),\n \\qquad\n \\norm{Qe^{-\\mathrm{i}tH_f}\\psi}^2\n =t^2\\norm{B\\psi}^2+O(|t|^3).\n\\]\nThese are finite Taylor expansions for fixed $H_f$ and $\\psi$.\nThus a nonzero $B$ supplies a state with nonzero short-time leakage.\nThe two-dimensional example\n\\[\n H_f=\\begin{pmatrix}0&\\overline\\gamma\\\\\\gamma&0\\end{pmatrix},\n \\qquad P=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\n\\]\nhas compressed Hamiltonian zero but detail probability\n$\\sin^2(|\\gamma|t)$ from the first basis state. It fails exact\nrefinement compatibility whenever $\\gamma\\neq0$.","0&0\\end{pmatrix}\n\\]\nhas compressed Hamiltonian zero but detail probability\n$\\sin^2(|\\gamma|t)$ from the first basis state. It fails exact\nrefinement compatibility whenever $\\gamma\\neq0$.","For $z$ outside the spectrum of $H_Q=(QH_fQ)|_{\\Ran Q}$, the\ncoarse Schur self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_fQ(z-H_Q)^{-1}QH_fP.\n \\label{a-finite-self-energy}\n\\end{equation}\nCompatibility makes it identically zero by setting $B=0$. This zero\nis an algebraic statement, not a uniform bound on a resolvent near\nits spectrum. A nonzero coupling requires the energy-dependent Schur\nterm when the detail sector is eliminated. The spectral part of this\nmonograph develops that elimination and its time-domain memory.","Refinement fixes the norm of an embedded packet. It does not fix an\ninteraction matrix element. The distinction becomes visible when one\ncoarse packet couples to many descendants rather than to one already\nnormalized aggregate packet. In this section $\\hbar=1$, so the\nHamiltonian has units of inverse time.","After $w$ refinements a packet has $n=p^w$ orthonormal descendants.\nIts normalized radial vector is\n\\begin{equation}\n |R_{p,w}\\rangle=n^{-1/2}\\sum_{\\mathbf d}|\\mathbf d\\rangle.\n \\label{a-finite-radial-vector}\n\\end{equation}\nThe norm is $n^{-1}\\sum_{\\mathbf d}1=1$, so each descendant has\namplitude modulus $p^{-w/2}$ and Born probability $p^{-w}$. This\nreproduces \\eqref{a-finite-nested-refinement}. The words ``radial'' and\n``angular'' here refer to the uniform vector and its orthogonal\ncomplement in a finite descendant space. They do not introduce a\nspacetime radius or a rotation group.","Introduce a visible address $|c\\rangle$ orthogonal to all descendants.\nSuppose every descendant has the same coupling from $|c\\rangle$,\nwith complex matrix element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=","Introduce a visible address $|c\\rangle$ orthogonal to all descendants.\nSuppose every descendant has the same coupling from $|c\\rangle$,\nwith complex matrix element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis","Introduce a visible address $|c\\rangle$ orthogonal to all descendants.\nSuppose every descendant has the same coupling from $|c\\rangle$,\nwith complex matrix element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coeffi","Introduce a visible address $|c\\rangle$ orthogonal to all descendants.\nSuppose every descendant has the same coupling from $|c\\rangle$,\nwith complex matrix element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","every descendant has the same coupling from $|c\\rangle$,\nwith complex matrix element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","element $\\gamma_{p,w}$. The Hermitian interaction\nis\n\\begin{equation}\n H_{\\mathrm{int}}=\n \\sum_{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","{\\mathbf d}\n \\left(\\gamma_{p,w}|\\mathbf d\\rangle\\langle c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","c|\n       +\\overline{\\gamma_{p,w}}|c\\rangle\\langle\\mathbf d|\\right).\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.",").\n \\label{a-finite-leaf-interaction}\n\\end{equation}\nThe conjugate reverse coefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","oefficient is necessary when\n$\\gamma_{p,w}$ is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","is not real. Define the collective coefficient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","cient\n\\begin{equation}\n G_{p,w}=\\sqrt n\\,\\gamma_{p,w}.\n \\label{a-finite-collective-coefficient}\n\\end{equation}\nThen $H_{\\mathrm{int}}|c\\rangle=G_{p,w}|R_{p,w}\\rangle$ and\n$H_{\\mathrm{int}}|R_{p,w}\\rangle=\n\\overline{G_{p,w}}|c\\rangle$. If\n$|A\\rangle=\\sum_{\\mathbf d}a_{\\mathbf d}|\\mathbf d\\rangle$\nis orthogonal to $|R_{p,w}\\rangle$, its coefficients satisfy\n$\\sum_{\\mathbf d}a_{\\mathbf d}=0$. Hence\n$H_{\\mathrm{int}}|A\\rangle=0$. Only the radial vector is bright\nunder this equal-leaf interaction.","This statement concerns the interaction block. A separate descendant\nHamiltonian can mix radial and angular vectors. To obtain an invariant\ntwo-dimensional radial subsystem, the full descendant Hamiltonian\nmust preserve the radial line, and the full Hamiltonian must leave its\nangular complement uncoupled from $|c\\rangle$. A degenerate hidden\nenergy $\\epsilon I$ is one sufficient choice.","Assume those invariance conditions and order the basis as\n$(|c\\rangle,|R_{p,w}\\rangle)$. With visible energy $E_c$ and\nradial energy $\\epsilon$, the exact bright block is\n\\begin{equation}\n H_{\\mathrm{bright}}=\n \\begin{pmatrix}E_c&\\overline G_{p,w}","Assume those invariance conditions and order the basis as\n$(|c\\rangle,|R_{p,w}\\rangle)$. With visible energy $E_c$ and\nradial energy $\\epsilon$, the exact bright block is\n\\begin{equation}\n H_{\\mathrm{bright}}=\n \\begin{pmatrix}E_c&\\overline G_{p,w}\\\\G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).","G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equat","G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is def","G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalent","G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance","G_{p,w}&\\epsilon\\end{pmatrix}.\n \\label{a-finite-bright-block}\n\\end{equation}\nSet $\\overline E=(E_c+\\epsilon)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","n)/2$,\n$\\delta=(E_c-\\epsilon)/2$, and\n$\\Omega=(\\delta^2+|G_{p,w}|^2)^{1/2}$. Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.",". Then\n$(H_{\\mathrm{bright}}-\\overline E I)^2=\\Omega^2I$, giving\n\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","\\begin{equation}\n e^{-\\mathrm{i}tH_{\\mathrm{bright}}}\n =e^{-\\mathrm{i}t\\overline E}\n \\left[I\\cos(\\Omega t)\n -\\mathrm{i}\\frac{H_{\\mathrm{bright}}-\\overline E I}{\\Omega}\n              \\sin(\\Omega t)\\right].\n \\label{a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","a-finite-bright-propagator}\n\\end{equation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","uation}\nAt $\\Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","Omega=0$ this formula is understood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","tood by continuous extension.\nStarting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","Starting in $|c\\rangle$, the radial probability is therefore\n\\begin{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","in{equation}\n \\Pr(R_{p,w};t)=\n \\frac{|G_{p,w}|^2}{\\delta^2+|G_{p,w}|^2}\n       \\sin^2\\!\\left(t\\sqrt{\\delta^2+|G_{p,w}|^2}\\right).\n \\label{a-finite-radial-probability}\n\\end{equation}\nThe probability is defined to be zero when $\\Omega=0$, equivalently\nwhen both $\\delta$ and $G_{p,w}$ vanish. On resonance $E_c=\\epsilon$ it reduces to\n$\\sin^2(|G_{p,w}|t)$. The individual descendant probabilities are\n$\\Pr(R_{p,w};t)/p^w$ when the hidden amplitude stays radial.","Eliminating the hidden radial level gives\n\\begin{equation}\n \\Sigma_{p,w}(z)=\\frac{|G_{p,w}|^2}{z-\\epsilon},\n \\qquad z\\neq\\epsilon.\n \\label{a-finite-radial-self-energy}\n\\end{equation}\nThis is the exact finite Schur term. The full visible Green function,\nwhen its denominator is nonzero, is\n$[z-E_c-\\Sigma_{p,w}(z)]^{-1}$. It is distinct from the self-energy\nitself, and the locations of its poles depend on both sectors.","Let $g\\in\\C$ be fixed and choose\n$\\gamma_{p,w}=g p^{-\\alpha w}$ for a real exponent $\\alpha$.\nThen\n\\begin{equation}\n G_{p,w}=g p^{(1/2-\\alpha)w},\\qquad\n |G_{p,w}|^2=|g|^2p^{(1-2\\alpha)w}.\n \\label{a-finite-coupling-exponent}\n\\end{equation}\nThe exact branching multiplicity produces the factor $p^{w/2}$.\nThus the collective coefficient grows with $w$ for $\\alpha<1/2$,\nis independent of $w$ for $\\alpha=1/2$, and decays for\n$\\alpha>1/2$. The same cases govern the operator norm of this\nequal-leaf interaction: its two radial eigenvalues are\n$\\pm|G_{p,w}|$, so $\\norm{H_{\\mathrm{int}}}=|G_{p,w}|$.\nThey do not determine the boundedness of a different infinite operator\nwith many interacting depth sectors.","d choose\n$\\gamma_{p,w}=g p^{-\\alpha w}$ for a real exponent $\\alpha$.\nThen\n\\begin{equation}\n G_{p,w}=g p^{(1/2-\\alpha)w},\\qquad\n |G_{p,w}|^2=|g|^2p^{(1-2\\alpha)w}.\n \\label{a-finite-coupling-exponent}\n\\end{equation}\nThe exact branching multiplicity produces the factor $p^{w/2}$.\nThus the collective coefficient grows with $w$ for $\\alpha<1/2$,\nis independent of $w$ for $\\alpha=1/2$, and decays for\n$\\alpha>1/2$. The same cases govern the operator norm of this\nequal-leaf interaction: its two radial eigenvalues are\n$\\pm|G_{p,w}|$, so $\\norm{H_{\\mathrm{int}}}=|G_{p,w}|$.\nThey do not determine the boundedness of a different infinite operator\nwith many interacting depth sectors.",")w},\\qquad\n |G_{p,w}|^2=|g|^2p^{(1-2\\alpha)w}.\n \\label{a-finite-coupling-exponent}\n\\end{equation}\nThe exact branching multiplicity produces the factor $p^{w/2}$.\nThus the collective coefficient grows with $w$ for $\\alpha<1/2$,\nis independent of $w$ for $\\alpha=1/2$, and decays for\n$\\alpha>1/2$. The same cases govern the operator norm of this\nequal-leaf interaction: its two radial eigenvalues are\n$\\pm|G_{p,w}|$, so $\\norm{H_{\\mathrm{int}}}=|G_{p,w}|$.\nThey do not determine the boundedness of a different infinite operator\nwith many interacting depth sectors.","}|^2=|g|^2p^{(1-2\\alpha)w}.\n \\label{a-finite-coupling-exponent}\n\\end{equation}\nThe exact branching multiplicity produces the factor $p^{w/2}$.\nThus the collective coefficient grows with $w$ for $\\alpha<1/2$,\nis independent of $w$ for $\\alpha=1/2$, and decays for\n$\\alpha>1/2$. The same cases govern the operator norm of this\nequal-leaf interaction: its two radial eigenvalues are\n$\\pm|G_{p,w}|$, so $\\norm{H_{\\mathrm{int}}}=|G_{p,w}|$.\nThey do not determine the boundedness of a different infinite operator\nwith many interacting depth sectors.","\\begin{center}\n\\begin{tabular}{@{}lccc@{}}\n\\toprule\nPer-leaf choice & $\\alpha$ & $|G_{p,w}|$ & $|G_{p,w}|^2$","\\midrule\nConstant matrix element & $0$ & $|g|p^{w/2}$ & $|g|^2p^w$","Constant collective coupling & $1/2$ & $|g|$ & $|g|^2$","Uniform measure coefficient & $1$ & $|g|p^{-w/2}$ & $|g|^2p^{-w}$","The measure coefficient $\\alpha=1$ assigns the number $p^{-w}$ to\neach per-leaf matrix element after factoring out $g$. It gives a\ncollective amplitude of size $p^{-w/2}$ and a self-energy numerator of\nsize $p^{-w}$. The self-energy estimate is uniform only where\n$|z-\\epsilon|$ stays bounded below. Likewise\n\\[\n \\Pr(R_{p,w};t)\\leq |g|^2t^2p^{-w}\n \\quad\\text{when }\\alpha=1\n\\]\nfollows from \\eqref{a-finite-radial-probability} and\n$|\\sin u|\\leq|u|$. It is a fixed-time bound. On resonance a time\nof order $p^{w/2}/|g|$ can still transfer order-one probability.","=1$ assigns the number $p^{-w}$ to\neach per-leaf matrix element after factoring out $g$. It gives a\ncollective amplitude of size $p^{-w/2}$ and a self-energy numerator of\nsize $p^{-w}$. The self-energy estimate is uniform only where\n$|z-\\epsilon|$ stays bounded below. Likewise\n\\[\n \\Pr(R_{p,w};t)\\leq |g|^2t^2p^{-w}\n \\quad\\text{when }\\alpha=1\n\\]\nfollows from \\eqref{a-finite-radial-probability} and\n$|\\sin u|\\leq|u|$. It is a fixed-time bound. On resonance a time\nof order $p^{w/2}/|g|$ can still transfer order-one probability.","oring out $g$. It gives a\ncollective amplitude of size $p^{-w/2}$ and a self-energy numerator of\nsize $p^{-w}$. The self-energy estimate is uniform only where\n$|z-\\epsilon|$ stays bounded below. Likewise\n\\[\n \\Pr(R_{p,w};t)\\leq |g|^2t^2p^{-w}\n \\quad\\text{when }\\alpha=1\n\\]\nfollows from \\eqref{a-finite-radial-probability} and\n$|\\sin u|\\leq|u|$. It is a fixed-time bound. On resonance a time\nof order $p^{w/2}/|g|$ can still transfer order-one probability.","A model that directly couples $|c\\rangle$ to one normalized hidden\npacket with matrix element $g p^{-w}$ instead has a collective\ncoefficient of that same size and a self-energy numerator\n$|g|^2p^{-2w}$. It is a valid two-level model, but it is not the\nequal-leaf model \\eqref{a-finite-leaf-interaction}. The number of\ndescendant addresses alone does not select between these interaction\ngeometries. A physical refinement law must specify the coupling,\nthe retained hidden modes, their energies, and the observation scale.","Fix a prime $p$. The compact additive group $\\Z_p$ has normalized Haar\nmeasure $\\mu_p$, with\n\\[\n B_{m,a}=a+p^m\\Z_p,\\qquad 0\\leq a<p^m,\\qquad\n \\mu_p(B_{m,a})=p^{-m}.\n\\]\nWrite $\\HH_p=L^2(\\Z_p,\\mu_p;\\C)$. We use the complex inner product\n$\\inner{f}{g}=\\int\\overline f g\\,d\\mu_p$, linear in its second argument.\nThe $p$-adic coordinate belongs to the measurable space. Amplitudes and the\nBorn probabilities use ordinary complex Hilbert space.","At depth $m$, the finite space $\\HH_{p,m}=\\ell^2(\\Z/p^m\\Z)$ has counting\ninner product and orthonormal basis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c","At depth $m$, the finite space $\\HH_{p,m}=\\ell^2(\\Z/p^m\\Z)$ has counting\ninner product and orthonormal basis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end","At depth $m$, the finite space $\\HH_{p,m}=\\ell^2(\\Z/p^m\\Z)$ has counting\ninner product and orthonormal basis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after t","At depth $m$, the finite space $\\HH_{p,m}=\\ell^2(\\Z/p^m\\Z)$ has counting\ninner product and orthonormal basis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","t and orthonormal basis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","asis $e_a$. Define\n\\begin{equation}\n \\mathcal U_{p,m}e_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","_a=p^{m/2}\\mathbf1_{B_{m,a}},\n \\qquad \\mathcal V_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","_m=\\operatorname{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","{span}\n \\{\\mathbf1_{B_{m,a}}:0\\leq a<p^m\\}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","}.\n \\label{a-limit-cylinder-embedding}\n\\end{equation}\nThe cylinders are disjoint. Consequently\n\\[\n \\inner{\\mathcal U_{p,m}e_a}{\\mathcal U_{p,m}e_b}\n =p^m\\mu_p(B_{m,a}\\cap B_{m,b})=\\delta_{ab}.\n\\]\nThe same indicators span $\\mathcal V_m$, so $\\mathcal U_{p,m}$ is unitary\nonto this subspace. For $c=\\sum_a c_ae_a$, its image has constant value\n$p^{m/2}c_a$ on $B_{m,a}$. In particular,\n\\begin{equation}\n \\int_{B_{m,a}}|\\mathcal U_{p,m}c|^2\\,d\\mu_p=|c_a|^2.\n \\label{a-limit-born-normalization}\n\\end{equation}\nThere is no extra probability factor after this normalization.","The finite refinement map is\n\\[\n V_{p,m}e_a=\\frac1{\\sqrt p}\n \\sum_{d=0}^{p-1}e_{a+d p^m}.\n\\]\nSince $B_{m,a}=\\bigsqcup_d B_{m+1,a+d p^m}$, it satisfies\n\\begin{equation}\n \\mathcal U_{p,m+1}V_{p,m}=\\mathcal U_{p,m}.\n \\label{a-limit-common-embedding}\n\\end{equation}\nIndeed, the coefficient on each child indicator is\n$p^{-1/2}p^{(m+1)/2}=p^{m/2}$, precisely the coefficient on the parent.\nThus the refinement changes the address resolution of an embedded state\nwithout changing the function in $\\HH_p$.","]\nSince $B_{m,a}=\\bigsqcup_d B_{m+1,a+d p^m}$, it satisfies\n\\begin{equation}\n \\mathcal U_{p,m+1}V_{p,m}=\\mathcal U_{p,m}.\n \\label{a-limit-common-embedding}\n\\end{equation}\nIndeed, the coefficient on each child indicator is\n$p^{-1/2}p^{(m+1)/2}=p^{m/2}$, precisely the coefficient on the parent.\nThus the refinement changes the address resolution of an embedded state\nwithout changing the function in $\\HH_p$.","An equivalent finite convention uses the weighted norm\n\\[\n \\norm{f}_{m,\\mathrm{Haar}}^2=p^{-m}\\sum_a|f(a)|^2.\n\\]\nHere $f(a)$ is a density amplitude rather than a counting amplitude, and\nthe embedding into $\\HH_p$ sends $f$ to\n$\\sum_a f(a)\\mathbf1_{B_{m,a}}$. Its refinement repeats the parent value\non all children, without $p^{-1/2}$. The coordinate change\n$f(a)=p^{m/2}c_a$ identifies the two conventions. The factors\n$p^{-m}$ in the measure and $p^{m/2}$ in the counting embedding are a\nsingle normalization expressed in two places.","convention uses the weighted norm\n\\[\n \\norm{f}_{m,\\mathrm{Haar}}^2=p^{-m}\\sum_a|f(a)|^2.\n\\]\nHere $f(a)$ is a density amplitude rather than a counting amplitude, and\nthe embedding into $\\HH_p$ sends $f$ to\n$\\sum_a f(a)\\mathbf1_{B_{m,a}}$. Its refinement repeats the parent value\non all children, without $p^{-1/2}$. The coordinate change\n$f(a)=p^{m/2}c_a$ identifies the two conventions. The factors\n$p^{-m}$ in the measure and $p^{m/2}$ in the counting embedding are a\nsingle normalization expressed in two places.","norm\n\\[\n \\norm{f}_{m,\\mathrm{Haar}}^2=p^{-m}\\sum_a|f(a)|^2.\n\\]\nHere $f(a)$ is a density amplitude rather than a counting amplitude, and\nthe embedding into $\\HH_p$ sends $f$ to\n$\\sum_a f(a)\\mathbf1_{B_{m,a}}$. Its refinement repeats the parent value\non all children, without $p^{-1/2}$. The coordinate change\n$f(a)=p^{m/2}c_a$ identifies the two conventions. The factors\n$p^{-m}$ in the measure and $p^{m/2}$ in the counting embedding are a\nsingle normalization expressed in two places.","For $f\\in\\HH_p$, define\n\\begin{equation}\n (\\mathsf E_m f)(x)\n =p^m\\int_{x+p^m\\Z_p} f(y)\\,d\\mu_p(y).\n \\label{a-limit-expectation}\n\\end{equation}\nThe integral is defined since $L^2$ is contained in $L^1$ on this\nprobability space. It produces a function constant on each depth-$m$\ncylinder.","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projec","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are neste","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller ra","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by takin","\\begin{proposition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoint","osition}[Nested cylinder projections]\n\\label{a-limit-expectation-algebra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","bra}\nThe operator $\\mathsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","athsf E_m$ is the orthogonal projection onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","ction onto $\\mathcal V_m$.\nFor $m,n\\geq0$,\n\\begin{equation}\n \\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","\\mathsf E_m^*=\\mathsf E_m,\\qquad\n \\mathsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","athsf E_m^2=\\mathsf E_m,\\qquad\n \\mathsf E_m\\mathsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","hsf E_n=\\mathsf E_{\\min(m,n)}.\n \\label{a-limit-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","it-projection-product}\n\\end{equation}\nMoreover $\\mathsf E_m f\\to f$ in $L^2$ for every $f\\in\\HH_p$.\n\\end{proposition}\n\\begin{proof}\nFor a cylinder-constant $g$, the integral of $f-\\mathsf E_m f$ over each\ndepth-$m$ cylinder is zero. Multiplying by the constant value of\n$\\overline g$ there and summing shows\n$\\inner{g}{f-\\mathsf E_mf}=0$. The image is in $\\mathcal V_m$, and averaging\na function in $\\mathcal V_m$ returns the function. These statements\ncharacterize the orthogonal projection and establish the first two\nidentities. The spaces are nested, because every cylinder partitions into\nits children. Orthogonal projections onto nested subspaces satisfy the\nthird identity: the larger projection fixes the smaller range, and the\nreverse order follows by taking adjoints.","For completeness, the union of these ranges is dense. Cylinders form a\ncountable clopen base for the topology on $\\Z_p$ and generate its Borel\nsigma-algebra. If $f\\in L^2$ is orthogonal to every cylinder indicator,\nthe finite complex measure $\\nu(A)=\\int_A f\\,d\\mu_p$ vanishes on\ncylinders and their finite unions. The class of sets on which $\\nu$ is\nzero contains the whole space, is closed under complements, and is\nclosed under countable disjoint unions. The elementary monotone-class\nextension from the cylinder algebra therefore gives $\\nu(A)=0$ for every\nBorel set. Applying this to the positive and negative level sets of the\nreal and imaginary parts of $f$ yields $f=0$ almost everywhere.\nThe orthogonal complement of $\\bigcup_m\\mathcal V_m$ is thus zero.","l\nsigma-algebra. If $f\\in L^2$ is orthogonal to every cylinder indicator,\nthe finite complex measure $\\nu(A)=\\int_A f\\,d\\mu_p$ vanishes on\ncylinders and their finite unions. The class of sets on which $\\nu$ is\nzero contains the whole space, is closed under complements, and is\nclosed under countable disjoint unions. The elementary monotone-class\nextension from the cylinder algebra therefore gives $\\nu(A)=0$ for every\nBorel set. Applying this to the positive and negative level sets of the\nreal and imaginary parts of $f$ yields $f=0$ almost everywhere.\nThe orthogonal complement of $\\bigcup_m\\mathcal V_m$ is thus zero.","mu_p$ vanishes on\ncylinders and their finite unions. The class of sets on which $\\nu$ is\nzero contains the whole space, is closed under complements, and is\nclosed under countable disjoint unions. The elementary monotone-class\nextension from the cylinder algebra therefore gives $\\nu(A)=0$ for every\nBorel set. Applying this to the positive and negative level sets of the\nreal and imaginary parts of $f$ yields $f=0$ almost everywhere.\nThe orthogonal complement of $\\bigcup_m\\mathcal V_m$ is thus zero.","hes on\ncylinders and their finite unions. The class of sets on which $\\nu$ is\nzero contains the whole space, is closed under complements, and is\nclosed under countable disjoint unions. The elementary monotone-class\nextension from the cylinder algebra therefore gives $\\nu(A)=0$ for every\nBorel set. Applying this to the positive and negative level sets of the\nreal and imaginary parts of $f$ yields $f=0$ almost everywhere.\nThe orthogonal complement of $\\bigcup_m\\mathcal V_m$ is thus zero.","sets on which $\\nu$ is\nzero contains the whole space, is closed under complements, and is\nclosed under countable disjoint unions. The elementary monotone-class\nextension from the cylinder algebra therefore gives $\\nu(A)=0$ for every\nBorel set. Applying this to the positive and negative level sets of the\nreal and imaginary parts of $f$ yields $f=0$ almost everywhere.\nThe orthogonal complement of $\\bigcup_m\\mathcal V_m$ is thus zero.","Given $\\varepsilon>0$, choose $g\\in\\mathcal V_M$ with\n$\\norm{f-g}_2<\\varepsilon$. For $m\\geq M$, the projection fixes $g$, so\n$\\norm{f-\\mathsf E_m f}_2\n=\\norm{(I-\\mathsf E_m)(f-g)}_2\\leq\\norm{f-g}_2<\\varepsilon$.\nThis proves strong convergence without exchanging an infinite integral\nwith a limit.\n\\end{proof}","The completion of the isometric direct system\n$(\\HH_{p,m},V_{p,m})$ is therefore $\\HH_p$: equation\n\\eqref{a-limit-common-embedding} gives a single isometric map on the\nalgebraic direct limit, and its image is the dense union of the\n$\\mathcal V_m$. For $m\\leq M$, with\n$V_{p,m\\to M}=V_{p,M-1}\\cdots V_{p,m}$, one also has\n\\begin{equation}\n \\mathsf E_m\\big|_{\\mathcal V_M}\n =\\mathcal U_{p,M}V_{p,m\\to M}V_{p,m\\to M}^*\n \\mathcal U_{p,M}^*.\n \\label{a-limit-finite-projection}\n\\end{equation}\nBoth sides are the orthogonal projection in $\\mathcal V_M$ onto\n$\\mathcal V_m$. This is the exact relation between the finite matrices\nand the common infinite Hilbert space.","gives a single isometric map on the\nalgebraic direct limit, and its image is the dense union of the\n$\\mathcal V_m$. For $m\\leq M$, with\n$V_{p,m\\to M}=V_{p,M-1}\\cdots V_{p,m}$, one also has\n\\begin{equation}\n \\mathsf E_m\\big|_{\\mathcal V_M}\n =\\mathcal U_{p,M}V_{p,m\\to M}V_{p,m\\to M}^*\n \\mathcal U_{p,M}^*.\n \\label{a-limit-finite-projection}\n\\end{equation}\nBoth sides are the orthogonal projection in $\\mathcal V_M$ onto\n$\\mathcal V_m$. This is the exact relation between the finite matrices\nand the common infinite Hilbert space.","tric map on the\nalgebraic direct limit, and its image is the dense union of the\n$\\mathcal V_m$. For $m\\leq M$, with\n$V_{p,m\\to M}=V_{p,M-1}\\cdots V_{p,m}$, one also has\n\\begin{equation}\n \\mathsf E_m\\big|_{\\mathcal V_M}\n =\\mathcal U_{p,M}V_{p,m\\to M}V_{p,m\\to M}^*\n \\mathcal U_{p,M}^*.\n \\label{a-limit-finite-projection}\n\\end{equation}\nBoth sides are the orthogonal projection in $\\mathcal V_M$ onto\n$\\mathcal V_m$. This is the exact relation between the finite matrices\nand the common infinite Hilbert space.","Use $\\mathsf F_0=\\mathsf E_0$ and\n$\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}$ for $m\\geq1$. These are the\ninfinite depth details called $\\mathsf D_m$ in the archived chapter.\nThey correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involvin","Use $\\mathsf F_0=\\mathsf E_0$ and\n$\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}$ for $m\\geq1$. These are the\ninfinite depth details called $\\mathsf D_m$ in the archived chapter.\nThey correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_","Use $\\mathsf F_0=\\mathsf E_0$ and\n$\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}$ for $m\\geq1$. These are the\ninfinite depth details called $\\mathsf D_m$ in the archived chapter.\nThey correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","$m\\geq1$. These are the\ninfinite depth details called $\\mathsf D_m$ in the archived chapter.\nThey correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","depth details called $\\mathsf D_m$ in the archived chapter.\nThey correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","correspond to the finite $D_m^{(M)}$ after the embedding\n$\\mathcal U_{p,M}$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","$. Equation \\eqref{a-limit-projection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","rojection-product} gives\n\\begin{equation}\n \\mathsf F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","F_m\\mathsf F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","F_n=\\delta_{mn}\\mathsf F_m,\n \\qquad \\sum_{m=0}^M\\mathsf F_m=\\mathsf E_M.\n \\label{a-limit-detail-algebra}\n\\end{equation}\nFor example, if $1\\leq m<n$, the four products in\n$(\\mathsf E_m-\\mathsf E_{m-1})(\\mathsf E_n-\\mathsf E_{n-1})$\nare $\\mathsf E_m,\\mathsf E_m,\\mathsf E_{m-1},\\mathsf E_{m-1}$ and cancel.\nFor $m=n$ the same expansion is\n$\\mathsf E_m-\\mathsf E_{m-1}$. The cases involving $m=0$ use\n$\\mathsf E_0\\mathsf E_n=\\mathsf E_0$.","Consequently\n\\begin{equation}\n \\HH_p=\\bigoplus_{m=0}^{\\infty}\\operatorname{Ran}\\mathsf F_m,\n \\qquad \\norm{f}_2^2=\\sum_{m=0}^{\\infty}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-parseval}\n\\end{equation}\nThe sum of the projections is $I$ in the strong operator topology.\nThe detail ranks are $d_0=1$ and $d_m=(p-1)p^{m-1}$ for $m\\geq1$,\nthe differences of the dimensions $p^m-p^{m-1}$.","For an explicit basis, put $\\omega_p=\\exp(2\\pi i/p)$ and define\n\\begin{equation}\n \\psi_{m,a,q}=p^{(m-1)/2}\\sum_{d=0}^{p-1}\n \\omega_p^{qd}\\mathbf1_{B_{m,a+d p^{m-1}}},\n \\quad\n \\begin{matrix}m\\geq1,\\quad 0\\leq a<p^{m-1},","For an explicit basis, put $\\omega_p=\\exp(2\\pi i/p)$ and define\n\\begin{equation}\n \\psi_{m,a,q}=p^{(m-1)/2}\\sum_{d=0}^{p-1}\n \\omega_p^{qd}\\mathbf1_{B_{m,a+d p^{m-1}}},\n \\quad\n \\begin{matrix}m\\geq1,\\quad 0\\leq a<p^{m-1},\\\\1\\leq q<p.\\end{matrix}\n \\label{a-limit-haar-wavelets}\n\\end{equation}\nEach has zero average over its parent, since\n$\\sum_d\\omega_p^{qd}=0$, and lies in $\\operatorname{Ran}\\mathsf F_m$.\nIts norm squared is $p^{m-1}p\\,p^{-m}=1$. Distinct parents have disjoint\nsupport, while two characters at the same parent have inner product\n$p^{-1}\\sum_d\\omega_p^{(q'-q)d}=\\delta_{qq'}$. The number of these\northonormal functions is exactly $d_m$, so they span that level.\nTogether with $\\psi_0=1$, they form a complete orthonormal basis by\n\\eqref{a-limit-parseval}.","1\\leq q<p.\\end{matrix}\n \\label{a-limit-haar-wavelets}\n\\end{equation}\nEach has zero average over its parent, since\n$\\sum_d\\omega_p^{qd}=0$, and lies in $\\operatorname{Ran}\\mathsf F_m$.\nIts norm squared is $p^{m-1}p\\,p^{-m}=1$. Distinct parents have disjoint\nsupport, while two characters at the same parent have inner product\n$p^{-1}\\sum_d\\omega_p^{(q'-q)d}=\\delta_{qq'}$. The number of these\northonormal functions is exactly $d_m$, so they span that level.\nTogether with $\\psi_0=1$, they form a complete orthonormal basis by\n\\eqref{a-limit-parseval}.","abel{a-limit-haar-wavelets}\n\\end{equation}\nEach has zero average over its parent, since\n$\\sum_d\\omega_p^{qd}=0$, and lies in $\\operatorname{Ran}\\mathsf F_m$.\nIts norm squared is $p^{m-1}p\\,p^{-m}=1$. Distinct parents have disjoint\nsupport, while two characters at the same parent have inner product\n$p^{-1}\\sum_d\\omega_p^{(q'-q)d}=\\delta_{qq'}$. The number of these\northonormal functions is exactly $d_m$, so they span that level.\nTogether with $\\psi_0=1$, they form a complete orthonormal basis by\n\\eqref{a-limit-parseval}.","h has zero average over its parent, since\n$\\sum_d\\omega_p^{qd}=0$, and lies in $\\operatorname{Ran}\\mathsf F_m$.\nIts norm squared is $p^{m-1}p\\,p^{-m}=1$. Distinct parents have disjoint\nsupport, while two characters at the same parent have inner product\n$p^{-1}\\sum_d\\omega_p^{(q'-q)d}=\\delta_{qq'}$. The number of these\northonormal functions is exactly $d_m$, so they span that level.\nTogether with $\\psi_0=1$, they form a complete orthonormal basis by\n\\eqref{a-limit-parseval}.","For $s>0$, define the Haar regularity norm\n\\[\n \\norm{f}_{\\mathfrak H_p^s}^2\n =\\sum_{m\\geq0}p^{2sm}\\norm{\\mathsf F_mf}_2^2,\n \\qquad\n \\mathfrak H_p^s=\\{f\\in\\HH_p:\\norm{f}_{\\mathfrak H_p^s}<\\infty\\}.\n\\]\nNo identification with another Sobolev convention is required here.\nThe definition itself gives the quantitative cutoff estimate.","\\begin{proposition}[Exact tail and regularity bound]\n\\label{a-limit-tail}\nFor $f\\in\\HH_p$ and every $M\\geq0$,\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ th","\\begin{proposition}[Exact tail and regularity bound]\n\\label{a-limit-tail}\nFor $f\\in\\HH_p$ and every $M\\geq0$,\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvecto","\\begin{proposition}[Exact tail and regularity bound]\n\\label{a-limit-tail}\nFor $f\\in\\HH_p$ and every $M\\geq0$,\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1","\\begin{proposition}[Exact tail and regularity bound]\n\\label{a-limit-tail}\nFor $f\\in\\HH_p$ and every $M\\geq0$,\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","[Exact tail and regularity bound]\n\\label{a-limit-tail}\nFor $f\\in\\HH_p$ and every $M\\geq0$,\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2^2\n =\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","\\label{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","{a-limit-tail-identity}\n\\end{equation}\nIf $f\\in\\mathfrak H_p^s$, then\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f}_2\n \\leq p^{-s(M+1)}\\norm{f}_{\\mathfrak H_p^s}.\n \\label{a-limit-tail-bound}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","nd}\n\\end{equation}\nThe convergence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","ence $\\mathsf E_M\\to I$ is strong and is not convergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","onvergence in\noperator norm.\n\\end{proposition}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","on}\n\\begin{proof}\nThe omitted details are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","ls are orthogonal, so Parseval gives the equality.\nFor $m>M$, $p^{2sm}\\geq p^{2s(M+1)}$. Multiplying the omitted summands\nby this lower bound proves the squared estimate, and taking square roots\ngives \\eqref{a-limit-tail-bound}. At every finite $M$ there is a unit\nvector in $\\operatorname{Ran}\\mathsf F_{M+1}$, on which\n$I-\\mathsf E_M$ acts as the identity. Thus\n$\\norm{I-\\mathsf E_M}=1$ for every $M$.\n\\end{proof}","The rate is a bound on states in the displayed regularity class. For an\narbitrary $L^2$ state the tail still tends to zero, but no common geometric\nrate follows. A finite array already in $\\mathcal V_L$ has zero tail once\n$M\\geq L$, which is a different statement from testing an infinite tail.","Let $(\\epsilon_m)_{m\\geq0}$ be any sequence of real numbers. Define\n\\begin{align}\n \\Dom(H_\\epsilon)\n &=\\left\\{f\\in\\HH_p:\n \\sum_{m\\geq0}|\\epsilon_m|^2\\norm{\\mathsf F_mf}_2^2<\\infty\\right\\},\n \\nonumber","Let $(\\epsilon_m)_{m\\geq0}$ be any sequence of real numbers. Define\n\\begin{align}\n \\Dom(H_\\epsilon)\n &=\\left\\{f\\in\\HH_p:\n \\sum_{m\\geq0}|\\epsilon_m|^2\\norm{\\mathsf F_mf}_2^2<\\infty\\right\\},\n \\nonumber\\\\\n H_\\epsilon f&=\\sum_{m\\geq0}\\epsilon_m\\mathsf F_mf.\n \\label{a-limit-hamiltonian-domain}\n\\end{align}\nThe series converges in $L^2$ exactly on this domain. It is not necessary\nto bound the energies uniformly, but it is necessary to state the domain.","H_\\epsilon f&=\\sum_{m\\geq0}\\epsilon_m\\mathsf F_mf.\n \\label{a-limit-hamiltonian-domain}\n\\end{align}\nThe series converges in $L^2$ exactly on this domain. It is not necessary\nto bound the energies uniformly, but it is necessary to state the domain.","\\begin{theorem}[Real level energies define a self-adjoint operator]\n\\label{a-limit-selfadjoint}\nThe operator in \\eqref{a-limit-hamiltonian-domain} is densely defined,\nclosed, and self-adjoint. The finite cylinder functions form an operator\ncore. It is bounded if and only if $\\sup_m|\\epsilon_m|<\\infty$, and its\nspectrum is\n\\begin{equation}\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees wit","\\begin{theorem}[Real level energies define a self-adjoint operator]\n\\label{a-limit-selfadjoint}\nThe operator in \\eqref{a-limit-hamiltonian-domain} is densely defined,\nclosed, and self-adjoint. The finite cylinder functions form an operator\ncore. It is bounded if and only if $\\sup_m|\\epsilon_m|<\\infty$, and its\nspectrum is\n\\begin{equation}\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore close…","iltonian-domain} is densely defined,\nclosed, and self-adjoint. The finite cylinder functions form an operator\ncore. It is bounded if and only if $\\sup_m|\\epsilon_m|<\\infty$, and its\nspectrum is\n\\begin{equation}\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","losed, and self-adjoint. The finite cylinder functions form an operator\ncore. It is bounded if and only if $\\sup_m|\\epsilon_m|<\\infty$, and its\nspectrum is\n\\begin{equation}\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","pectrum is\n\\begin{equation}\n \\sigma(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","(H_\\epsilon)=\\overline{\\{\\epsilon_m:m\\geq0\\}}.\n \\label{a-limit-spectrum}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","eorem}\n\\begin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","gin{proof}\nThe orthogonal decomposition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","composition \\eqref{a-limit-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","it-parseval} identifies $f$ with\nthe square-summable sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","able sequence $f_m=\\mathsf F_mf$. Finite sequences belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","belong\nto the domain, so it is dense. The real multipliers imply symmetry.\nTo identify the adjoint, suppose $g\\in\\Dom(H_\\epsilon^*)$ and write\n$k=H_\\epsilon^*g$. Testing the adjoint identity against an arbitrary\nvector supported on level $m$ gives $k_m=\\epsilon_m g_m$. Since $k$ is\nsquare summable, $\\sum_m|\\epsilon_m|^2\\norm{g_m}^2<\\infty$.\nThus $g\\in\\Dom(H_\\epsilon)$ and the adjoint agrees with $H_\\epsilon$.\nThe reverse inclusion is symmetry. This proves self-adjointness and\ntherefore closedness.","For $f\\in\\Dom(H_\\epsilon)$, the cylinder truncation $\\mathsf E_M f$\nsatisfies both\n$\\norm{f-\\mathsf E_M f}_2\\to0$ and\n$\\norm{H_\\epsilon(f-\\mathsf E_M f)}_2\\to0$, because the two squared norms\nare the tails of convergent nonnegative series. This is precisely density\nin the graph norm and proves the core assertion.\nIf the energies have supremum $C<\\infty$, Parseval gives\n$\\norm{H_\\epsilon f}_2\\leq C\\norm f_2$. Conversely, a unit vector on\nlevel $m$ has image norm $|\\epsilon_m|$, so boundedness forces a finite\nsupremum.","ion $\\mathsf E_M f$\nsatisfies both\n$\\norm{f-\\mathsf E_M f}_2\\to0$ and\n$\\norm{H_\\epsilon(f-\\mathsf E_M f)}_2\\to0$, because the two squared norms\nare the tails of convergent nonnegative series. This is precisely density\nin the graph norm and proves the core assertion.\nIf the energies have supremum $C<\\infty$, Parseval gives\n$\\norm{H_\\epsilon f}_2\\leq C\\norm f_2$. Conversely, a unit vector on\nlevel $m$ has image norm $|\\epsilon_m|$, so boundedness forces a finite\nsupremum.","satisfies both\n$\\norm{f-\\mathsf E_M f}_2\\to0$ and\n$\\norm{H_\\epsilon(f-\\mathsf E_M f)}_2\\to0$, because the two squared norms\nare the tails of convergent nonnegative series. This is precisely density\nin the graph norm and proves the core assertion.\nIf the energies have supremum $C<\\infty$, Parseval gives\n$\\norm{H_\\epsilon f}_2\\leq C\\norm f_2$. Conversely, a unit vector on\nlevel $m$ has image norm $|\\epsilon_m|$, so boundedness forces a finite\nsupremum.","If $z$ is outside the closed energy set, the sequence\n$(\\epsilon_m-z)^{-1}$ is bounded and defines an inverse for\n$H_\\epsilon-z$. Its image lies in the domain because\n$\\epsilon_m/(\\epsilon_m-z)=1+z/(\\epsilon_m-z)$ is bounded as well.\nThis proves that such $z$ is in the resolvent set. An energy\n$\\epsilon_m$ is an eigenvalue, while a limit point $z$ has unit level\nvectors $u_j$ with $\\norm{(H_\\epsilon-z)u_j}_2\\to0$.\nA bounded inverse would contradict this limit. Hence the closed energy\nset is exactly the spectrum.\n\\end{proof}","For bounded Borel $F:\\R\\to\\C$, define\n\\begin{equation}\n F(H_\\epsilon)f=\\sum_{m\\geq0}F(\\epsilon_m)\\mathsf F_mf.\n \\label{a-limit-borel-calculus}\n\\end{equation}\nThe series converges by Parseval and\n$\\norm{F(H_\\epsilon)f}_2\\leq\\sup_m|F(\\epsilon_m)|\\norm f_2$.\nThe same level decomposition verifies sums, products, and adjoints of\nthese multipliers directly. In particular,\n\\begin{align}\n U(t)&=e^{-itH_\\epsilon}\n =\\sum_{m\\geq0}e^{-it\\epsilon_m}\\mathsf F_m,\n \\label{a-limit-unitary}","For bounded Borel $F:\\R\\to\\C$, define\n\\begin{equation}\n F(H_\\epsilon)f=\\sum_{m\\geq0}F(\\epsilon_m)\\mathsf F_mf.\n \\label{a-limit-borel-calculus}\n\\end{equation}\nThe series converges by Parseval and\n$\\norm{F(H_\\epsilon)f}_2\\leq\\sup_m|F(\\epsilon_m)|\\norm f_2$.\nThe same level decomposition verifies sums, products, and adjoints of\nthese multipliers directly. In particular,\n\\begin{align}\n U(t)&=e^{-itH_\\epsilon}\n =\\sum_{m\\geq0}e^{-it\\epsilon_m}\\mathsf F_m,\n \\label{a-limit-unitary}\\\\\n (z-H_\\epsilon)^{-1}\n &=\\sum_{m\\geq0}(z-\\epsilon_m)^{-1}\\mathsf F_m,\n \\quad z\\notin\\sigma(H_\\epsilon).\n \\label{a-limit-resolvent}\n\\end{align}\nThe projections in these sums converge strongly, not absolutely in\noperator norm in general.","The modulus-one multipliers give $U(t)^*U(t)=I$ and\n$U(t+s)=U(t)U(s)$. They also give strong continuity:\n\\[\n \\norm{(U(t)-U(t_0))f}_2^2\n =\\sum_m|e^{-it\\epsilon_m}-e^{-it_0\\epsilon_m}|^2\n \\norm{\\mathsf F_mf}_2^2\\longrightarrow0.\n\\]\nTo justify this limit, first bound the omitted summands by\n$4\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2$, then use ordinary continuity in\nthe finite sum up to $M$. On $\\Dom(H_\\epsilon)$ the same argument,\nusing $|(e^{-it\\epsilon}-1)/t|\\leq|\\epsilon|$, proves\n\\[\n \\lim_{t\\to0}\\frac{U(t)f-f}{t}=-iH_\\epsilon f.\n\\]\nThus the declared self-adjoint operator is the generator. A general\n$L^2$ state evolves unitarily even if this derivative does not exist.","$U(t)^*U(t)=I$ and\n$U(t+s)=U(t)U(s)$. They also give strong continuity:\n\\[\n \\norm{(U(t)-U(t_0))f}_2^2\n =\\sum_m|e^{-it\\epsilon_m}-e^{-it_0\\epsilon_m}|^2\n \\norm{\\mathsf F_mf}_2^2\\longrightarrow0.\n\\]\nTo justify this limit, first bound the omitted summands by\n$4\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2$, then use ordinary continuity in\nthe finite sum up to $M$. On $\\Dom(H_\\epsilon)$ the same argument,\nusing $|(e^{-it\\epsilon}-1)/t|\\leq|\\epsilon|$, proves\n\\[\n \\lim_{t\\to0}\\frac{U(t)f-f}{t}=-iH_\\epsilon f.\n\\]\nThus the declared self-adjoint operator is the generator. A general\n$L^2$ state evolves unitarily even if this derivative does not exist.","ive strong continuity:\n\\[\n \\norm{(U(t)-U(t_0))f}_2^2\n =\\sum_m|e^{-it\\epsilon_m}-e^{-it_0\\epsilon_m}|^2\n \\norm{\\mathsf F_mf}_2^2\\longrightarrow0.\n\\]\nTo justify this limit, first bound the omitted summands by\n$4\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2$, then use ordinary continuity in\nthe finite sum up to $M$. On $\\Dom(H_\\epsilon)$ the same argument,\nusing $|(e^{-it\\epsilon}-1)/t|\\leq|\\epsilon|$, proves\n\\[\n \\lim_{t\\to0}\\frac{U(t)f-f}{t}=-iH_\\epsilon f.\n\\]\nThus the declared self-adjoint operator is the generator. A general\n$L^2$ state evolves unitarily even if this derivative does not exist.","mf}_2^2\\longrightarrow0.\n\\]\nTo justify this limit, first bound the omitted summands by\n$4\\sum_{m>M}\\norm{\\mathsf F_mf}_2^2$, then use ordinary continuity in\nthe finite sum up to $M$. On $\\Dom(H_\\epsilon)$ the same argument,\nusing $|(e^{-it\\epsilon}-1)/t|\\leq|\\epsilon|$, proves\n\\[\n \\lim_{t\\to0}\\frac{U(t)f-f}{t}=-iH_\\epsilon f.\n\\]\nThus the declared self-adjoint operator is the generator. A general\n$L^2$ state evolves unitarily even if this derivative does not exist.","Let $H_\\epsilon^{(M)}$ be the finite restriction of $H_\\epsilon$ to\n$\\mathcal V_M$. One useful approximation to $U(t)$ is its embedded\nfinite propagator, extended by zero on the omitted subspace:\n\\begin{equation}\n U_M^{\\mathrm{emb}}(t)\n =\\sum_{m=0}^{M}e^{-it\\epsilon_m}\\mathsf F_m\n =U(t)\\mathsf E_M.\n \\label{a-limit-embedded-propagator}\n\\end{equation}\nIt is unitary on $\\mathcal V_M$ and has kernel\n$\\mathcal V_M^\\perp$ as an operator on $\\HH_p$.","striction of $H_\\epsilon$ to\n$\\mathcal V_M$. One useful approximation to $U(t)$ is its embedded\nfinite propagator, extended by zero on the omitted subspace:\n\\begin{equation}\n U_M^{\\mathrm{emb}}(t)\n =\\sum_{m=0}^{M}e^{-it\\epsilon_m}\\mathsf F_m\n =U(t)\\mathsf E_M.\n \\label{a-limit-embedded-propagator}\n\\end{equation}\nIt is unitary on $\\mathcal V_M$ and has kernel\n$\\mathcal V_M^\\perp$ as an operator on $\\HH_p$.","\\Needspace{15\\baselineskip}\n\\begin{proposition}[Exact state error for a compatible cutoff]\n\\label{a-limit-unitary-error}\nFor every $f\\in\\HH_p$,\n\\begin{equation}\n \\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2\n =\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-unitary-tail}\n\\end{equation}\nThis equality is uniform for all real $t$. For\n$f\\in\\mathfrak H_p^s$, the right side is bounded by\n$p^{-s(M+1)}\\norm f_{\\mathfrak H_p^s}$.\nFor $f\\in\\Dom(H_\\epsilon)$ there is also the exact graph-norm identity\n\\begin{align}\n &\\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2^2\n +\\norm{H_\\epsilon(U(t)f-U_M^{\\mathrm{emb}}(t)f)}_2^2\n \\nonumber","lineskip}\n\\begin{proposition}[Exact state error for a compatible cutoff]\n\\label{a-limit-unitary-error}\nFor every $f\\in\\HH_p$,\n\\begin{equation}\n \\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2\n =\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-unitary-tail}\n\\end{equation}\nThis equality is uniform for all real $t$. For\n$f\\in\\mathfrak H_p^s$, the right side is bounded by\n$p^{-s(M+1)}\\norm f_{\\mathfrak H_p^s}$.\nFor $f\\in\\Dom(H_\\epsilon)$ there is also the exact graph-norm identity\n\\begin{align}\n &\\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2^2\n +\\norm{H_\\epsilon(U(t)f-U_M^{\\mathrm{emb}}(t)f)}_2^2\n \\nonumber","{a-limit-unitary-error}\nFor every $f\\in\\HH_p$,\n\\begin{equation}\n \\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2\n =\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-unitary-tail}\n\\end{equation}\nThis equality is uniform for all real $t$. For\n$f\\in\\mathfrak H_p^s$, the right side is bounded by\n$p^{-s(M+1)}\\norm f_{\\mathfrak H_p^s}$.\nFor $f\\in\\Dom(H_\\epsilon)$ there is also the exact graph-norm identity\n\\begin{align}\n &\\norm{U(t)f-U_M^{\\mathrm{emb}}(t)f}_2^2\n +\\norm{H_\\epsilon(U(t)f-U_M^{\\mathrm{emb}}(t)f)}_2^2\n \\nonumber\\\\\n &\\hspace{2em}=\n \\sum_{m>M}(1+|\\epsilon_m|^2)\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-graph-tail}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nThe difference is the omitted orthogonal sum\n$\\sum_{m>M}e^{-it\\epsilon_m}\\mathsf F_mf$. Its multipliers have modulus\none, so Parseval gives \\eqref{a-limit-unitary-tail}. Applying\n$H_\\epsilon$ multiplies the omitted level by $\\epsilon_m$ and gives the\nsecond identity. The domain hypothesis ensures that this second series\nis finite. The regularity estimate is Proposition \\ref{a-limit-tail}.\n\\end{proof}","&\\hspace{2em}=\n \\sum_{m>M}(1+|\\epsilon_m|^2)\\norm{\\mathsf F_mf}_2^2.\n \\label{a-limit-graph-tail}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nThe difference is the omitted orthogonal sum\n$\\sum_{m>M}e^{-it\\epsilon_m}\\mathsf F_mf$. Its multipliers have modulus\none, so Parseval gives \\eqref{a-limit-unitary-tail}. Applying\n$H_\\epsilon$ multiplies the omitted level by $\\epsilon_m$ and gives the\nsecond identity. The domain hypothesis ensures that this second series\nis finite. The regularity estimate is Proposition \\ref{a-limit-tail}.\n\\end{proof}","An alternative is to extend the finite Hamiltonian by the zero\nHamiltonian on the omitted subspace:\n\\[\n \\widetilde H_M=H_\\epsilon\\mathsf E_M\n =\\sum_{m\\leq M}\\epsilon_m\\mathsf F_m.\n\\]\nThis is a bounded self-adjoint operator on all of $\\HH_p$. Its\npropagator is a full-space unitary, but has a different tail:\n\\begin{equation}\n e^{-it\\widetilde H_M}\n =U_M^{\\mathrm{emb}}(t)+(I-\\mathsf E_M),\n \\qquad\n \\norm{(e^{-it\\widetilde H_M}-U(t))f}_2\n \\leq2\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-zero-hamiltonian-extension}\n\\end{equation}\nLikewise, for nonreal $z$,\n\\begin{align}\n (z-\\widetilde H_M)^{-1}\n &=\\mathsf E_M(z-H_\\epsilon)^{-1}\n   +z^{-1}(I-\\mathsf E_M),\\nonumber","eq M}\\epsilon_m\\mathsf F_m.\n\\]\nThis is a bounded self-adjoint operator on all of $\\HH_p$. Its\npropagator is a full-space unitary, but has a different tail:\n\\begin{equation}\n e^{-it\\widetilde H_M}\n =U_M^{\\mathrm{emb}}(t)+(I-\\mathsf E_M),\n \\qquad\n \\norm{(e^{-it\\widetilde H_M}-U(t))f}_2\n \\leq2\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-zero-hamiltonian-extension}\n\\end{equation}\nLikewise, for nonreal $z$,\n\\begin{align}\n (z-\\widetilde H_M)^{-1}\n &=\\mathsf E_M(z-H_\\epsilon)^{-1}\n   +z^{-1}(I-\\mathsf E_M),\\nonumber","athsf F_m.\n\\]\nThis is a bounded self-adjoint operator on all of $\\HH_p$. Its\npropagator is a full-space unitary, but has a different tail:\n\\begin{equation}\n e^{-it\\widetilde H_M}\n =U_M^{\\mathrm{emb}}(t)+(I-\\mathsf E_M),\n \\qquad\n \\norm{(e^{-it\\widetilde H_M}-U(t))f}_2\n \\leq2\\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-zero-hamiltonian-extension}\n\\end{equation}\nLikewise, for nonreal $z$,\n\\begin{align}\n (z-\\widetilde H_M)^{-1}\n &=\\mathsf E_M(z-H_\\epsilon)^{-1}\n   +z^{-1}(I-\\mathsf E_M),\\nonumber\\\\\n \\norm{((z-\\widetilde H_M)^{-1}-(z-H_\\epsilon)^{-1})f}_2\n &\\leq\\left(|z|^{-1}+|\\operatorname{Im}z|^{-1}\\right)\n \\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-strong-resolvent}\n\\end{align}\nThus $\\widetilde H_M$ converges to $H_\\epsilon$ in the strong resolvent\nsense. These statements specify operators on one common Hilbert space.\nThey do not compare matrices of changing dimensions without an embedding.","\\norm{((z-\\widetilde H_M)^{-1}-(z-H_\\epsilon)^{-1})f}_2\n &\\leq\\left(|z|^{-1}+|\\operatorname{Im}z|^{-1}\\right)\n \\norm{(I-\\mathsf E_M)f}_2.\n \\label{a-limit-strong-resolvent}\n\\end{align}\nThus $\\widetilde H_M$ converges to $H_\\epsilon$ in the strong resolvent\nsense. These statements specify operators on one common Hilbert space.\nThey do not compare matrices of changing dimensions without an embedding.","The embedded propagators converge strongly, uniformly in time on each\nfixed state. Their operator norm error nevertheless satisfies\n\\[\n \\norm{U(t)-U_M^{\\mathrm{emb}}(t)}=1\n \\qquad\\text{for every }M,t.\n\\]\nThe fully unitary extensions also converge strongly with the bound in\n\\eqref{a-limit-zero-hamiltonian-extension}; norm convergence of these\nextensions needs a separate condition on the omitted phases. Neither\nassertion is a general norm-limit theorem for arbitrary quantum dynamics.","\\begin{example}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolu","\\begin{example}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHam","\\begin{example}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rath","\\begin{example}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last l","\\begin{example}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","e}[Convergent states and finite energy derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","derivatives]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","ves]\n\\label{a-limit-geometric-example}\nChoose one unit vector $u_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","_m\\in\\operatorname{Ran}\\mathsf F_m$ at every\nlevel, let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}",", let $0<q<1$, and set\n\\[\n f_q=\\sum_{m\\geq0}\\sqrt{(1-q)q^m}\\,u_m,\n \\qquad \\epsilon_m=2^m.\n\\]\nThen $\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","$\\norm{f_q}_2=1$ and\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)f_q}_2^2=q^{M+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","+1}.\n \\label{a-limit-geometric-state-tail}\n\\end{equation}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","ion}\nHowever, $f_q\\in\\Dom(H_\\epsilon)$ exactly when $4q<1$:\n\\[\n \\norm{H_\\epsilon f_q}_2^2\n =(1-q)\\sum_{m\\geq0}(4q)^m.\n\\]\nFor $q<1/4$, its graph tail is exactly\n\\begin{equation}\n q^{M+1}+\\frac{(1-q)(4q)^{M+1}}{1-4q}.\n \\label{a-limit-geometric-graph-tail}\n\\end{equation}\nFor $q=1/8$, the state norm tail decays faster than the graph norm tail.\nFor $q=1/2$, the normalized state still has a convergent $L^2$ cylinder\napproximation and a well-defined unitary evolution, but is outside the\nHamiltonian domain. All these conclusions concern genuinely infinite\nseries, rather than a finite last level with its tail set to zero.\n\\end{example}","The omitted embedded resolvent has the exact norm\n\\begin{equation}\n \\norm{(I-\\mathsf E_M)(z-H_\\epsilon)^{-1}}\n =\\sup_{m>M}|z-\\epsilon_m|^{-1}.\n \\label{a-limit-resolvent-tail-norm}\n\\end{equation}\nThe upper bound follows by Parseval. For the lower bound, test on a unit\nvector at any omitted level and take the supremum. Since every level is\nnonzero and finite dimensional, this proves the following equivalence.","\\begin{proposition}[Exact compactness criterion]\n\\label{a-limit-compact-resolvent}\nThe operator $H_\\epsilon$ has compact resolvent if and only if\n$|\\epsilon_m|\\to\\infty$. In that case the embedded finite resolvents\nconverge in operator norm. This is an assertion about the embedded\nresolvents, rather than the zero-Hamiltonian extensions of\n\\eqref{a-limit-strong-resolvent}.\n\\end{proposition}\n\\begin{proof}\nIf the absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vec","\\begin{proposition}[Exact compactness criterion]\n\\label{a-limit-compact-resolvent}\nThe operator $H_\\epsilon$ has compact resolvent if and only if\n$|\\epsilon_m|\\to\\infty$. In that case the embedded finite resolvents\nconverge in operator norm. This is an assertion about the embedded\nresolvents, rather than the zero-Hamiltonian extensions of\n\\eqref{a-limit-strong-resolvent}.\n\\end{proposition}\n\\begin{proof}\nIf the absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonrea","limit-strong-resolvent}.\n\\end{proposition}\n\\begin{proof}\nIf the absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","g-resolvent}.\n\\end{proposition}\n\\begin{proof}\nIf the absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","egin{proof}\nIf the absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","absolute energies tend to infinity, the right side of\n\\eqref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","ref{a-limit-resolvent-tail-norm} tends to zero. The retained part of the\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","e\nresolvent has finite rank, so the resolvent is a norm limit of finite-rank\noperators and is compact. Conversely, if the absolute energies do not\ntend to infinity, there are infinitely many distinct levels $m_j$ with\n$|\\epsilon_{m_j}|\\leq C$ for some $C$. Unit vectors $u_j$ on these levels\nare orthogonal. For a fixed nonreal $z$, their resolvent images are also\northogonal and have norm at least $(|z|+C)^{-1}$.\nThey have no convergent subsequence. The resolvent is therefore not\ncompact.\n\\end{proof}","When $|\\epsilon_m|\\to\\infty$, the omitted part of the resolvent for\n$\\widetilde H_M$ in \\eqref{a-limit-strong-resolvent} still carries the\nconstant multiplier $z^{-1}$. It does not converge in norm to the omitted\nresolvent of $H_\\epsilon$. Strong resolvent convergence and norm\nconvergence of an embedded finite-rank resolvent are distinct claims.","For $\\beta>0$, the multiplier $e^{-\\beta H_\\epsilon}$ is bounded precisely\nwhen the energies are bounded below. If there is no lower bound, the\nlevelwise exponential is an unbounded operator with domain\n$\\{f:\\sum_m e^{-2\\beta\\epsilon_m}\\norm{\\mathsf F_mf}_2^2<\\infty\\}$.\nThus a finite Euclidean intertwining identity does not by itself supply a\nbounded infinite heat semigroup.\nWhen bounded below, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. I","For $\\beta>0$, the multiplier $e^{-\\beta H_\\epsilon}$ is bounded precisely\nwhen the energies are bounded below. If there is no lower bound, the\nlevelwise exponential is an unbounded operator with domain\n$\\{f:\\sum_m e^{-2\\beta\\epsilon_m}\\norm{\\mathsf F_mf}_2^2<\\infty\\}$.\nThus a finite Euclidean intertwining identity does not by itself supply a\nbounded infinite heat semigroup.\nWhen bounded below, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nserie","For $\\beta>0$, the multiplier $e^{-\\beta H_\\epsilon}$ is bounded precisely\nwhen the energies are bounded below. If there is no lower bound, the\nlevelwise exponential is an unbounded operator with domain\n$\\{f:\\sum_m e^{-2\\beta\\epsilon_m}\\norm{\\mathsf F_mf}_2^2<\\infty\\}$.\nThus a finite Euclidean intertwining identity does not by itself supply a\nbounded infinite heat semigroup.\nWhen bounded below, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential en","silon}$ is bounded precisely\nwhen the energies are bounded below. If there is no lower bound, the\nlevelwise exponential is an unbounded operator with domain\n$\\{f:\\sum_m e^{-2\\beta\\epsilon_m}\\norm{\\mathsf F_mf}_2^2<\\infty\\}$.\nThus a finite Euclidean intertwining identity does not by itself supply a\nbounded infinite heat semigroup.\nWhen bounded below, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in…","igroup.\nWhen bounded below, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","elow, the positive operator has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","tor has trace\n\\begin{equation}\n \\operatorname{Tr}(e^{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","{-\\beta H_\\epsilon})\n =e^{-\\beta\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","a\\epsilon_0}\n +\\sum_{m\\geq1}(p-1)p^{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","{m-1}e^{-\\beta\\epsilon_m},\n \\label{a-limit-thermal-trace}\n\\end{equation}\npossibly infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","y infinite. This follows by summing its nonnegative diagonal\nentries in the complete Haar basis. It is trace class exactly when this\nseries converges. For $\\epsilon_m=c m$ with $c>0$, the condition is\n$\\beta c>\\log p$, by the geometric-series test. For\n$\\epsilon_m=p^{\\alpha m}$ with $\\alpha>0$, it converges for every\n$\\beta>0$, since the exponential energy term eventually dominates the\nlinear-in-$m$ logarithm of the multiplicity. A thermodynamic partition\nfunction therefore needs an energy choice and a summability condition in\naddition to Hilbert-space reconstruction.","Let $\\mathcal P$ be a finite prime set and let $\\KK_{\\mathrm{tor}}$ be a\nfinite-dimensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qqu","Let $\\mathcal P$ be a finite prime set and let $\\KK_{\\mathrm{tor}}$ be a\nfinite-dimensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every","Let $\\mathcal P$ be a finite prime set and let $\\KK_{\\mathrm{tor}}$ be a\nfinite-dimensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n\\end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which te","Let $\\mathcal P$ be a finite prime set and let $\\KK_{\\mathrm{tor}}$ be a\nfinite-dimensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n\\end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which tends strongly to zero in each term,\nwhile all other projection factors have norm at most…","mathrm{tor}}$ be a\nfinite-dimensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n\\end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which tends strongly to zero in each term,\nwhile all other projection factors have norm at most one. Finite sums of\nelementary tensors are dense, and…","imensional complex space for the retained torsion and dyadic sign\nregisters. Set\n\\begin{equation}\n \\HH_{\\mathcal P}^{\\infty}\n =\\KK_{\\mathrm{tor}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n\\end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which tends strongly to zero in each term,\nwhile all other projection factors have norm at most one. Finite sums of\nelementary tensors are dense, and the projections are contra…","}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}L^2(\\Z_p,\\mu_p).\n \\label{a-limit-multiprime-space}\n\\end{equation}\nThe corresponding finite register embeds by\n$I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathcal U_{p,h_p}$.\nFor a multi-depth $\\mathbf h=(h_p)$, define\n\\[\n \\mathsf E_{\\mathbf h}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\\bigotimes_p\\mathsf E_{p,h_p}.\n\\]\nFactorwise projection identities give\n\\begin{equation}\n \\mathsf E_{\\mathbf h}\\mathsf E_{\\mathbf k}\n =\\mathsf E_{\\min(\\mathbf h,\\mathbf k)},\n \\qquad \\mathsf E_{\\mathbf h}f\\to f\n \\quad\\text{as every }h_p\\to\\infty.\n \\label{a-limit-rectangular-projections}\n\\end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which tends strongly to zero in each term,\nwhile all other projection factors have norm at most one. Finite sums of\nelementary tensors are dense, and the projections are contractions. This\nextends convergence to every $f$.","end{equation}\nTo justify the second statement, first take an elementary tensor. The\ndifference of the tensor projections and the identity is a finite\ntelescoping sum, one factor of which tends strongly to zero in each term,\nwhile all other projection factors have norm at most one. Finite sums of\nelementary tensors are dense, and the projections are contractions. This\nextends convergence to every $f$.","Refinement in distinct coordinates is path independent because\n\\[\n (V_p\\otimes I)(I\\otimes V_q)\n =(I\\otimes V_q)(V_p\\otimes I)=V_p\\otimes V_q.\n\\]\nThe identity on $\\KK_{\\mathrm{tor}}$ leaves the separately retained sign\nchannel untouched by the averaging. This is a finite set of primes.\nAn infinite tensor product, a restricted adelic product, and a model\nincluding the Archimedean place each require further choices.","Define the joint detail projections\n\\begin{equation}\n \\mathsf F_{\\mathbf r}\n =I_{\\KK_{\\mathrm{tor}}}\\otimes\n \\bigotimes_{p\\in\\mathcal P}\\mathsf F_{p,r_p},\n \\qquad \\mathbf r\\in\\N^{\\mathcal P}.\n \\label{a-limit-joint-details}\n\\end{equation}\nThey are pairwise orthogonal and have finite-dimensional ranges.\nRectangular sums equal $\\mathsf E_{\\mathbf h}$, so their strong sum is\nthe identity. This permits interactions between primes while preserving\nthe exact depth filtration.","\\begin{theorem}[Self-adjoint packet blocks and compatible finite dynamics]\n\\label{a-limit-block-hamiltonian}\nFor every $\\mathbf r$, let $A_{\\mathbf r}$ be a self-adjoint operator on\nthe finite-dimensional space\n$\\KK_{\\mathbf r}=\\operatorname{Ran}\\mathsf F_{\\mathbf r}$.\nOn $\\HH_{\\mathcal P}^{\\infty}=\\bigoplus_{\\mathbf r}\\KK_{\\mathbf r}$ define\n\\begin{align}\n \\Dom(H_A)&=\\left\\{f:\\sum_{\\mathbf r}\n \\norm{A_{\\mathbf r}\\mathsf F_{\\mathbf r}f}^2<\\infty\\right\\},\n \\nonumber","\\begin{theorem}[Self-adjoint packet blocks and compatible finite dynamics]\n\\label{a-limit-block-hamiltonian}\nFor every $\\mathbf r$, let $A_{\\mathbf r}$ be a self-adjoint operator on\nthe finite-dimensional space\n$\\KK_{\\mathbf r}=\\operatorname{Ran}\\mathsf F_{\\mathbf r}$.\nOn $\\HH_{\\mathcal P}^{\\infty}=\\bigoplus_{\\mathbf r}\\KK_{\\mathbf r}$ define\n\\begin{align}\n \\Dom(H_A)&=\\left\\{f:\\sum_{\\mathbf r}\n \\norm{A_{\\mathbf r}\\mathsf F_{\\mathbf r}f}^2<\\infty\\right\\},\n \\nonumber\\\\\n H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf…","H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and","H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*","H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_","H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $","H_A f&=\\sum_{\\mathbf r}A_{\\mathbf r}\\mathsf F_{\\mathbf r}f.\n \\label{a-limit-block-domain}\n\\end{align}\nThen $H_A$ is self-adjoint, and finite rectangular packet sums form a\ncore. Every rectangular range reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain conditi…","ge reduces $H_A$, and its finite restriction\n$H_A^{(\\mathbf h)}$ satisfies exact embedding intertwining with every\nlarger rectangular restriction. For bounded Borel $F$,\n\\begin{equation}\n F(H_A)f=\\sum_{\\mathbf r}F(A_{\\mathbf r})\\mathsf F_{\\mathbf r}f,\n \\qquad\n F(H_A)\\mathsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","thsf E_{\\mathbf h}\\to F(H_A)\n \\quad\\text{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","xt{strongly}.\n \\label{a-limit-block-calculus}\n\\end{equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","equation}\nIn particular,\n\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","\\begin{equation}\n \\norm{e^{-itH_A}(I-\\mathsf E_{\\mathbf h})f}\n =\\norm{(I-\\mathsf E_{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","{\\mathbf h})f}.\n \\label{a-limit-block-unitary-error}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","orem}\n\\begin{proof}\nFinite rectangular sums belong to the domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","domain and are dense. Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.",". Symmetry\nholds by symmetry on every block. If $g$ belongs to the adjoint domain\nand $k=H_A^*g$, testing against an arbitrary vector in\n$\\KK_{\\mathbf r}$ gives\n$\\mathsf F_{\\mathbf r}k=A_{\\mathbf r}\\mathsf F_{\\mathbf r}g$.\nSquare summability of $k$ is precisely the domain condition in\n\\eqref{a-limit-block-domain}. Thus $H_A^*=H_A$. Truncating the two\nsquare-summable sequences $f_{\\mathbf r}$ and\n$A_{\\mathbf r}f_{\\mathbf r}$ proves the graph-core assertion.","The rectangular projection selects a collection of whole blocks, so it\ncommutes with $H_A$ on the domain and reduces the operator.\nThe restriction to each retained block is literally the same matrix at\nevery subsequent depth. This proves the finite intertwining, and\nfinite-dimensional spectral calculus on each block gives\n\\eqref{a-limit-block-calculus}. Boundedness of $F$ bounds its omitted\nimage by $\\norm F_\\infty\\norm{(I-\\mathsf E_{\\mathbf h})f}$, proving\nstrong convergence. Finally, each block exponential is unitary, so\northogonality of the omitted blocks gives\n\\eqref{a-limit-block-unitary-error}.\n\\end{proof}","rectangular projection selects a collection of whole blocks, so it\ncommutes with $H_A$ on the domain and reduces the operator.\nThe restriction to each retained block is literally the same matrix at\nevery subsequent depth. This proves the finite intertwining, and\nfinite-dimensional spectral calculus on each block gives\n\\eqref{a-limit-block-calculus}. Boundedness of $F$ bounds its omitted\nimage by $\\norm F_\\infty\\norm{(I-\\mathsf E_{\\mathbf h})f}$, proving\nstrong convergence. Finally, each block exponential is unitary, so\northogonality of the omitted blocks gives\n\\eqref{a-limit-block-unitary-error}.\n\\end{proof}","For example, on a joint depth packet one may take\n\\[\n A_{\\mathbf r}\n =\\left(\\sum_a\\epsilon_{a,\\mathbf r}T_a\\right)\n \\otimes I_{\\mathrm{phase},\\mathbf r},\n \\qquad \\epsilon_{a,\\mathbf r}\\in\\R,\\quad T_a=T_a^*.\n\\]\nThe matrices $T_a$ act on the sign and torsion factors. Their mutual\ncommutation is unnecessary because their sum is self-adjoint on the\nfinite block. The coefficients assigned to an existing packet are fixed\nunder refinement. An extension acting wholly inside a new detail\ncomplement has no matrix element between that complement and the old\nrange. A cross term between them would be a different model.","There is also an exact construction for any \\emph{specified}\nself-adjoint compatible finite family along a cofinal nested sequence.\nSuppose finite-dimensional spaces $\\mathcal V_0\\subset\\mathcal V_1\n\\subset\\cdots$ have dense union, and self-adjoint $K_M$ satisfy\n$K_{M+1}f=K_Mf$ for $f\\in\\mathcal V_M$.\nSelf-adjointness implies that $\\mathcal V_M^\\perp\\cap\\mathcal V_{M+1}$\nis invariant: for $g$ there and $f\\in\\mathcal V_M$,\n$\\inner{f}{K_{M+1}g}=\\inner{K_{M+1}f}{g}=0$.\nThe initial space $\\mathcal V_0$ and the successive difference spaces\ntherefore have stable self-adjoint finite blocks.\nTheir orthogonal direct sum with the maximal square-summability domain\nis self-adjoint by the same adjoint proof above. This is an editorial\ncompatible-family construction, not an assertion that arbitrary finite\nHamiltonians are compatible. The identities $K_{M+1}|_{\\mathcal V_M}=K_M$\nare its substantive hypothesis.","s also an exact construction for any \\emph{specified}\nself-adjoint compatible finite family along a cofinal nested sequence.\nSuppose finite-dimensional spaces $\\mathcal V_0\\subset\\mathcal V_1\n\\subset\\cdots$ have dense union, and self-adjoint $K_M$ satisfy\n$K_{M+1}f=K_Mf$ for $f\\in\\mathcal V_M$.\nSelf-adjointness implies that $\\mathcal V_M^\\perp\\cap\\mathcal V_{M+1}$\nis invariant: for $g$ there and $f\\in\\mathcal V_M$,\n$\\inner{f}{K_{M+1}g}=\\inner{K_{M+1}f}{g}=0$.\nThe initial space $\\mathcal V_0$ and the successive difference spaces\ntherefore have stable self-adjoint finite blocks.\nTheir orthogonal direct sum with the maximal square-summability domain\nis self-adjoint by the same adjoint proof above. This is an editorial\ncompatible-family construction, not an assertion that arbitrary finite\nHamiltonians are compatible. The identities $K_{M+1}|_{\\mathcal V_M}=K_M$\nare its substantive hypothesis.","truction for any \\emph{specified}\nself-adjoint compatible finite family along a cofinal nested sequence.\nSuppose finite-dimensional spaces $\\mathcal V_0\\subset\\mathcal V_1\n\\subset\\cdots$ have dense union, and self-adjoint $K_M$ satisfy\n$K_{M+1}f=K_Mf$ for $f\\in\\mathcal V_M$.\nSelf-adjointness implies that $\\mathcal V_M^\\perp\\cap\\mathcal V_{M+1}$\nis invariant: for $g$ there and $f\\in\\mathcal V_M$,\n$\\inner{f}{K_{M+1}g}=\\inner{K_{M+1}f}{g}=0$.\nThe initial space $\\mathcal V_0$ and the successive difference spaces\ntherefore have stable self-adjoint finite blocks.\nTheir orthogonal direct sum with the maximal square-summability domain\nis self-adjoint by the same adjoint proof above. This is an editorial\ncompatible-family construction, not an assertion that arbitrary finite\nHamiltonians are compatible. The identities $K_{M+1}|_{\\mathcal V_M}=K_M$\nare its substantive hypothesis.","e family along a cofinal nested sequence.\nSuppose finite-dimensional spaces $\\mathcal V_0\\subset\\mathcal V_1\n\\subset\\cdots$ have dense union, and self-adjoint $K_M$ satisfy\n$K_{M+1}f=K_Mf$ for $f\\in\\mathcal V_M$.\nSelf-adjointness implies that $\\mathcal V_M^\\perp\\cap\\mathcal V_{M+1}$\nis invariant: for $g$ there and $f\\in\\mathcal V_M$,\n$\\inner{f}{K_{M+1}g}=\\inner{K_{M+1}f}{g}=0$.\nThe initial space $\\mathcal V_0$ and the successive difference spaces\ntherefore have stable self-adjoint finite blocks.\nTheir orthogonal direct sum with the maximal square-summability domain\nis self-adjoint by the same adjoint proof above. This is an editorial\ncompatible-family construction, not an assertion that arbitrary finite\nHamiltonians are compatible. The identities $K_{M+1}|_{\\mathcal V_M}=K_M$\nare its substantive hypothesis.","inner{K_{M+1}f}{g}=0$.\nThe initial space $\\mathcal V_0$ and the successive difference spaces\ntherefore have stable self-adjoint finite blocks.\nTheir orthogonal direct sum with the maximal square-summability domain\nis self-adjoint by the same adjoint proof above. This is an editorial\ncompatible-family construction, not an assertion that arbitrary finite\nHamiltonians are compatible. The identities $K_{M+1}|_{\\mathcal V_M}=K_M$\nare its substantive hypothesis.","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\oti","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equa","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two d","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states.","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one an","Let $Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two","$Z_{\\mathrm{sign}}$ have eigenvalues $+1,-1$ on a dyadic sign qubit,\nand let $B=B^*$ act on the remaining factors. Assume there are orthonormal\nstates $|b_0\\rangle,|b_1\\rangle$ with\n$B|b_0\\rangle=0$ and $B|b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensio…","b_1\\rangle=|b_1\\rangle$.\nOne concrete choice is an odd-prime torsion involution times a fixed\nodd-prime detail projector, with the torsion state in its $+1$ eigenspace.\nFor real $g\\neq0$, use\n\\[\n H_{\\mathrm{int}}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglem…","}=gZ_{\\mathrm{sign}}\\otimes B,\n \\qquad\n |\\Psi(0)\\rangle=|+_x\\rangle\\otimes\n \\frac{|b_0\\rangle+|b_1\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can","\\rangle}{\\sqrt2}.\n\\]\nEvolution under this explicit interaction alone gives\n\\begin{equation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alon…","quation}\n |\\Psi(t)\\rangle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition th…","angle=\\frac1{\\sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the resul…","sqrt2}\n \\left(|+_x\\rangle\\otimes|b_0\\rangle\n +e^{-igtZ_{\\mathrm{sign}}}|+_x\\rangle\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the result.","e\\otimes|b_1\\rangle\\right).\n \\label{a-limit-entangled-state}\n\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the result.","\\end{equation}\nTracing out the orthogonal $b$ states yields the equally weighted mixture\nof the two displayed sign states. Their overlap is $\\cos(gt)$.\nA mixture of two unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the result.","wo unit vectors with equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the result.","th equal weights and overlap magnitude\n$c$ has eigenvalues $(1+c)/2$ and $(1-c)/2$: its trace is one and its\ndeterminant on their two-dimensional span is $(1-c^2)/4$.\nThus the sign entropy in bits is\n\\begin{equation}\n S_{\\mathrm{sign}}(t)\n =h_2\\left(\\frac{1+|\\cos(gt)|}{2}\\right),\n \\quad h_2(u)=-u\\log_2u-(1-u)\\log_2(1-u),\n \\label{a-limit-entanglement-entropy}\n\\end{equation}\nwith $0\\log_2 0=0$. At $t_* =\\pi/(2|g|)$ the two sign states are\northogonal and the entropy is one bit. A fixed level projector is\ncompatible under refinement, so this interaction can entangle the prime\nfactors while preserving the cylinder embedding.\nThe tensor factorization alone creates no entanglement. It is this\nchosen generator and initial superposition that produce the result.","Let $V$ be an isometry from a coarse space to a finite fine space, set\n$P=VV^*$ and $Q=I-P$, and let $H_0=H_0^*$ satisfy\n$H_0V=VH_c$. Let $B=B^*$ be purely off diagonal:\n\\[\n PBP=0,\\qquad QBQ=0.\n\\]\nFor a real $\\lambda$, define $H_\\lambda=H_0+\\lambda B$. In the source\nmodel $\\lambda=p^{-w}$ for an assigned weight $w\\geq1$.","\\begin{proposition}[Defect, leakage, and a second-order self-energy]\n\\label{a-limit-leakage-bounds}\nUnder these hypotheses,\n\\begin{align}\n \\norm{H_\\lambda V-VH_c}&=|\\lambda|\\norm{BV},\n \\label{a-limit-defect-norm}\\\\\n \\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n\\end{align}\nFor $z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})$, the coarse self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t","\\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n\\end{align}\nFor $z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})$, the coarse self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-","\\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n\\end{align}\nFor $z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})$, the coarse self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integ","\\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n\\end{align}\nFor $z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})$, the coarse self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|","\\norm{Qe^{-itH_\\lambda}P}\n &\\leq\\min\\{1,|t\\lambda|\\norm{QBP}\\}.\n \\label{a-limit-duhamel-bound}\n\\end{align}\nFor $z\\notin\\sigma(QH_\\lambda Q|_{Q\\HH})$, the coarse self-energy is\n\\begin{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit…","n{equation}\n \\Sigma_P(z)=PH_\\lambda Q\n (z-QH_\\lambda Q|_{Q\\HH})^{-1}QH_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","_\\lambda P,\n \\label{a-limit-feshbach-definition}\n\\end{equation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","quation}\nwhere the inverse acts only on $Q\\HH$, and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","and\n\\begin{equation}\n \\norm{\\Sigma_P(z)}\n \\leq\\frac{|\\lambda|^2\\norm{PBQ}^2}\n {\\operatorname{dist}(z,\\sigma(QH_\\lambda Q|_{Q\\HH}))}.\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.",".\n \\label{a-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","-limit-feshbach-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe defect is $\\lambda BV$, so the first identity is immediate.\nSelf-adjointness of $H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","$H_0$ makes its invariant coarse range reducing.\nHence $Qe^{-itH_0}P=0$, and Duhamel's identity gives\n\\[\n Qe^{-itH_\\lambda}P\n =-i\\lambda\\int_0^t\n Qe^{-i(t-s)H_\\lambda}B e^{-isH_0}P\\,ds.\n\\]\nThe off-diagonal assumption and reduction by $H_0$ give\n$B e^{-isH_0}P=QBP e^{-isH_0}P$. The two propagators and the exterior\nprojections are contractions. Integrating the norm bound over an\ninterval of length $|t|$ gives the second expression in\n\\eqref{a-limit-duhamel-bound}. The bound one follows directly from\nunitarity and the norms of the projections.","The coarse/detail blocks of $H_0$ vanish, so\n$PH_\\lambda Q=\\lambda PBQ$ and $QH_\\lambda P=\\lambda QBP$.\nThe detail compression is self-adjoint. Diagonalizing it bounds the\nresolvent norm by the reciprocal spectral distance. Since\n$QBP=(PBQ)^*$, submultiplicativity gives\n\\eqref{a-limit-feshbach-bound}.\n\\end{proof}","The small-time leading term is $-it\\lambda QBP$. More precisely,\n\\begin{equation}\n \\norm{Qe^{-itH_\\lambda}P+it\\lambda QBP}\n \\leq\\frac{|\\lambda|t^2}{2}\n \\bigl(\\norm{H_\\lambda}+\\norm{H_0}\\bigr)\\norm{QBP}.\n \\label{a-limit-shorttime-remainder}\n\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","mbda QBP$. More precisely,\n\\begin{equation}\n \\norm{Qe^{-itH_\\lambda}P+it\\lambda QBP}\n \\leq\\frac{|\\lambda|t^2}{2}\n \\bigl(\\norm{H_\\lambda}+\\norm{H_0}\\bigr)\\norm{QBP}.\n \\label{a-limit-shorttime-remainder}\n\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","\\bigl(\\norm{H_\\lambda}+\\norm{H_0}\\bigr)\\norm{QBP}.\n \\label{a-limit-shorttime-remainder}\n\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","H_\\lambda}+\\norm{H_0}\\bigr)\\norm{QBP}.\n \\label{a-limit-shorttime-remainder}\n\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","}.\n \\label{a-limit-shorttime-remainder}\n\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","\\end{equation}\nTo verify this, subtract the constant integrand $QBP$ in Duhamel's\nformula and use $\\norm{e^{-iuK}-I}\\leq|u|\\norm K$ for a finite\nself-adjoint $K$. The two deviations are bounded by\n$(|t-s|\\norm{H_\\lambda}+|s|\\norm{H_0})\\norm{QBP}$ on the oriented\nintegration interval. Their integrals are both $t^2/2$.\nFor $|\\lambda|\\leq1$, this is a uniform $O(t^2|\\lambda|)$ remainder.\nA generic $O(t^2\\lambda^2)$ remainder would omit the first-order\ninteraction with the background Hamiltonian.","For the stipulated assignment $\\lambda=p^{-w}$, the operator defect and\nthis leakage upper bound scale linearly with $p^{-w}$. The self-energy\nbound is quadratic in $p^{-w}$ away from the detail spectrum. Under the\npurely off-diagonal hypothesis, $QH_\\lambda Q=QH_0Q$, so the denominator\ndoes not depend on $\\lambda$. Approaching a detail resonance removes\nthat uniform estimate. These are exact finite quantum statements for a\nchosen coupling law. An integral valuation packet does not, by itself,\nforce every physical interaction strength to equal $p^{-w}$.","quadratic in $p^{-w}$ away from the detail spectrum. Under the\npurely off-diagonal hypothesis, $QH_\\lambda Q=QH_0Q$, so the denominator\ndoes not depend on $\\lambda$. Approaching a detail resonance removes\nthat uniform estimate. These are exact finite quantum statements for a\nchosen coupling law. An integral valuation packet does not, by itself,\nforce every physical interaction strength to equal $p^{-w}$.","Let $N\\geq3$ and $h\\geq0$. Let $C_N$ be the physical cycle and let\n$T_{p,h}$ be the rooted finite $p$-ary tree of prefixes of length at most\n$h$. The product graph is\n\\[\n \\mathcal G=C_N\\square T_{p,h}.\n\\]\nA vertex is $v=(x,\\mathbf d)$, with $x\\in\\Z/N\\Z$ and $\\mathbf d$ a\nprefix. An edge changes either $x$ by one on the cycle or the prefix by\none parent-child step. The graph has\n$N\\sum_{r=0}^h p^r$ vertices. It includes all tree depths, unlike a\ncylinder register which contains only the residues at its last depth.","Let $\\vec E$ denote the directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-wal","Let $\\vec E$ denote the directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because","Let $\\vec E$ denote the directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U","Let $\\vec E$ denote the directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*","Let $\\vec E$ denote the directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","directed arcs and let $\\tau_2=2$,\n$\\tau_p=p-1$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","$ for odd $p$. The finite arc Hilbert space is\n\\[\n \\HH_{\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","\\mathrm{arc}}=\\ell^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","^2(\\vec E)\\otimes\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","times\\C^{\\tau_p}.\n\\]\nFor each vertex of degree $d_v$, define the normalized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","malized outgoing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","oing vector\n$s_v=d_v^{-1/2}(1,\\ldots,1)$ and the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","nd the Grover coin\n$G_v=2|s_v\\rangle\\langle s_v|-I$.\nIts square is $I$, because $|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","$|s_v\\rangle\\langle s_v|$ is a projection.\nTake any local unitary $R_v$ on the torsion or sign factor and put\n\\[\n C=\\bigoplus_v(G_v\\otimes R_v).\n\\]\nThe reversal shift $S$ sends the arc $v\\to u$ to $u\\to v$, retaining\nthe internal register. It is a permutation with $S^*=S$ and $S^2=I$.\nThe one-tick walk\n\\begin{equation}\n U_{p,h,N}=SC\n \\label{a-limit-walk-unitary}\n\\end{equation}\nis unitary because $U_{p,h,N}^*U_{p,h,N}=C^*S^*SC=C^*C=I$.\nHere locality concerns the graph edges, with no continuum limit assumed.","Write $\\psi^n(v,u)$ for the internal amplitude on an arc leaving $v$\nbefore the coin, and $\\phi^n=C\\psi^n$ for its post-coin value. Define\n\\[\n \\rho_v^n=\\sum_{u\\sim v}\\norm{\\psi^n(v,u)}^2,\n \\qquad F_{v\\to u}^n=\\norm{\\phi^n(v,u)}^2.\n\\]\nThe local unitary coin gives\n$\\sum_uF_{v\\to u}^n=\\rho_v^n$.\nThe reversal shift gives\n$\\rho_v^{n+1}=\\sum_u F_{u\\to v}^n$. Subtraction proves\n\\begin{equation}\n \\rho_v^{n+1}-\\rho_v^n\n =\\sum_{u\\sim v}(F_{u\\to v}^n-F_{v\\to u}^n).\n \\label{a-limit-walk-continuity}\n\\end{equation}\nFor every set of vertices $A$, summing this equation cancels every\ninternal directed flux once with its opposite occurrence. Hence\n\\begin{equation}\n \\sum_{v\\in A}(\\rho_v^{n+1}-\\rho_v^n)\n =\\sum_{\\substack{u\\notin","Write $\\psi^n(v,u)$ for the internal amplitude on an arc leaving $v$\nbefore the coin, and $\\phi^n=C\\psi^n$ for its post-coin value. Define\n\\[\n \\rho_v^n=\\sum_{u\\sim v}\\norm{\\psi^n(v,u)}^2,\n \\qquad F_{v\\to u}^n=\\norm{\\phi^n(v,u)}^2.\n\\]\nThe local unitary coin gives\n$\\sum_uF_{v\\to u}^n=\\rho_v^n$.\nThe reversal shift gives\n$\\rho_v^{n+1}=\\sum_u F_{u\\to v}^n$. Subtraction proves\n\\begin{equation}\n \\rho_v^{n+1}-\\rho_v^n\n =\\sum_{u\\sim v}(F_{u\\to v}^n-F_{v\\to u}^n).\n \\label{a-limit-walk-continuity}\n\\end{equation}\nFor every set of vertices $A$, summing this equation cancels every\ninternal directed flux once with its opposite occurrence. Hence\n\\begin{equation}\n \\sum_{v\\in A}(\\rho_v^{n+1}-\\rho_v^n)\n =\\sum_{\\substack{u\\notin A,\\ v\\in A","fine\n\\[\n \\rho_v^n=\\sum_{u\\sim v}\\norm{\\psi^n(v,u)}^2,\n \\qquad F_{v\\to u}^n=\\norm{\\phi^n(v,u)}^2.\n\\]\nThe local unitary coin gives\n$\\sum_uF_{v\\to u}^n=\\rho_v^n$.\nThe reversal shift gives\n$\\rho_v^{n+1}=\\sum_u F_{u\\to v}^n$. Subtraction proves\n\\begin{equation}\n \\rho_v^{n+1}-\\rho_v^n\n =\\sum_{u\\sim v}(F_{u\\to v}^n-F_{v\\to u}^n).\n \\label{a-limit-walk-continuity}\n\\end{equation}\nFor every set of vertices $A$, summing this equation cancels every\ninternal directed flux once with its opposite occurrence. Hence\n\\begin{equation}\n \\sum_{v\\in A}(\\rho_v^{n+1}-\\rho_v^n)\n =\\sum_{\\substack{u\\notin A,\\ v\\in A","in gives\n$\\sum_uF_{v\\to u}^n=\\rho_v^n$.\nThe reversal shift gives\n$\\rho_v^{n+1}=\\sum_u F_{u\\to v}^n$. Subtraction proves\n\\begin{equation}\n \\rho_v^{n+1}-\\rho_v^n\n =\\sum_{u\\sim v}(F_{u\\to v}^n-F_{v\\to u}^n).\n \\label{a-limit-walk-continuity}\n\\end{equation}\nFor every set of vertices $A$, summing this equation cancels every\ninternal directed flux once with its opposite occurrence. Hence\n\\begin{equation}\n \\sum_{v\\in A}(\\rho_v^{n+1}-\\rho_v^n)\n =\\sum_{\\substack{u\\notin A,\\ v\\in A\\\\u\\sim v}}F_{u\\to v}^n\n -\\sum_{\\substack{v\\in A,\\ u\\notin A\\\\u\\sim v}}F_{v\\to u}^n.\n \\label{a-limit-walk-coarse-balance}\n\\end{equation}\nTaking $A$ to be all vertices gives conservation of total probability.\nThe coarse identity uses a subset of vertices and does not presume\namplitude-level Haar refinement.","u\\sim v}}F_{v\\to u}^n.\n \\label{a-limit-walk-coarse-balance}\n\\end{equation}\nTaking $A$ to be all vertices gives conservation of total probability.\nThe coarse identity uses a subset of vertices and does not presume\namplitude-level Haar refinement.","\\begin{proposition}[Exact finite propagation]\n\\label{a-limit-walk-cone}\nIf the initial outgoing amplitudes are supported at $v_0$, then\n\\begin{equation}\n d_{C_N}(x,x_0)+d_T(\\mathbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n\\end{equation}\nFor two prefixes, writing $\\operatorname{lcp}$ for the length of their\nlongest common prefix,\n\\[\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n\\]\nIn particular, each tick allows at most one physical edge or one tree\nedge, and $\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n$.\n\\end{proposition}\n\\begin{proof}\nEvery path in a Cartesian product graph uses a number of physical edges\nat least the physical distance and a number of tree edges at least the\ntree distance. Their sum is a l","\\begin{proposition}[Exact finite propagation]\n\\label{a-limit-walk-cone}\nIf the initial outgoing amplitudes are supported at $v_0$, then\n\\begin{equation}\n d_{C_N}(x,x_0)+d_T(\\mathbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n\\end{equation}\nFor two prefixes, writing $\\operatorname{lcp}$ for the length of their\nlongest common prefix,\n\\[\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n\\]\nIn particular, each tick allows at most one physical edge or one tree\nedge, and $\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n$.\n\\end{proposition}\n\\begin{proof}\nEvery path in a Cartesian product graph uses a number of physical edges\nat least the physical distance and a number of tree edges at least the\ntree distance. Their sum is a lower bound on path length. Following a\nshortest physical path and then a shortest tree path attains that sum,\nso it is the product graph distance. In a","\\begin{proposition}[Exact finite propagation]\n\\label{a-limit-walk-cone}\nIf the initial outgoing amplitudes are supported at $v_0$, then\n\\begin{equation}\n d_{C_N}(x,x_0)+d_T(\\mathbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n\\end{equation}\nFor two prefixes, writing $\\operatorname{lcp}$ for the length of their\nlongest common prefix,\n\\[\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n\\]\nIn particular, each tick allows at most one physical edge or one tree\nedge, and $\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n$.\n\\end{proposition}\n\\begin{proof}\nEvery path in a Cartesian product graph uses a number of physical edges\nat least the physical distance and a number of tree edges at least the\ntree distance. Their sum is a lower bound on path length. Following a\nshortest physical path and then a shortest tree path attains that sum,\nso it is the product graph distance. In a rooted tree the shortest path\nrises to the longest common pr","\\begin{proposition}[Exact finite propagation]\n\\label{a-limit-walk-cone}\nIf the initial outgoing amplitudes are supported at $v_0$, then\n\\begin{equation}\n d_{C_N}(x,x_0)+d_T(\\mathbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n\\end{equation}\nFor two prefixes, writing $\\operatorname{lcp}$ for the length of their\nlongest common prefix,\n\\[\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n\\]\nIn particular, each tick allows at most one physical edge or one tree\nedge, and $\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n$.\n\\end{proposition}\n\\begin{proof}\nEvery path in a Cartesian product graph uses a number of physical edges\nat least the physical distance and a number of tree edges at least the\ntree distance. Their sum is a lower bound on path length. Following a\nshortest physical path and then a shortest tree path attains that sum,\nso it is the product graph distance. In a rooted tree the shortest path\nrises to the longest common prefix and then descends, giving the\ndisplayed formula.","thbf d,\\mathbf d_0)>n\n \\quad\\Longrightarrow\\quad\\rho_{(x,\\mathbf d)}^n=0.\n \\label{a-limit-product-cone}\n\\end{equation}\nFor two prefixes, writing $\\operatorname{lcp}$ for the length of their\nlongest common prefix,\n\\[\n d_T(\\mathbf d,\\mathbf d_0)\n =|\\mathbf d|+|\\mathbf d_0|-2\\operatorname{lcp}(\\mathbf d,\\mathbf d_0).\n\\]\nIn particular, each tick allows at most one physical edge or one tree\nedge, and $\\bigl||\\mathbf d|-|\\mathbf d_0|\\bigr|\\leq n$.\n\\end{proposition}\n\\begin{proof}\nEvery path in a Cartesian product graph uses a number of physical edges\nat least the physical distance and a number of tree edges at least the\ntree distance. Their sum is a lower bound on path length. Following a\nshortest physical path and then a shortest tree path attains that sum,\nso it is the product graph distance. In a rooted tree the shortest path\nrises to the longest common prefix and then descends, giving the\ndisplayed formula.","The claim at $n=0$ is the support hypothesis. A coin mixes outgoing\narcs only at their common source, so it does not enlarge the vertex\nsupport. The shift moves the source across one edge. A state supported\nin the radius-$n$ ball is therefore supported in the radius-$(n+1)$\nball after a tick. Induction proves the result. Since each graph edge\nchanges only one factor, the individual distance and depth bounds follow.\n\\end{proof}","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{e","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p","\\begin{example}[A boundary coin breaks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","ks the canonical arc embedding]\n\\label{a-limit-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","it-walk-refinement-counterexample}\nTake $h\\geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","geq1$ and $N\\geq3$, and embed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","mbed the old arcs of\n$C_N\\square T_{p,h}$ in $C_N\\square T_{p,h+1}$ by identical labels,\nwith zero amplitude on new arcs. Denote this isometry by $J$.\nAt an old leaf, the old degree is $d=3$: two cycle neighbors and one\nparent. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","nt. The larger graph adds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","ds $p$ children, changing the degree to $d+p$.\nFor an old outgoing arc basis vector $e_j$, the new Grover coin gives\namplitude $2/(d+p)$ on each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","each of the $p$ new outgoing child arcs. An\ninternal unitary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","ary preserves its internal norm. The subsequent reversal\nshift sends these amplitudes to new arcs at the children, outside the old\narc range. Therefore, for a unit internal state $\\chi$,\n\\begin{equation}\n \\norm{(I-JJ^*)U_{p,h+1,N}J(e_j\\otimes\\chi)}\n =\\frac{2\\sqrt p}{3+p}>0.\n \\label{a-limit-walk-refinement-leakage}\n\\end{equation}\nThe state $JU_{p,h,N}(e_j\\otimes\\chi)$ has zero amplitude there, so\n$U_{p,h+1,N}J\\neq JU_{p,h,N}$.\n\\end{example}","The exact result comprises complex $L^2$ cylinder reconstruction and\ndynamics for declared real level energies or self-adjoint packet blocks.\nState convergence uses an $L^2$ tail; graph norm convergence uses an\noperator-domain tail. Compact resolvents require energies escaping every\nbounded set, and finite thermal traces require multiplicity summability.\nEach claim has a distinct hypothesis.","The source programs check finite restrictions of these formulas. Code 45\nuses three towers,\n\\[\n (p,M)=(2,7),\\ (3,5),\\ (5,3),\n\\]\nand a fixed random seed. Their terminal zero tails do not certify an\ninfinite domain or compactness; the proofs supply those conclusions.\nCode 40 tests specified dyadic/ternary interactions and weighted\noff-diagonal couplings. Code 42 tests finite product walks, whose\nunitarity does not imply compatibility with the Haar tower.","Refinement & An isometry $V$ has $V^*V=I$. Its old-range projector is $P=VV^*$ and its new-detail projector is $Q=I-P$. Exact dynamical compatibility is a separate identity, $H'V=VH$.","Hierarchy & The finite chapter introduces cumulative projectors $E_r$ and their details. The infinite chapter writes $\\mathsf E_m$ for conditional expectations and $\\mathsf F_m=\\mathsf E_m-\\mathsf E_{m-1}$ for depth details, with the constant sector declared separately.","Spectral parameter & Schur formulas use $z-H$ and require the hidden resolvent $(z-QHQ)^{-1}$. Real-energy formulas are restricted to their regular chart. They are not evaluated at hidden poles without an extension argument.","\\item G. Teschl,\n\\emph{Jacobi Operators and Completely Integrable Nonlinear Lattices},\nMathematical Surveys and Monographs 72, American Mathematical Society, 2000.\n\\href{https://www.mat.univie.ac.at/~gerald/ftp/book-jac/jacop.pdf}{Author's online edition},\n\\S\\S1.1 and 2.1.\nJacobi continued-fraction and Weyl-function context.\nThe sign of a Weyl function changes when $(J-z)^{-1}$ is replaced by $(z-J)^{-1}$.","\\item S. V. Kozyrev,\n``Wavelet theory as $p$-adic spectral analysis,''\n\\emph{Izvestiya: Mathematics} 66(2), 367--376 (2002).\n\\href{https://doi.org/10.1070/IM2002v066n02ABEH000381}{doi:10.1070/IM2002v066n02ABEH000381}.\nPublished English Theorems 2 and 7 inspected.\nThe $\\mathbb Q_p$ Vladimirov setting supplies wavelet context;\nthe present $\\mathbb Z_p$ level-diagonal construction is proved separately.","We use complex Hilbert spaces with inner product linear in the second\nargument and set $\\hbar=1$. All matrices in the finite sections act on\ntheir stated coordinate spaces. In particular, the inverse of a low\nchannel block is a low channel operator, rather than a full space matrix.","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks sat","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{al","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_","Fix an integer $p\\geq2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","2$ and $M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","$M\\geq3$, and put $N=pM$. Primality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","rimality is not\nneeded for the identities in this section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","section. For $0\\leq r<p$, define the\nisometry $J_r:\\C^M\\to\\C^N$ by\n\\[\n J_r|k\\rangle=|pk+r\\rangle,\\qquad 0\\leq k<M.\n\\]\nThen $J_r^*J_s=\\delta_{rs}I_M$ and $\\sum_rJ_rJ_r^*=I_N$. The standard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","ndard\nanalysis map $\\mathcal A$ sends $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","nds $|pk+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","+r\\rangle$ to\n$|r\\rangle\\otimes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","mes|k\\rangle$. This is a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","a permutation unitary. For any\noperator $X$ on $\\C^N$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber","$, its residue blocks $X_{rs}=J_r^*XJ_s$ reconstruct it:\n\\begin{equation}\n X=\\sum_{r,s=0}^{p-1}J_rX_{rs}J_s^*,\\qquad\n (\\mathcal A X\\mathcal A^*)_{rs}=X_{rs}.\n \\label{a-spectral-block-reconstruction}\n\\end{equation}\nThe first equality follows by inserting $\\sum_rJ_rJ_r^*$ on both sides\nof $X$. If $H=H^*$, its blocks satisfy $H_{rs}^*=H_{sr}$. If $U$ is\nunitary, the same insertion gives\n\\begin{align}\n \\sum_rU_{rs}^*U_{rt}&=J_s^*U^*UJ_t=\\delta_{st}I_M,\\nonumber\\\\\n \\sum_sU_{rs}U_{ts}^*&=J_r^*UU^*J_t=\\delta_{rt}I_M.\n \\label{a-spectral-paraunitarity}\n\\end{align}\nThese are the block unitary identities. Every coordinate is retained.","For finite $H$, expansion of the matrix exponential proves\n\\[\n e^{-it\\mathcal A H\\mathcal A^*}\n   =\\mathcal A e^{-itH}\\mathcal A^*.\n\\]\nIndeed $(\\mathcal A H\\mathcal A^*)^n=\\mathcal A H^n\\mathcal A^*$,\nand the series converges in matrix norm. The same transformation of\neach Kraus operator gives a unitarily equivalent quantum channel:\nif $\\Phi(\\rho)=\\sum_\\mu K_\\mu\\rho K_\\mu^*$ with\n$\\sum_\\mu K_\\mu^*K_\\mu=I$, then\n\\[\n K_\\mu^{\\rm poly}=\\mathcal A K_\\mu\\mathcal A^*,\\qquad\n \\sum_\\mu(K_\\mu^{\\rm poly})^*K_\\mu^{\\rm poly}=I.\n\\]\nThis is a change of coordinates in a given channel. It does not supply\na new dissipative dynamics.","xpansion of the matrix exponential proves\n\\[\n e^{-it\\mathcal A H\\mathcal A^*}\n   =\\mathcal A e^{-itH}\\mathcal A^*.\n\\]\nIndeed $(\\mathcal A H\\mathcal A^*)^n=\\mathcal A H^n\\mathcal A^*$,\nand the series converges in matrix norm. The same transformation of\neach Kraus operator gives a unitarily equivalent quantum channel:\nif $\\Phi(\\rho)=\\sum_\\mu K_\\mu\\rho K_\\mu^*$ with\n$\\sum_\\mu K_\\mu^*K_\\mu=I$, then\n\\[\n K_\\mu^{\\rm poly}=\\mathcal A K_\\mu\\mathcal A^*,\\qquad\n \\sum_\\mu(K_\\mu^{\\rm poly})^*K_\\mu^{\\rm poly}=I.\n\\]\nThis is a change of coordinates in a given channel. It does not supply\na new dissipative dynamics.","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Probability currents between residue channels]\n\\label{a-spectral-channel-current}\nSuppose $i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{pr","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Probability currents between residue channels]\n\\label{a-spectral-channel-current}\nSuppose $i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a pr","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Probability currents between residue channels]\n\\label{a-spectral-channel-current}\nSuppose $i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Call","\\Needspace{16\\baselineskip}\n\\begin{proposition}[Probability currents between residue channels]\n\\label{a-spectral-channel-current}\nSuppose $i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Calling it an\nenergy would require an additional energy observable and its balance law.","nts between residue channels]\n\\label{a-spectral-channel-current}\nSuppose $i\\dot\\psi_r=\\sum_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Calling it an\nenergy would require an additional energy observable and its balance law.","um_sH_{rs}\\psi_s$ with $H_{rs}^*=H_{sr}$.\nThe channel weights $w_r=\\norm{\\psi_r}^2$ satisfy\n\\[\n \\dot w_r=\\sum_s\\mathcal J_{r\\leftarrow s},\\qquad\n \\mathcal J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Calling it an\nenergy would require an additional energy observable and its balance law.","al J_{r\\leftarrow s}\n   =2\\operatorname{Im}\\langle\\psi_r,H_{rs}\\psi_s\\rangle,\n \\qquad\n \\mathcal J_{r\\leftarrow s}=-\\mathcal J_{s\\leftarrow r}.\n\\]\nConsequently $\\sum_rw_r$ is constant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Calling it an\nenergy would require an additional energy observable and its balance law.","onstant.\n\\end{proposition}\n\\begin{proof}\nDifferentiating the norm gives\n$\\dot w_r=2\\operatorname{Re}\\langle\\psi_r,-i\\sum_sH_{rs}\\psi_s\\rangle$,\nwhich is the stated sum of imaginary parts. Self-adjoint block symmetry\nimplies\n$\\langle\\psi_s,H_{sr}\\psi_r\\rangle\n=\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}$.\nThe currents therefore cancel in pairs, including a zero diagonal current.\n\\end{proof}\nHere $w_r$ is a probability or squared amplitude weight. Calling it an\nenergy would require an additional energy observable and its balance law.","Let $S|n\\rangle=|n+1\\pmod N\\rangle$ and\n\\begin{equation}\n H=\\mu I+\\kappa(S+S^*),\\qquad \\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","tion}\n H=\\mu I+\\kappa(S+S^*),\\qquad \\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","mu I+\\kappa(S+S^*),\\qquad \\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","(S+S^*),\\qquad \\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","\\mu,\\kappa\\in\\R.\n \\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","\\label{a-spectral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","ctral-free-cycle}\n\\end{equation}\nUse the coarse Fourier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","rier matrix\n$(F_M)_{jk}=M^{-1/2}e^{-2\\pi ijk/M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,","M}$ and\n$\\theta_j=2\\pi j/M$. After $\\mathcal A$, Fourier transformation in $k$,\nand the permutation from residue major to momentum major ordering, the\noperator becomes a direct sum of $p\\times p$ symbols. To see their entries,\nnote that $S$ sends $(r,k)$ to $(r+1,k)$ for $r<p-1$ and sends\n$(p-1,k)$ to $(0,k+1)$. The coarse shift has Fourier eigenvalue\n$e^{-i\\theta_j}$. Thus define\n\\begin{equation}\n P_p(\\theta)e_r=\n \\begin{cases}\n e_{r+1},&r<p-1,\\\\\n e^{-i\\theta}e_0,&r=p-1.\n \\end{cases}\n \\qquad\n \\mathsf H_p(\\theta)=\\mu I_p+\\kappa(P_p(\\theta)+P_p(\\theta)^*).\n \\label{a-spectral-bloch-symbol}\n\\end{equation}\nThe transformed $H$ is $\\bigoplus_{j=0}^{M-1}\\mathsf H_p(\\theta_j)$.\nIn particular, $P_p(\\theta)^p=e^{-i\\theta}I_p$.","e^{-i\\theta}e_0,&r=p-1.\n \\end{cases}\n \\qquad\n \\mathsf H_p(\\theta)=\\mu I_p+\\kappa(P_p(\\theta)+P_p(\\theta)^*).\n \\label{a-spectral-bloch-symbol}\n\\end{equation}\nThe transformed $H$ is $\\bigoplus_{j=0}^{M-1}\\mathsf H_p(\\theta_j)$.\nIn particular, $P_p(\\theta)^p=e^{-i\\theta}I_p$.","\\begin{proposition}[Exact folded cyclic spectrum]\n\\label{a-spectral-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eq","\\begin{proposition}[Exact folded cyclic spectrum]\n\\label{a-spectral-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine F","\\begin{proposition}[Exact folded cyclic spectrum]\n\\label{a-spectral-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenv","\\begin{proposition}[Exact folded cyclic spectrum]\n\\label{a-spectral-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ p","\\begin{proposition}[Exact folded cyclic spectrum]\n\\label{a-spectral-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity st","al-folded-bands}\nFor $q=0,\\ldots,p-1$, the eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","e eigenvalues of\n$\\mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","mathsf H_p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","p(\\theta)$ are\n\\begin{equation}\n E_q(\\theta)=\\mu+2\\kappa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","pa\n     \\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right).\n \\label{a-spectral-band-formula}\n\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","\\end{equation}\nAt $\\theta=\\theta_j$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","$, the union of these $pM$ eigenvalues, counting\nmultiplicity, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","y, is exactly the spectrum of \\eqref{a-spectral-free-cycle}.\n\\end{proposition}\n\\begin{proof}\nPut $\\alpha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","ha_q=(\\theta+2\\pi q)/p$ and let the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","t the $r$th component of\n$v_q$ be $p^{-1/2}e^{i\\alpha_qr}$. On the interior components,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","onents,\n$P_p(\\theta)v_q=e^{-i\\alpha_q}v_q$. The wrapped component has the\nsame value because $e^{ip\\alpha_q}=e^{i\\theta}$. The $v_q$ are\northonormal by the finite character sum. Applying\n$\\mu I+\\kappa(P_p+P_p^*)$ gives \\eqref{a-spectral-band-formula}.\nEvery fine Fourier index is uniquely $k=j+qM$. Its angle satisfies\n$2\\pi k/(pM)=(\\theta_j+2\\pi q)/p$. Substitution into the fine cyclic\neigenvalue $\\mu+2\\kappa\\cos(2\\pi k/N)$ proves the multiplicity statement.\n\\end{proof}","The same construction works for any circulant polynomial\n$H=\\sum_\\ell c_\\ell S^\\ell$, with symbol\n$\\sum_\\ell c_\\ell P_p(\\theta)^\\ell$. It preserves all bands, whether or\nnot their energies coincide. For $N=p^h$, digit reversal relates this\nstandard residue split to the newest-digit adic split developed earlier.\nIt is still necessary to distinguish the retained residue-zero sites\nfrom the retained Haar average channel. They are different subspaces.","Let a finite Hilbert space be the orthogonal sum\n$\\HH_L\\oplus\\HH_D$, and write a self-adjoint Hamiltonian as\n\\begin{equation}\n H=\\begin{pmatrix}A&B","Let a finite Hilbert space be the orthogonal sum\n$\\HH_L\\oplus\\HH_D$, and write a self-adjoint Hamiltonian as\n\\begin{equation}\n H=\\begin{pmatrix}A&B\\\\B^*&D\\end{pmatrix},\\qquad A=A^*,\\quad D=D^*.\n \\label{a-spectral-block-hamiltonian}\n\\end{equation}\nHere $B:\\HH_D\\to\\HH_L$. Let $\\iota_L$ embed $\\HH_L$ into the full\nspace. This notation fixes the coordinate type of every compression.","B^*&D\\end{pmatrix},\\qquad A=A^*,\\quad D=D^*.\n \\label{a-spectral-block-hamiltonian}\n\\end{equation}\nHere $B:\\HH_D\\to\\HH_L$. Let $\\iota_L$ embed $\\HH_L$ into the full\nspace. This notation fixes the coordinate type of every compression.","\\begin{theorem}[Finite Feshbach identity]\n\\label{a-spectral-schur-theorem}\nIf $z\\notin\\sigma(D)$, define\n\\begin{equation}\n \\Sigma(z)=B(z-D)^{-1}B^*,\\qquad\n \\mathcal S(z)=z-A-\\Sigma(z).\n \\label{a-spectral-schur-definition}\n\\end{equation}\nThen $z-H$ is invertible exactly when $\\mathcal S(z)$ is invertible.\nOn this common regular set,\n\\begin{align}\n \\iota_L^*(z-H)^{-1}\\iota_L&=\\mathcal S(z)^{-1},\\nonumber","\\begin{theorem}[Finite Feshbach identity]\n\\label{a-spectral-schur-theorem}\nIf $z\\notin\\sigma(D)$, define\n\\begin{equation}\n \\Sigma(z)=B(z-D)^{-1}B^*,\\qquad\n \\mathcal S(z)=z-A-\\Sigma(z).\n \\label{a-spectral-schur-definition}\n\\end{equation}\nThen $z-H$ is invertible exactly when $\\mathcal S(z)$ is invertible.\nOn this common regular set,\n\\begin{align}\n \\iota_L^*(z-H)^{-1}\\iota_L&=\\mathcal S(z)^{-1},\\nonumber\\\\\n \\det(z-H)&=\\det(z-D)\\det\\mathcal S(z).\n \\label{a-spectral-schur-identities}\n\\end{align}\nFor $E\\in\\R\\setminus\\sigma(D)$, an eigenvector of $H$ is equivalent to\na nonzero solution of $\\mathcal S(E)u=0$, with hidden component\n$v=(E-D)^{-1}B^*u$.\n\\end{theorem}\n\\begin{proof}\nPut $C=z-D$. Direct block multiplication gives\n\\[\n z-H=\n \\begin{pmatrix}I&-BC^{-1}","\\det(z-H)&=\\det(z-D)\\det\\mathcal S(z).\n \\label{a-spectral-schur-identities}\n\\end{align}\nFor $E\\in\\R\\setminus\\sigma(D)$, an eigenvector of $H$ is equivalent to\na nonzero solution of $\\mathcal S(E)u=0$, with hidden component\n$v=(E-D)^{-1}B^*u$.\n\\end{theorem}\n\\begin{proof}\nPut $C=z-D$. Direct block multiplication gives\n\\[\n z-H=\n \\begin{pmatrix}I&-BC^{-1}","\\det(z-H)&=\\det(z-D)\\det\\mathcal S(z).\n \\label{a-spectral-schur-identities}\n\\end{align}\nFor $E\\in\\R\\setminus\\sigma(D)$, an eigenvector of $H$ is equivalent to\na nonzero solution of $\\mathcal S(E)u=0$, with hidden component\n$v=(E-D)^{-1}B^*u$.\n\\end{theorem}\n\\begin{proof}\nPut $C=z-D$. Direct block multiplication gives\n\\[\n z-H=\n \\begin{pmatrix}I&-BC^{-1}\\\\0&I\\end{pmatrix}\n \\begin{pmatrix}\\mathcal S(z)&0\\\\0&C\\end{pmatrix}\n \\begin{pmatrix}I&0\\\\-C^{-1}B^*&I\\end{pmatrix}.\n\\]\nBoth outer factors are invertible and have determinant one. This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","-C^{-1}B^*&I\\end{pmatrix}.\n\\]\nBoth outer factors are invertible and have determinant one. This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","ter factors are invertible and have determinant one. This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","nvertible and have determinant one. This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","erminant one. This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","This proves\nthe invertibility equivalence and determinant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","eterminant formula. Inverting the\nfactors gives the stated upper left block. Alternatively, in\n$(z-H)(u,v)^{\\mathsf T}=(f,0)^{\\mathsf T}$, the lower equation gives\n$v=C^{-1}B^*u$ and the upper equation becomes $\\mathcal S(z)u=f$.\nThe same elimination with $f=0$ proves the eigenvector statement.\nIf $u=0$ at such an $E$, invertibility of $E-D$ forces $v=0$, so a\nnonzero full eigenvector indeed has $u\\ne0$.\n\\end{proof}","If $P_L=\\iota_L\\iota_L^*$, the full space compression is\n$P_L(z-H)^{-1}P_L=\\iota_L\\mathcal S(z)^{-1}\\iota_L^*$.\nWhen a Haar transform $\\mathcal W$ first exposes low and detail channels,\nthe embedding in the original coordinates is\n$\\iota_L=\\mathcal W^*(|0\\rangle\\otimes I)$.\nThe inverse on $\\HH_L$ must be embedded in this way before it is compared\nwith a full space matrix.","The determinant equality is initially a rational identity on the regular\nset. Multiplication by $\\det(z-D)$ cancels its denominators and yields the\nfull characteristic polynomial, which extends across $\\sigma(D)$.\nThis extension does not make the displayed inverse of $z-D$ exist at\na pole. At a resonant energy the uneliminated equations determine which\nhidden modes couple, survive, or cancel. A resonant subspace can be kept\nas part of an enlarged visible block.","Let $D=\\sum_\\lambda\\lambda E_\\lambda$ be its spectral resolution. Then\n\\begin{equation}\n \\Sigma(z)=\\sum_\\lambda\\frac{B E_\\lambda B^*}{z-\\lambda}.\n \\label{a-spectral-self-energy-residues}\n\\end{equation}\nEvery residue $B E_\\lambda B^*$ is positive semidefinite. Its rank records\nwhich directions of the hidden eigenspace reach the visible space.\nA vanishing residue removes the corresponding pole of $\\Sigma$, even\nthough $z-D$ is still singular there.","ambda E_\\lambda$ be its spectral resolution. Then\n\\begin{equation}\n \\Sigma(z)=\\sum_\\lambda\\frac{B E_\\lambda B^*}{z-\\lambda}.\n \\label{a-spectral-self-energy-residues}\n\\end{equation}\nEvery residue $B E_\\lambda B^*$ is positive semidefinite. Its rank records\nwhich directions of the hidden eigenspace reach the visible space.\nA vanishing residue removes the corresponding pole of $\\Sigma$, even\nthough $z-D$ is still singular there.","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cau","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwar","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivia","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the reso","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in","\\begin{proposition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed …","sition}[Upper half-plane sign and the exact visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm …","visible norm]\n\\label{a-spectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\ran…","ectral-resolvent-sign}\nFor $z=x+iy$ with $y>0$,\n\\[\n \\operatorname{Im}\\Sigma(z)\n  =-yB\\bigl((x-D)^2+y^2I\\bigr)^{-1}B^*\\leq0,\n \\qquad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$…","uad\n \\operatorname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","torname{Im}\\mathcal S(z)\\geq yI.\n\\]\nIn particular $\\mathcal S(z)$ is invertible and\n$\\norm{\\mathcal S(z)^{-1}}\\leq y^{-1}$.\nFor real $E\\notin\\sigma(D)$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}",")$,\n\\[\n \\Sigma'(E)=-B(E-D)^{-2}B^*.\n\\]\nIf $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","If $(u,v)$ is a normalized full eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","ll eigenvector at $E$, then\n\\begin{equation}\n 1=\\langle u,(I-\\Sigma'(E))u\\rangle.\n \\label{a-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","-spectral-visible-norm}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor real $\\lambda$,\n$\\operatorname{Im}(z-\\lambda)^{-1}=-y/((x-\\lambda)^2+y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","y^2)$.\nApplying this identity to the spectral sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","l sum gives the first formula and\nthen the bound on $\\operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","operatorname{Im}\\mathcal S$.\nFor any $u$, Cauchy--Schwarz implies\n$y\\norm{u}^2\\leq|\\langle u,\\mathcal S(z)u\\rangle|\n\\leq\\norm{u}\\norm{\\mathcal S(z)u}$.\nThus $\\mathcal S(z)$ has trivial kernel and an inverse of the stated\nnorm in finite dimension. Differentiating the resolvent gives the minus\nsign in $\\Sigma'$. The reconstructed hidden norm is\n$\\norm{v}^2=\\langle u,B(E-D)^{-2}B^*u\\rangle$.\nAdding $\\norm{u}^2$ proves \\eqref{a-spectral-visible-norm}.\n\\end{proof}","For a one-dimensional visible block this says that an eigenstate's\nvisible probability is $[1-\\Sigma'(E)]^{-1}$. It is at most one, as\nrequired. Eliminating a normalized eigenvector therefore changes its\nnormalization metric as well as its spectral equation. Solving only a\nnonlinear energy equation does not provide that probability weight.","\\begin{theorem}[Causal memory with the initial hidden forcing]\n\\label{a-spectral-memory-theorem}\nThe finite system\n$i\\dot u=Au+Bv$, $i\\dot v=Dv+B^*u$, with initial data $(u_0,v_0)$,\nis equivalent to\n\\begin{align}\n i\\dot u(t)&=Au(t)+B e^{-itD}v_0\n              -i\\int_0^tK(t-s)u(s)\\,ds,\\nonumber","\\begin{theorem}[Causal memory with the initial hidden forcing]\n\\label{a-spectral-memory-theorem}\nThe finite system\n$i\\dot u=Au+Bv$, $i\\dot v=Dv+B^*u$, with initial data $(u_0,v_0)$,\nis equivalent to\n\\begin{align}\n i\\dot u(t)&=Au(t)+B e^{-itD}v_0\n              -i\\int_0^tK(t-s)u(s)\\,ds,\\nonumber\\\\\n K(\\tau)&=B e^{-i\\tau D}B^*,\\qquad \\tau\\geq0,\n \\label{a-spectral-memory-equation}\n\\end{align}\ntogether with reconstruction\n\\begin{equation}\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n\\end{equation}\nFor $\\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:\n\\begin{equation}\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial","K(\\tau)&=B e^{-i\\tau D}B^*,\\qquad \\tau\\geq0,\n \\label{a-spectral-memory-equation}\n\\end{align}\ntogether with reconstruction\n\\begin{equation}\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n\\end{equation}\nFor $\\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:\n\\begin{equation}\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes th","K(\\tau)&=B e^{-i\\tau D}B^*,\\qquad \\tau\\geq0,\n \\label{a-spectral-memory-equation}\n\\end{align}\ntogether with reconstruction\n\\begin{equation}\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n\\end{equation}\nFor $\\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:\n\\begin{equation}\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a","K(\\tau)&=B e^{-i\\tau D}B^*,\\qquad \\tau\\geq0,\n \\label{a-spectral-memory-equation}\n\\end{align}\ntogether with reconstruction\n\\begin{equation}\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n\\end{equation}\nFor $\\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:\n\\begin{equation}\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqr…","}\ntogether with reconstruction\n\\begin{equation}\n v(t)=e^{-itD}v_0-i\\int_0^te^{-i(t-s)D}B^*u(s)\\,ds.\n \\label{a-spectral-hidden-reconstruction}\n\\end{equation}\nFor $\\operatorname{Im}z>0$, the self-energy is its exact one-sided transform:\n\\begin{equation}\n \\Sigma(z)=-i\\int_0^\\infty e^{iz\\tau}K(\\tau)\\,d\\tau.\n \\label{a-spectral-memory-transform}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","m}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","d{theorem}\n\\begin{proof}\nDifferentiate $e^{itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","itD}v(t)$ and integrate the hidden equation. This gives\n\\eqref{a-spectral-hidden-reconstruction}, whose substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","substitution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","titution in the\nvisible equation gives \\eqref{a-spectral-memory-equation}. Conversely\nthe reconstruction differentiates to the hidden equation and supplies\nthe prescribed initial value, so it recovers the full evolution.\nSince $D$ is self-adjoint, $\\norm{K(\\tau)}\\leq\\norm{B}^2$.\nThe factor $e^{-\\operatorname{Im}z\\,\\tau}$ makes the transform absolutely\nconvergent. On a spectral eigenvalue of $D$,\n$\\int_0^\\infty e^{i(z-\\lambda)\\tau}\\,d\\tau=i/(z-\\lambda)$.\nThe finite spectral sum then proves \\eqref{a-spectral-memory-transform}.\n\\end{proof}","The memory is causal because it uses only $s\\leq t$. An initially empty\nhidden sector removes the forcing $B e^{-itD}v_0$, not the convolution.\nIn finite dimension $K$ is a finite sum of oscillatory terms. Its decay,\na Markov approximation, or irreversible damping requires additional\nlimits or assumptions. The closed full dynamics preserves\n$\\norm{u(t)}^2+\\norm{v(t)}^2$, while the visible norm alone can decrease\nand later return. Energy-dependent Schur reduction is consequently\ndifferent from replacing $H$ by a fixed smaller self-adjoint matrix.","e^{-itD}v_0$, not the convolution.\nIn finite dimension $K$ is a finite sum of oscillatory terms. Its decay,\na Markov approximation, or irreversible damping requires additional\nlimits or assumptions. The closed full dynamics preserves\n$\\norm{u(t)}^2+\\norm{v(t)}^2$, while the visible norm alone can decrease\nand later return. Energy-dependent Schur reduction is consequently\ndifferent from replacing $H$ by a fixed smaller self-adjoint matrix.","Let $T_{p,h}$ be a rooted $p$-ary tree of depth $h\\geq1$. Vertices at\ndepth $r$ have onsite energy $\\epsilon_r\\in\\R$. Every parent-to-child\nedge from level $r$ to $r+1$ has real coupling $\\gamma_r/\\sqrt p$,\nwhere $\\gamma_r\\in\\R$. The Hamiltonian is the corresponding real\nsymmetric adjacency matrix with onsite diagonal. The level symmetry\nand this branch normalization are structural assumptions of the model.","Define normalized level states\n\\[\n |r\\rangle=p^{-r/2}\\sum_{|v|=r}|v\\rangle,\n \\qquad r=0,\\ldots,h.\n\\]\nThey are orthonormal because different levels are disjoint and level\n$r$ contains $p^r$ vertices. A direct action calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","ad r=0,\\ldots,h.\n\\]\nThey are orthonormal because different levels are disjoint and level\n$r$ contains $p^r$ vertices. A direct action calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","dots,h.\n\\]\nThey are orthonormal because different levels are disjoint and level\n$r$ contains $p^r$ vertices. A direct action calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","level\n$r$ contains $p^r$ vertices. A direct action calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","ontains $p^r$ vertices. A direct action calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","on calculation gives\n\\begin{equation}\n H|r\\rangle=\\gamma_{r-1}|r-1\\rangle\n              +\\epsilon_r|r\\rangle+\\gamma_r|r+1\\rangle,\n \\label{a-spectral-radial-action}\n\\end{equation}\nwith omitted terms at the endpoints. For example, the coefficient at\neach child at level $r+1$ is\n$p^{-r/2}\\gamma_r/\\sqrt p=\\gamma_r p^{-(r+1)/2}$.\nAt each parent at level $r-1$, the $p$ child contributions instead give\n$p\\,p^{-r/2}\\gamma_{r-1}/\\sqrt p\n=\\gamma_{r-1}p^{-(r-1)/2}$.","Let $R:\\C^{h+1}\\to\\ell^2(T_{p,h})$ send $e_r$ to $|r\\rangle$.\nEquation \\eqref{a-spectral-radial-action} proves $HR=RJ_h$, where\n\\begin{equation}\n J_h=\\begin{pmatrix}\n \\epsilon_0&\\gamma_0&&&","Let $R:\\C^{h+1}\\to\\ell^2(T_{p,h})$ send $e_r$ to $|r\\rangle$.\nEquation \\eqref{a-spectral-radial-action} proves $HR=RJ_h$, where\n\\begin{equation}\n J_h=\\begin{pmatrix}\n \\epsilon_0&\\gamma_0&&&\\\\\n \\gamma_0&\\epsilon_1&\\gamma_1&&\\\\\n &\\gamma_1&\\epsilon_2&\\ddots&\\\\\n &&\\ddots&\\ddots&\\gamma_{h-1}\\\\\n &&&\\gamma_{h-1}&\\epsilon_h\n \\end{pmatrix}.\n \\label{a-spectral-jacobi-matrix}\n\\end{equation}\nBecause $H$ is self-adjoint, the orthogonal complement of $\\Ran R$ is\nalso invariant: for $w\\perp\\Ran R$,\n$\\langle Re_r,Hw\\rangle=\\langle HRe_r,w\\rangle=0$.\nHence this is a reducing decomposition, rather than only a compressed\napproximation. The root belongs to the radial block.","&&&\\gamma_{h-1}&\\epsilon_h\n \\end{pmatrix}.\n \\label{a-spectral-jacobi-matrix}\n\\end{equation}\nBecause $H$ is self-adjoint, the orthogonal complement of $\\Ran R$ is\nalso invariant: for $w\\perp\\Ran R$,\n$\\langle Re_r,Hw\\rangle=\\langle HRe_r,w\\rangle=0$.\nHence this is a reducing decomposition, rather than only a compressed\napproximation. The root belongs to the radial block.","There is a more precise form of the complementary decomposition. For\neach vertex $u$ at depth $a<h$, choose an orthonormal basis of vectors\n$c=(c_0,\\ldots,c_{p-1})\\in\\C^p$ with $\\sum_dc_d=0$. For\n$s=0,\\ldots,h-a-1$, set\n\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring ch","There is a more precise form of the complementary decomposition. For\neach vertex $u$ at depth $a<h$, choose an orthonormal basis of vectors\n$c=(c_0,\\ldots,c_{p-1})\\in\\C^p$ with $\\sum_dc_d=0$. For\n$s=0,\\ldots,h-a-1$, set\n\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","choose an orthonormal basis of vectors\n$c=(c_0,\\ldots,c_{p-1})\\in\\C^p$ with $\\sum_dc_d=0$. For\n$s=0,\\ldots,h-a-1$, set\n\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","ldots,c_{p-1})\\in\\C^p$ with $\\sum_dc_d=0$. For\n$s=0,\\ldots,h-a-1$, set\n\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","0$. For\n$s=0,\\ldots,h-a-1$, set\n\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","\\begin{equation}\n |u,c;s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","s\\rangle\n =p^{-s/2}\\sum_{d=0}^{p-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","-1}c_d\n       \\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}|u d w\\rangle.\n \\label{a-spectral-angular-chain}\n\\end{equation}\nFor fixed $u,c$, these are unit vectors on successive levels. They obey\nthe Jacobi recurrence with onsite energies\n$\\epsilon_{a+1},\\ldots,\\epsilon_h$ and couplings\n$\\gamma_{a+1},\\ldots,\\gamma_{h-1}$. At $s=0$ the upward amplitude into\n$u$ is proportional to $\\sum_dc_d$, so it vanishes. At $s>0$ the same\nbranch-counting calculation as in \\eqref{a-spectral-radial-action}\nproduces the neighboring chain vectors.","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogona","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform ra","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contrib","\\begin{proposition}[Complete angular-chain decomposition]\n\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute","\\label{a-spectral-angular-completeness}\nThe radial chain and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orth…","in and all the chains \\eqref{a-spectral-angular-chain}\nform an orthogonal decomposition of $\\ell^2(T_{p,h})$. A chain starting\nat depth $r$ has length $h-r+1$ and appears with multiplicity\n$(p-1)p^{r-1}$, for $1\\leq r\\leq h$.\n\\end{proposition}\n\\begin{proof}\nWithin one chain, different $s$ have disjoint level supports. Chains\nbased at the same $u$ and different contrast vectors have inner product\n$\\langle c,c'\\rangle=0$ at a common level. If two vertices are in disjoint\nsubtrees, their chains have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimens…","ns have disjoint support. If one vertex lies below\nanother, the higher chain is constant, up to its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","its fixed ancestral\ncoefficient, on all children below the lower vertex at a common level.\nThe lower chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","r chain has zero sum over its own child contrasts, so their\ninner product is zero. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","o. The same zero-sum calculation makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","makes every\nangular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","angular vector orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","ctor orthogonal to the uniform radial level vector.\nAt a fixed depth $b$, the radial vector contributes one dimension.\nThe angular vectors present there contribute\n$\\sum_{r=1}^b(p-1)p^{r-1}=p^b-1$ dimensions. They are orthogonal and\ntherefore form a basis of that $p^b$-dimensional level space. Taking\nall levels proves completeness. There are $p^{r-1}$ choices of the\nvertex $u$ at depth $r-1$ and $p-1$ contrasts at each, giving the\nstated multiplicity.\n\\end{proof}","The rank $(p-1)p^{r-1}$ thus counts chains born at level $r$ or\nnew Haar directions on a level. A single level's detail space is\nusually transported to the next level; it is the entire angular\nchain that is invariant under the tree Hamiltonian. The full dark\ndimension is\n\\[\n \\sum_{r=1}^h(p-1)p^{r-1}(h-r+1)\n =\\sum_{b=0}^hp^b-(h+1).\n\\]\nEvery one of these states is invisible to a root-localized symmetric\nsource because it is orthogonal to the radial reducing block.","For a tail $J_{r:h}$, put\n$g_r(z)=\\langle r,(z-J_{r:h})^{-1}r\\rangle$. On a common regular\ndomain, scalar Schur elimination gives\n\\begin{align}\n g_h(z)&=(z-\\epsilon_h)^{-1},\\nonumber","For a tail $J_{r:h}$, put\n$g_r(z)=\\langle r,(z-J_{r:h})^{-1}r\\rangle$. On a common regular\ndomain, scalar Schur elimination gives\n\\begin{align}\n g_h(z)&=(z-\\epsilon_h)^{-1},\\nonumber\\\\\n g_r(z)&=\\frac1{z-\\epsilon_r-\\gamma_r^2g_{r+1}(z)},\n          \\qquad r<h.\n \\label{a-spectral-continued-fraction}\n\\end{align}\nDefining $\\Sigma_h=0$ and\n$\\Sigma_r=\\gamma_r^2/(z-\\epsilon_{r+1}-\\Sigma_{r+1})$ gives\n\\begin{equation}\n \\langle\\mathrm{root},(z-H)^{-1}\\mathrm{root}\\rangle\n   =g_0(z)=\\frac1{z-\\epsilon_0-\\Sigma_0(z)}.\n \\label{a-spectral-root-green}\n\\end{equation}\nThe equality to the tree resolvent follows from the exact reducing\ndecomposition. Equations \\eqref{a-spectral-continued-fraction} are\nrational identities. Nested denominators can become singular at a\nreal energy where a canceled final rational expression is finite.\nOne should use the resolvent or the canceled rational function at\nsuch a point, rather than perform division by zero in the nested chart.","g_r(z)&=\\frac1{z-\\epsilon_r-\\gamma_r^2g_{r+1}(z)},\n          \\qquad r<h.\n \\label{a-spectral-continued-fraction}\n\\end{align}\nDefining $\\Sigma_h=0$ and\n$\\Sigma_r=\\gamma_r^2/(z-\\epsilon_{r+1}-\\Sigma_{r+1})$ gives\n\\begin{equation}\n \\langle\\mathrm{root},(z-H)^{-1}\\mathrm{root}\\rangle\n   =g_0(z)=\\frac1{z-\\epsilon_0-\\Sigma_0(z)}.\n \\label{a-spectral-root-green}\n\\end{equation}\nThe equality to the tree resolvent follows from the exact reducing\ndecomposition. Equations \\eqref{a-spectral-continued-fraction} are\nrational identities. Nested denominators can become singular at a\nreal energy where a canceled final rational expression is finite.\nOne should use the resolvent or the canceled rational function at\nsuch a point, rather than perform division by zero in the nested chart.","The Jacobi coefficients in \\eqref{a-spectral-jacobi-matrix} do not\ndepend on $p$. Thus the root Green function and radial propagator\nare prime independent when the same $\\epsilon_r,\\gamma_r$ are used.\nThe complete tree spectrum retains the $p$-dependent angular\nmultiplicities. If the per-edge coupling is instead a fixed $g_r$,\nthe radial coupling is $\\sqrt p\\,g_r$ and its response depends on $p$.\nPrime independence is therefore a consequence of the declared\nnormalization, not a universality statement about arbitrary trees.","As a conditional finite model, take\n$A=H_D(k)=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation","As a conditional finite model, take\n$A=H_D(k)=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific","As a conditional finite model, take\n$A=H_D(k)=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific model.\nIts mass, couplings,","As a conditional finite model, take\n$A=H_D(k)=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific model.\nIts mass, couplings, and hidden sector have been supplied. No physical\ninteraction or measured dispersion follows solely from t","As a conditional finite model, take\n$A=H_D(k)=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific model.\nIts mass, couplings, and hidden sector have been supplied. No physical\ninteraction or measured dispersion follows solely from the tree split.","=k\\sigma_z+m_0\\sigma_x$ with $k,m_0\\in\\R$ and\n$D=J_{1:h}\\otimes I_2$. Couple the visible spinor identically to the\nfirst site of the tail by $B=\\gamma_0(\\langle1|\\otimes I_2)$.\nIts self-energy is scalar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific model.\nIts mass, couplings, and hidden sector have been supplied. No physical\ninteraction or measured dispersion follows solely from the tree split.","lar in spin space:\n\\[\n \\Sigma_D(z)=\\sigma(z)I_2,\\qquad\n \\sigma(z)=\\gamma_0^2\\langle1,(z-J_{1:h})^{-1}1\\rangle.\n\\]\nThe determinant formula from Theorem~\\ref{a-spectral-schur-theorem} is\n\\begin{equation}\n \\det(z-H_{\\rm full})\n =\\det(z-D)\\bigl((z-\\sigma(z))^2-k^2-m_0^2\\bigr).\n \\label{a-spectral-visible-determinant}\n\\end{equation}\nConsequently any eigenvalue in the regular chart obeys\n\\begin{equation}\n (E-\\sigma(E))^2=k^2+m_0^2.\n \\label{a-spectral-visible-branch}\n\\end{equation}\nThe formula is a dressed dispersion relation of this specific model.\nIts mass, couplings, and hidden sector have been supplied. No physical\ninteraction or measured dispersion follows solely from the tree split.","For a nondegenerate visible spin direction with normalized low-space\ndirection $u_*$, the visible weight of its normalized full eigenstate is\n$[1-\\sigma'(E)]^{-1}$. If $k=m_0=0$, the spin eigenspace is degenerate,\nbut the same normalization relation holds for each chosen unit vector\nin that eigenspace.","If every internal tail hopping is nonzero, the first tail site is cyclic.\nOne can see this without an abstract cyclicity theorem: a tail eigenvector\nwith zero first component has zero second component by the first row,\nand successive rows force all later components to vanish. Therefore\nevery tail eigenvector has nonzero first component. At an eigenvalue\n$\\lambda$ of that tail, solvability of\n$(\\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail\neigenspace. With $\\gamma_0\\ne0$ and identical spin coupling this forces\n$u=0$. The visible equation would then require $Bv=0$, which forces\n$v=0$ on the tail eigenspace. Thus in this irreducible case the full\ncoupled spectrum avoids the tail eigenvalues. With a vanishing hopping,\ncoincident visible and hidden energies can occur and the regular-chart\nqualification in \\eqref{a-spectral-visible-branch} is essential.","this without an abstract cyclicity theorem: a tail eigenvector\nwith zero first component has zero second component by the first row,\nand successive rows force all later components to vanish. Therefore\nevery tail eigenvector has nonzero first component. At an eigenvalue\n$\\lambda$ of that tail, solvability of\n$(\\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail\neigenspace. With $\\gamma_0\\ne0$ and identical spin coupling this forces\n$u=0$. The visible equation would then require $Bv=0$, which forces\n$v=0$ on the tail eigenspace. Thus in this irreducible case the full\ncoupled spectrum avoids the tail eigenvalues. With a vanishing hopping,\ncoincident visible and hidden energies can occur and the regular-chart\nqualification in \\eqref{a-spectral-visible-branch} is essential.","nvector\nwith zero first component has zero second component by the first row,\nand successive rows force all later components to vanish. Therefore\nevery tail eigenvector has nonzero first component. At an eigenvalue\n$\\lambda$ of that tail, solvability of\n$(\\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail\neigenspace. With $\\gamma_0\\ne0$ and identical spin coupling this forces\n$u=0$. The visible equation would then require $Bv=0$, which forces\n$v=0$ on the tail eigenspace. Thus in this irreducible case the full\ncoupled spectrum avoids the tail eigenvalues. With a vanishing hopping,\ncoincident visible and hidden energies can occur and the regular-chart\nqualification in \\eqref{a-spectral-visible-branch} is essential.","nd component by the first row,\nand successive rows force all later components to vanish. Therefore\nevery tail eigenvector has nonzero first component. At an eigenvalue\n$\\lambda$ of that tail, solvability of\n$(\\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail\neigenspace. With $\\gamma_0\\ne0$ and identical spin coupling this forces\n$u=0$. The visible equation would then require $Bv=0$, which forces\n$v=0$ on the tail eigenspace. Thus in this irreducible case the full\ncoupled spectrum avoids the tail eigenvalues. With a vanishing hopping,\ncoincident visible and hidden energies can occur and the regular-chart\nqualification in \\eqref{a-spectral-visible-branch} is essential.","irst row,\nand successive rows force all later components to vanish. Therefore\nevery tail eigenvector has nonzero first component. At an eigenvalue\n$\\lambda$ of that tail, solvability of\n$(\\lambda-D)v=B^*u$ requires its right side to be orthogonal to the tail\neigenspace. With $\\gamma_0\\ne0$ and identical spin coupling this forces\n$u=0$. The visible equation would then require $Bv=0$, which forces\n$v=0$ on the tail eigenspace. Thus in this irreducible case the full\ncoupled spectrum avoids the tail eigenvalues. With a vanishing hopping,\ncoincident visible and hidden energies can occur and the regular-chart\nqualification in \\eqref{a-spectral-visible-branch} is essential.","The scalar causal kernel is\n$K(\\tau)=\\gamma_0^2\\langle1,e^{-i\\tau J_{1:h}}1\\rangle$, multiplying\n$I_2$. It uses the same radial coefficients as the rational self-energy.\nIts exact causal relation is \\eqref{a-spectral-memory-transform}.\n\\adicfigure{08_radial_self_energy}{A branch-normalized finite tree has\nthe same root response as its radial Jacobi chain. The complex-energy\nresolvent avoids real poles. The oscillatory memory is the return\namplitude of a finite hidden chain.}","We now eliminate intermediate residue sites in\n\\eqref{a-spectral-free-cycle}, assuming $\\kappa\\ne0$. The retained\nsites are $C=\\{0,p,\\ldots,(M-1)p\\}$ and the eliminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","\\eqref{a-spectral-free-cycle}, assuming $\\kappa\\ne0$. The retained\nsites are $C=\\{0,p,\\ldots,(M-1)p\\}$ and the eliminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","re $C=\\{0,p,\\ldots,(M-1)p\\}$ and the eliminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","dots,(M-1)p\\}$ and the eliminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","}$ and the eliminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","liminated sites comprise\nthe $p-1$ interior sites between adjacent retained sites. Identify\n$\\ell^2(C)$ with $\\C^M$ and let $T|j\\rangle=|j+1\\pmod M\\rangle$.\nSet\n\\begin{equation}\n x=\\frac{z-\\mu}{\\kappa},\\qquad\n R_n(x)=2T_n(x/2),\n \\label{a-spectral-normalized-energy}\n\\end{equation}\nwhere $T_n$ is the first-kind Chebyshev polynomial. The second-kind\npolynomials $U_n$ satisfy $U_{-1}=0$, $U_0=1$ and\n$U_{n+1}(t)=2tU_n(t)-U_{n-1}(t)$.","The stationary equation on an interior site is\n$\\psi_{n+1}=x\\psi_n-\\psi_{n-1}$. The eliminated block of $z-H$ in\none cell is $\\kappa K_{p-1}(x)$, where $K_n(x)$ has diagonal $x$\nand neighboring off-diagonal entries $-1$.","\\begin{lemma}[Dirichlet path determinant and endpoint entries]\n\\label{a-spectral-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. I","\\begin{lemma}[Dirichlet path determinant and endpoint entries]\n\\label{a-spectral-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1","\\begin{lemma}[Dirichlet path determinant and endpoint entries]\n\\label{a-spectral-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","t and endpoint entries]\n\\label{a-spectral-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","l{a-spectral-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","-path-inverse}\nFor $n\\geq1$, $\\det K_n(x)=U_n(x/2)$. If this determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","determinant is nonzero,\n\\[\n (K_n(x)^{-1})_{11}=(K_n(x)^{-1})_{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","{nn}\n       =\\frac{U_{n-1}(x/2)}{U_n(x/2)},\\qquad\n (K_n(x)^{-1})_{1n}=\\frac1{U_n(x/2)}.\n\\]\nFor $n=1$ these entries all denote the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","the same value $1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","$1/x$.\n\\end{lemma}\n\\begin{proof}\nExpansion along the last row gives the determinant recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","recurrence\n$d_n=xd_{n-1}-d_{n-2}$, with $d_0=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","=1$ and $d_1=x$.\nThis is exactly the recurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","ecurrence and initial data for $U_n(x/2)$.\nDeleting the first or last row and column leaves $K_{n-1}$, proving\nthe diagonal inverse entries by cofactors. For the endpoint entry,\nthe cofactor consists of the product of $n-1$ entries $-1$ along the\nunique path from the first endpoint to the last. Its cofactor sign is\n$(-1)^{n+1}$, so the product has total sign\n$(-1)^{n+1}(-1)^{n-1}=1$. Division by $d_n$ gives the stated value.\n\\end{proof}","\\begin{theorem}[Exact free-cycle decimation]\n\\label{a-spectral-chebyshev-schur}\nFor $U_{p-1}(x/2)\\ne0$, the Schur operator on the retained sublattice is\n\\begin{equation}\n \\mathcal S_p(z)=\\frac{\\kappa}{U_{p-1}(x/2)}\n          \\bigl[R_p(x)I_M-(T+T^*)\\bigr].\n \\label{a-spectral-decimation-formula}\n\\end{equation}\nEquivalently, $\\mathcal S_p(z)=zI-H_{\\rm eff}^{(p)}(z)$, where\n\\begin{align}\n H_{\\rm eff}^{(p)}(z)&=\\mu_p(z)I+\\kappa_p(z)(T+T^*),\\nonumber","\\begin{theorem}[Exact free-cycle decimation]\n\\label{a-spectral-chebyshev-schur}\nFor $U_{p-1}(x/2)\\ne0$, the Schur operator on the retained sublattice is\n\\begin{equation}\n \\mathcal S_p(z)=\\frac{\\kappa}{U_{p-1}(x/2)}\n          \\bigl[R_p(x)I_M-(T+T^*)\\bigr].\n \\label{a-spectral-decimation-formula}\n\\end{equation}\nEquivalently, $\\mathcal S_p(z)=zI-H_{\\rm eff}^{(p)}(z)$, where\n\\begin{align}\n H_{\\rm eff}^{(p)}(z)&=\\mu_p(z)I+\\kappa_p(z)(T+T^*),\\nonumber\\\\\n \\kappa_p(z)&=\\frac{\\kappa}{U_{p-1}(x/2)},\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe eliminated sites split into $M$ disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal…","\\kappa_p(z)&=\\frac{\\kappa}{U_{p-1}(x/2)},\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe eliminated sites split into $M$ disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituti","\\kappa_p(z)&=\\frac{\\kappa}{U_{p-1}(x/2)},\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe eliminated sites split into $M$ disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\th","\\kappa_p(z)&=\\frac{\\kappa}{U_{p-1}(x/2)},\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe eliminated sites split into $M$ disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polyn",",\\qquad\n \\mu_p(z)=z-\\frac{\\kappa R_p(x)}{U_{p-1}(x/2)}.\n \\label{a-spectral-effective-coefficients}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe eliminated sites split into $M$ disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging…","disjoint Dirichlet paths of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","s of length\n$p-1$. Hence their resolvent exists exactly when\n$U_{p-1}(x/2)\\ne0$. A retained site touches one endpoint on each of\nits two neighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","eighboring paths. Each path contributes the onsite\nself-energy $\\kappa U_{p-2}/U_{p-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","-1}$ by\nLemma~\\ref{a-spectral-path-inverse}; the two contributions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","ibutions add.\nThe path joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","joining two adjacent retained sites contributes off-diagonal\nself-energy $\\kappa/U_{p-1}$. Thus\n\\[\n \\mathcal S_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","_p(z)\n =\\kappa\\left(x-\\frac{2U_{p-2}(x/2)}{U_{p-1}(x/2)}\\right)I\n       -\\frac{\\kappa}{U_{p-1}(x/2)}(T+T^*).\n\\]\nThe identity\n$xU_{p-1}(x/2)-2U_{p-2}(x/2)=2T_p(x/2)$ follows from the\nthree-term recurrence and $U_p-U_{p-2}=2T_p$.\nFor the latter identity, substituting $x/2=\\cos\\theta$ gives\n$[\\sin((p+1)\\theta)-\\sin((p-1)\\theta)]/\\sin\\theta\n=2\\cos(p\\theta)$, and polynomial equality extends it to all $x$.\nThis proves \\eqref{a-spectral-decimation-formula} including its\nprefactor. Rearranging its diagonal gives\n\\eqref{a-spectral-effective-coefficients}.\n\\end{proof}","The normalized effective energy is\n$(z-\\mu_p(z))/\\kappa_p(z)=R_p(x)$. The parameter $z$ remains fixed\nthrough the elimination. Since $\\mu_p$ and $\\kappa_p$ depend on it,\nthis is an energy-dependent spectral closure, not an energy-independent\nHamiltonian flow. For $p=2$, $R_2(x)=x^2-2$ and $U_1(x/2)=x$, giving\n\\begin{equation}\n H_{\\rm eff}^{(2)}(z)\n =\\mu I+\\frac{\\kappa^2}{z-\\mu}(2I+T+T^*),\\qquad z\\ne\\mu.\n \\label{a-spectral-dyadic-effective}\n\\end{equation}\nThis decimation retains every other site. It does not retain the\naverage of each neighboring pair.","The sine formula $U_{p-1}(\\cos\\theta)=\\sin(p\\theta)/\\sin\\theta$\nshows that the excluded normalized energies are\n\\begin{equation}\n x_j=2\\cos(j\\pi/p),\\qquad\n z_j=\\mu+2\\kappa\\cos(j\\pi/p),\\quad 1\\leq j<p.\n \\label{a-spectral-dirichlet-resonances}\n\\end{equation}\nThese are the energies of one eliminated path. Indeed its $j$th\neigenvector has interior components $\\sin(j\\pi r/p)$,\n$r=1,\\ldots,p-1$, and the nearest-neighbor recurrence gives precisely\n$z_j$. Both boundary values at $r=0,p$ vanish. The eigenvectors are\northogonal and there are $p-1$ of them, so these exhaust the path spectrum.","heta)/\\sin\\theta$\nshows that the excluded normalized energies are\n\\begin{equation}\n x_j=2\\cos(j\\pi/p),\\qquad\n z_j=\\mu+2\\kappa\\cos(j\\pi/p),\\quad 1\\leq j<p.\n \\label{a-spectral-dirichlet-resonances}\n\\end{equation}\nThese are the energies of one eliminated path. Indeed its $j$th\neigenvector has interior components $\\sin(j\\pi r/p)$,\n$r=1,\\ldots,p-1$, and the nearest-neighbor recurrence gives precisely\n$z_j$. Both boundary values at $r=0,p$ vanish. The eigenvectors are\northogonal and there are $p-1$ of them, so these exhaust the path spectrum.","e excluded normalized energies are\n\\begin{equation}\n x_j=2\\cos(j\\pi/p),\\qquad\n z_j=\\mu+2\\kappa\\cos(j\\pi/p),\\quad 1\\leq j<p.\n \\label{a-spectral-dirichlet-resonances}\n\\end{equation}\nThese are the energies of one eliminated path. Indeed its $j$th\neigenvector has interior components $\\sin(j\\pi r/p)$,\n$r=1,\\ldots,p-1$, and the nearest-neighbor recurrence gives precisely\n$z_j$. Both boundary values at $r=0,p$ vanish. The eigenvectors are\northogonal and there are $p-1$ of them, so these exhaust the path spectrum.","(j\\pi/p),\\qquad\n z_j=\\mu+2\\kappa\\cos(j\\pi/p),\\quad 1\\leq j<p.\n \\label{a-spectral-dirichlet-resonances}\n\\end{equation}\nThese are the energies of one eliminated path. Indeed its $j$th\neigenvector has interior components $\\sin(j\\pi r/p)$,\n$r=1,\\ldots,p-1$, and the nearest-neighbor recurrence gives precisely\n$z_j$. Both boundary values at $r=0,p$ vanish. The eigenvectors are\northogonal and there are $p-1$ of them, so these exhaust the path spectrum.","The polynomial map $R_p$ is defined at every $x_j$, while the Schur\ncoefficients in \\eqref{a-spectral-effective-coefficients} are not.\nCancelling denominators in a determinant can cross this chart boundary;\ndividing by them in an eigenvector reconstruction cannot. A correct\ndescription there keeps the resonant cell modes or solves the full\nfinite system. The fixed poles of the eliminated path are not all\nautomatically eigenvalues of the full cyclic Hamiltonian.","Let $Q$ project onto the eliminated sites. On the regular chart,\n\\[\n \\det(z-H)=\\det(z-QHQ)\\det\\mathcal S_p(z),\\qquad\n \\det(z-QHQ)=\\bigl[\\kappa^{p-1}U_{p-1}(x/2)\\bigr]^M.\n\\]\nTheir product is a polynomial identity after cancellation. It records\nboth the hidden path factor and the coarse Schur factor, including\ntheir cancellations at the resonant energies.","Define $A(x)=\\begin{pmatrix}x&-1\\\\1&0\\end{pmatrix}$. Its determinant\nis one and induction gives\n\\begin{equation}\n A(x)^n=\\begin{pmatrix}\n U_n(x/2)&-U_{n-1}(x/2)","1&0\\end{pmatrix}$. Its determinant\nis one and induction gives\n\\begin{equation}\n A(x)^n=\\begin{pmatrix}\n U_n(x/2)&-U_{n-1}(x/2)\\\\\n U_{n-1}(x/2)&-U_{n-2}(x/2)\n \\end{pmatrix},\\quad n\\geq1.\n \\label{a-spectral-transfer-power}\n\\end{equation}\nFor $n=1$ this uses $U_{-1}=0$. Multiplication by $A(x)$ advances\nthe first column through the Chebyshev recurrence and places the\nnegative previous first column in the second. The trace is $R_n(x)$.\nSampling the fine recurrence every $p$ steps therefore uses the\nunimodular transfer matrix $A(x)^p$. Cayley--Hamilton gives the coarse\nrecurrence $\\chi_{j+1}-R_p(x)\\chi_j+\\chi_{j-1}=0$ for\n$\\chi_j=\\psi_{pj}$. It explains the normalized spectral polynomial.\nThe path-inverse proof above additionally supplies the prefactor needed\nfor the resolvent.","U_{n-1}(x/2)&-U_{n-2}(x/2)\n \\end{pmatrix},\\quad n\\geq1.\n \\label{a-spectral-transfer-power}\n\\end{equation}\nFor $n=1$ this uses $U_{-1}=0$. Multiplication by $A(x)$ advances\nthe first column through the Chebyshev recurrence and places the\nnegative previous first column in the second. The trace is $R_n(x)$.\nSampling the fine recurrence every $p$ steps therefore uses the\nunimodular transfer matrix $A(x)^p$. Cayley--Hamilton gives the coarse\nrecurrence $\\chi_{j+1}-R_p(x)\\chi_j+\\chi_{j-1}=0$ for\n$\\chi_j=\\psi_{pj}$. It explains the normalized spectral polynomial.\nThe path-inverse proof above additionally supplies the prefactor needed\nfor the resolvent.","1.\n \\label{a-spectral-transfer-power}\n\\end{equation}\nFor $n=1$ this uses $U_{-1}=0$. Multiplication by $A(x)$ advances\nthe first column through the Chebyshev recurrence and places the\nnegative previous first column in the second. The trace is $R_n(x)$.\nSampling the fine recurrence every $p$ steps therefore uses the\nunimodular transfer matrix $A(x)^p$. Cayley--Hamilton gives the coarse\nrecurrence $\\chi_{j+1}-R_p(x)\\chi_j+\\chi_{j-1}=0$ for\n$\\chi_j=\\psi_{pj}$. It explains the normalized spectral polynomial.\nThe path-inverse proof above additionally supplies the prefactor needed\nfor the resolvent.","equation}\nFor $n=1$ this uses $U_{-1}=0$. Multiplication by $A(x)$ advances\nthe first column through the Chebyshev recurrence and places the\nnegative previous first column in the second. The trace is $R_n(x)$.\nSampling the fine recurrence every $p$ steps therefore uses the\nunimodular transfer matrix $A(x)^p$. Cayley--Hamilton gives the coarse\nrecurrence $\\chi_{j+1}-R_p(x)\\chi_j+\\chi_{j-1}=0$ for\n$\\chi_j=\\psi_{pj}$. It explains the normalized spectral polynomial.\nThe path-inverse proof above additionally supplies the prefactor needed\nfor the resolvent.","The folded branches from Proposition~\\ref{a-spectral-folded-bands},\nin normalized coordinates, satisfy\n\\begin{equation}\n x_{p,q}(\\theta)=2\\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right),\n \\qquad R_p(x_{p,q}(\\theta))=2\\cos\\theta.\n \\label{a-spectral-branch-relation}\n\\end{equation}\nThe second equality is just $T_p(\\cos\\alpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","from Proposition~\\ref{a-spectral-folded-bands},\nin normalized coordinates, satisfy\n\\begin{equation}\n x_{p,q}(\\theta)=2\\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right),\n \\qquad R_p(x_{p,q}(\\theta))=2\\cos\\theta.\n \\label{a-spectral-branch-relation}\n\\end{equation}\nThe second equality is just $T_p(\\cos\\alpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","ded-bands},\nin normalized coordinates, satisfy\n\\begin{equation}\n x_{p,q}(\\theta)=2\\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right),\n \\qquad R_p(x_{p,q}(\\theta))=2\\cos\\theta.\n \\label{a-spectral-branch-relation}\n\\end{equation}\nThe second equality is just $T_p(\\cos\\alpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","oordinates, satisfy\n\\begin{equation}\n x_{p,q}(\\theta)=2\\cos\\!\\left(\\frac{\\theta+2\\pi q}{p}\\right),\n \\qquad R_p(x_{p,q}(\\theta))=2\\cos\\theta.\n \\label{a-spectral-branch-relation}\n\\end{equation}\nThe second equality is just $T_p(\\cos\\alpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","_p(\\cos\\alpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","lpha)=\\cos(p\\alpha)$.\nFor a coarse normalized energy in the open interval $(-2,2)$,\nchoose its angle in $(0,\\pi)$. These formulas supply the $p$ distinct\nreal roots of $R_p(x)=2\\cos\\theta$. At the endpoints, some roots\ncoincide and algebraic multiplicities must be retained. Globally the\nbranches exchange labels when the angle crosses a cell boundary.\nThere is no single-valued inverse of $R_p$ on $[-2,2]$.","For the fine cycle, $x_k=2\\cos(2\\pi k/(pM))$ maps to\n$2\\cos(2\\pi k/M)$. Each coarse Fourier label has exactly $p$ fine\npreimages. If two labels already have the same numerical energy,\ntheir multiplicities add as usual. Band folding retains the branch\nlabel and its vector, whereas normalized decimation identifies all\nof these labels at the same coarse energy. Their spectral relation\ndoes not make elimination an invertible operator change of basis.","\\begin{theorem}[Chebyshev composition and Schur composition]\n\\label{a-spectral-semigroup-theorem}\nFor all positive integers $m,n$, the polynomial identities\n\\begin{align}\n R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\\nonumber","\\begin{theorem}[Chebyshev composition and Schur composition]\n\\label{a-spectral-semigroup-theorem}\nFor all positive integers $m,n$, the polynomial identities\n\\begin{align}\n R_n(R_m(x))&=R_{mn}(x)=R_m(R_n(x)),\\nonumber\\\\\n U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\th","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\s","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPol","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends","U_{mn-1}(t)&=U_{n-1}(T_m(t))\\,U_{m-1}(t)\n \\label{a-spectral-polynomial-semigroup}\n\\end{align}\nhold over $\\C$. In a free cycle with $N=mnL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}…","nL$, $L\\geq3$, successive\n$m$- then $n$-site decimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu…","ecimation at a fixed $z$ agrees with direct\n$mn$-site decimation whenever its intermediate Schur inverses exist.\nIts final coefficients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives…","icients are\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the display…","re\n\\begin{equation}\n \\kappa_{mn}(z)=\\frac{\\kappa}{U_{mn-1}(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed comple…","(x/2)},\\qquad\n \\mu_{mn}(z)=z-\\frac{\\kappa R_{mn}(x)}{U_{mn-1}(x/2)}.\n \\label{a-spectral-iterated-coefficients}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFor $x=2\\cos\\theta$, first-kind composition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues…","mposition gives\n$R_n(R_m(x))=2\\cos(mn\\theta)=R_{mn}(x)$.\nIt holds on an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues to apply. These are rational identities on a common\nregular chart. They need not be interpreted as self-adjoint matrices at\ncomplex $z$.\n\\end{proof}","an interval and is therefore an identity of polynomials,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues to apply. These are rational identities on a common\nregular chart. They need not be interpreted as self-adjoint matrices at\ncomplex $z$.\n\\end{proof}","ls,\nincluding complex $x$. For $t=\\cos\\theta$ away from zeros of the sine\ndenominators,\n\\[\n U_{n-1}(T_m(t))U_{m-1}(t)\n =\\frac{\\sin(mn\\theta)}{\\sin(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues to apply. These are rational identities on a common\nregular chart. They need not be interpreted as self-adjoint matrices at\ncomplex $z$.\n\\end{proof}","(m\\theta)}\n       \\frac{\\sin(m\\theta)}{\\sin\\theta}\n =U_{mn-1}(t).\n\\]\nPolynomial equality extends it through those zeros and to all $t$.\nAfter the first regular elimination, the normalized variable is\n$x_m=(z-\\mu_m)/\\kappa_m=R_m(x)$. A second one gives\n\\[\n \\kappa_{m,n}\n =\\frac{\\kappa_m}{U_{n-1}(x_m/2)}\n =\\frac{\\kappa}\n   {U_{m-1}(x/2)U_{n-1}(T_m(x/2))}\n =\\frac{\\kappa}{U_{mn-1}(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues to apply. These are rational identities on a common\nregular chart. They need not be interpreted as self-adjoint matrices at\ncomplex $z$.\n\\end{proof}","(x/2)}.\n\\]\nSimilarly\n$\\mu_{m,n}=z-\\kappa_{m,n}R_n(R_m(x))$, which gives\n\\eqref{a-spectral-iterated-coefficients}.\nAt each stage the matrices remain scalar nearest-neighbor circulants\nwith the displayed complex coefficients, so their algebraic Schur\nformula continues to apply. These are rational identities on a common\nregular chart. They need not be interpreted as self-adjoint matrices at\ncomplex $z$.\n\\end{proof}","The product identity in \\eqref{a-spectral-polynomial-semigroup} makes\nthe exceptional denominators explicit. A nonzero final denominator\nimplies both denominators along that two-stage chart are nonzero.\nAt a zero of the final denominator the polynomial trace composition\nis still valid, but the divided Schur formulas require a new chart or\nretained resonant modes. Iterating one prime $p$ through depth $h$,\nwhere the cycle has the required divisibility, replaces $p$ by $p^h$\nin \\eqref{a-spectral-iterated-coefficients}. Different prime orders\ngive the same regular free-chain result. A nontranslation-invariant\nonsite potential, an interacting Hamiltonian, or a field-dependent\nconstraint elimination needs its own closure analysis.","roduct identity in \\eqref{a-spectral-polynomial-semigroup} makes\nthe exceptional denominators explicit. A nonzero final denominator\nimplies both denominators along that two-stage chart are nonzero.\nAt a zero of the final denominator the polynomial trace composition\nis still valid, but the divided Schur formulas require a new chart or\nretained resonant modes. Iterating one prime $p$ through depth $h$,\nwhere the cycle has the required divisibility, replaces $p$ by $p^h$\nin \\eqref{a-spectral-iterated-coefficients}. Different prime orders\ngive the same regular free-chain result. A nontranslation-invariant\nonsite potential, an interacting Hamiltonian, or a field-dependent\nconstraint elimination needs its own closure analysis.","kes\nthe exceptional denominators explicit. A nonzero final denominator\nimplies both denominators along that two-stage chart are nonzero.\nAt a zero of the final denominator the polynomial trace composition\nis still valid, but the divided Schur formulas require a new chart or\nretained resonant modes. Iterating one prime $p$ through depth $h$,\nwhere the cycle has the required divisibility, replaces $p$ by $p^h$\nin \\eqref{a-spectral-iterated-coefficients}. Different prime orders\ngive the same regular free-chain result. A nontranslation-invariant\nonsite potential, an interacting Hamiltonian, or a field-dependent\nconstraint elimination needs its own closure analysis.","exceptional denominators explicit. A nonzero final denominator\nimplies both denominators along that two-stage chart are nonzero.\nAt a zero of the final denominator the polynomial trace composition\nis still valid, but the divided Schur formulas require a new chart or\nretained resonant modes. Iterating one prime $p$ through depth $h$,\nwhere the cycle has the required divisibility, replaces $p$ by $p^h$\nin \\eqref{a-spectral-iterated-coefficients}. Different prime orders\ngive the same regular free-chain result. A nontranslation-invariant\nonsite potential, an interacting Hamiltonian, or a field-dependent\nconstraint elimination needs its own closure analysis.","Let $M_0\\in SL_2(\\C)$ and $x=\\operatorname{tr}M_0$. No\ndiagonalizability assumption is needed. Cayley--Hamilton gives\n$M_0^2-xM_0+I=0$, hence\n\\begin{equation}\n M_0^n=U_{n-1}(x/2)M_0-U_{n-2}(x/2)I,\\qquad n\\geq1,\n \\label{a-spectral-sl2-power}\n\\end{equation}\nwith $U_{-1}=0$. For $n=1$ it is the identity $M_0=M_0$, and for\n$n=2$ it is Cayley--Hamilton. Multiplying by $M_0$ and replacing\n$M_0^2$ by $xM_0-I$ proves the induction step. Taking traces yields\n\\begin{equation}\n \\operatorname{tr}(M_0^n)=R_n(\\operatorname{tr}M_0).\n \\label{a-spectral-sl2-trace}\n\\end{equation}\nThe coefficient identity is the same\n$xU_{n-1}-2U_{n-2}=2T_n$ used above.","ty assumption is needed. Cayley--Hamilton gives\n$M_0^2-xM_0+I=0$, hence\n\\begin{equation}\n M_0^n=U_{n-1}(x/2)M_0-U_{n-2}(x/2)I,\\qquad n\\geq1,\n \\label{a-spectral-sl2-power}\n\\end{equation}\nwith $U_{-1}=0$. For $n=1$ it is the identity $M_0=M_0$, and for\n$n=2$ it is Cayley--Hamilton. Multiplying by $M_0$ and replacing\n$M_0^2$ by $xM_0-I$ proves the induction step. Taking traces yields\n\\begin{equation}\n \\operatorname{tr}(M_0^n)=R_n(\\operatorname{tr}M_0).\n \\label{a-spectral-sl2-trace}\n\\end{equation}\nThe coefficient identity is the same\n$xU_{n-1}-2U_{n-2}=2T_n$ used above.","q1,\n \\label{a-spectral-sl2-power}\n\\end{equation}\nwith $U_{-1}=0$. For $n=1$ it is the identity $M_0=M_0$, and for\n$n=2$ it is Cayley--Hamilton. Multiplying by $M_0$ and replacing\n$M_0^2$ by $xM_0-I$ proves the induction step. Taking traces yields\n\\begin{equation}\n \\operatorname{tr}(M_0^n)=R_n(\\operatorname{tr}M_0).\n \\label{a-spectral-sl2-trace}\n\\end{equation}\nThe coefficient identity is the same\n$xU_{n-1}-2U_{n-2}=2T_n$ used above.","If $w_{k+1}=M_0w_k$, each residue subsequence satisfies\n\\[\n w_{nq+j}=M_0^j(M_0^n)^q w_0,\\qquad 0\\leq j<n.\n\\]\nThus ordinary matrix powering produces the normalized trace semigroup,\nincluding matrices with repeated eigenvalues at trace $\\pm2$.\nThe trace alone does not determine $M_0$, its eigenvectors, or a\nlocal Hamiltonian realization. In the free chain, $M_0=A(x)$ has\nthe required nearest-neighbor realization. For a general transfer\nproblem only the matrix power and trace statements have been proved.","On $[-2,2]$ define the probability measure\n\\begin{equation}\n d\\nu(x)=\\frac{dx}{\\pi\\sqrt{4-x^2}}.\n \\label{a-spectral-arcsine-measure}\n\\end{equation}\nIt is the distribution of $2\\cos\\theta$ for a uniform angle on the\ncircle, or on $[0,\\pi]$. It is also the limiting normalized spectral\nmeasure of the free cycles: for a continuous test function $f$,\n\\[\n \\frac1N\\sum_{k=0}^{N-1}f(2\\cos(2\\pi k/N))\n \\longrightarrow\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos\\theta)\\,d\\theta.\n\\]\nThis is the Riemann-sum limit of a continuous periodic function.","\\Needspace{17\\baselineskip}\n\\begin{proposition}[Invariant free measure and exact mode routing]\n\\label{a-spectral-arcsine-invariance}\nFor every integer $p\\geq2$, $(R_p)_*\\nu=\\nu$. The map\n$\\mathcal K_pf=f\\circ R_p$ is an isometry on $L^2(\\nu)$. With\n\\[\n \\phi_0=1,\\qquad \\phi_n(x)=\\sqrt2\\,T_n(x/2),\\quad n\\geq1,\n\\]\nits action and adjoint on an orthonormal basis are\n\\begin{equation}\n \\mathcal K_p\\phi_n=\\phi_{pn},\\qquad\n \\mathcal K_p^*\\phi_m=\n \\begin{cases}\\phi_{m/p},&p\\mid m,","\\Needspace{17\\baselineskip}\n\\begin{proposition}[Invariant free measure and exact mode routing]\n\\label{a-spectral-arcsine-invariance}\nFor every integer $p\\geq2$, $(R_p)_*\\nu=\\nu$. The map\n$\\mathcal K_pf=f\\circ R_p$ is an isometry on $L^2(\\nu)$. With\n\\[\n \\phi_0=1,\\qquad \\phi_n(x)=\\sqrt2\\,T_n(x/2),\\quad n\\geq1,\n\\]\nits action and adjoint on an orthonormal basis are\n\\begin{equation}\n \\mathcal K_p\\phi_n=\\phi_{pn},\\qquad\n \\mathcal K_p^*\\phi_m=\n \\begin{cases}\\phi_{m/p},&p\\mid m,\\\\0,&p\\nmid m.\\end{cases}\n \\label{a-spectral-koopman-routing}\n\\end{equation}\nIn the first case $m=0$ is included, so $\\mathcal K_p^*\\phi_0=\\phi_0$.\n\\end{proposition}\n\\begin{proof}\nFor a bounded measurable $f$, use the full circle parameterization:\n\\begin{align*}\n \\int f(R_p(x))\\,d\\nu(x)\n &=\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos(p\\theta))\\,d\\theta","0,&p\\nmid m.\\end{cases}\n \\label{a-spectral-koopman-routing}\n\\end{equation}\nIn the first case $m=0$ is included, so $\\mathcal K_p^*\\phi_0=\\phi_0$.\n\\end{proposition}\n\\begin{proof}\nFor a bounded measurable $f$, use the full circle parameterization:\n\\begin{align*}\n \\int f(R_p(x))\\,d\\nu(x)\n &=\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos(p\\theta))\\,d\\theta","0,&p\\nmid m.\\end{cases}\n \\label{a-spectral-koopman-routing}\n\\end{equation}\nIn the first case $m=0$ is included, so $\\mathcal K_p^*\\phi_0=\\phi_0$.\n\\end{proposition}\n\\begin{proof}\nFor a bounded measurable $f$, use the full circle parameterization:\n\\begin{align*}\n \\int f(R_p(x))\\,d\\nu(x)\n &=\\frac1{2\\pi}\\int_0^{2\\pi}f(2\\cos(p\\theta))\\,d\\theta\\\\\n &=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n\\end{align*}\nThe last equality uses periodicity. Applying this equality to $|f|^2$\nproves the isometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nCo…","&=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n\\end{align*}\nThe last equality uses periodicity. Applying this equality to $|f|^2$\nproves the isometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval s","&=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n\\end{align*}\nThe last equality uses periodicity. Applying this equality to $|f|^2$\nproves the isometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Cheby","&=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n\\end{align*}\nThe last equality uses periodicity. Applying this equality to $|f|^2$\nproves the isometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\ma","&=\\frac1{2\\pi p}\\int_0^{2\\pi p}f(2\\cos u)\\,du\n =\\int f(x)\\,d\\nu(x).\n\\end{align*}\nThe last equality uses periodicity. Applying this equality to $|f|^2$\nproves the isometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\ph","ometry, first on bounded functions and then by truncation\non all of $L^2(\\nu)$. Pulling back to $[0,\\pi]$ makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","makes the $\\phi_n$\nthe normalized cosine functions, and elementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","lementary cosine orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","ne orthogonality\nproves their orthonormality. Their span is dense: Chebyshev polynomials\nspan ordinary polynomials; those approximate continuous functions\nuniformly, for example by Bernstein polynomials after rescaling to\n$[0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","0,1]$. The approximation follows from uniform continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","orm continuity and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","ty and the\nbinomial variance bound, by separating $|k/n-t|\\leq\\delta$ from its\ncomplement and then letting $n\\to\\infty$, $\\delta\\to0$.\nContinuous functions are dense for this finite measure, as one sees by\napproximating interval steps in the angle coordinate by continuous\nramps. This supplies completeness.\nFinally, Chebyshev composition gives $\\mathcal K_p\\phi_n=\\phi_{pn}$.\nFor every $n,m\\geq0$,\n$\\langle\\phi_n,\\mathcal K_p^*\\phi_m\\rangle\n=\\langle\\phi_{pn},\\phi_m\\rangle=\\delta_{pn,m}$.\nCompleteness yields the adjoint formula.\n\\end{proof}","$\\mathcal K_p$ is an isometry with proper range, rather than a unitary:\nfor example $\\phi_1$ is orthogonal to its range. Its adjoint discards\nnondivisible modes. Invariance of $\\nu$ is a statement about the\nnormalized free spectral distribution under $R_p$. It is not a proof\nthat a sequence of growing Hilbert spaces has converged to an interacting\nHamiltonian or that its quantum state converges. The direct-limit\noperator construction has separate hypotheses in the final part.","The mean logarithmic expansion of this map is also exact:\n\\begin{equation}\n \\int_{-2}^2\\log|R_p'(x)|\\,d\\nu(x)=\\log p.\n \\label{a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","this map is also exact:\n\\begin{equation}\n \\int_{-2}^2\\log|R_p'(x)|\\,d\\nu(x)=\\log p.\n \\label{a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","egin{equation}\n \\int_{-2}^2\\log|R_p'(x)|\\,d\\nu(x)=\\log p.\n \\label{a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","-2}^2\\log|R_p'(x)|\\,d\\nu(x)=\\log p.\n \\label{a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","\\log p.\n \\label{a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","a-spectral-log-expansion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","ion}\n\\end{equation}\nIndeed for $x=2\\cos\\theta$,\n$R_p'(x)=p\\sin(p\\theta)/\\sin\\theta$. Logarithms at its finitely\nmany zeros are integrable, since they have the local form\n$\\log|\\theta-\\theta_0|$ plus a bounded term. Hence the integral is\n$\\log p$ plus the difference of the integrals of\n$\\log|\\sin(p\\theta)|$ and $\\log\\sin\\theta$ over $[0,\\pi]$.\nThe substitution $u=p\\theta$ and $\\pi$-periodicity of\n$\\log|\\sin u|$ make these two integrals equal. This calculation\ndoes not assert the entropy of a different physical dynamics.","\\adicfigure{11_prime_renormalization}{The normalized maps\n$R_2$, $R_3$, and $R_5$ preserve the same arcsine spectral measure.\nThe accompanying bounded moment checks illustrate the exact measure\nidentity, while each Schur realization retains its own excluded poles.}","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficie","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spec","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equa","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain an","Fix an integer $L\\geq2$, write $\\omega_L=e^{2\\pi i/L}$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rota…","$, and let\n$F(z)=\\sum_{n=0}^\\infty a_nz^n$ be analytic for $|z|<\\rho$.\nThe whole disk is invariant under the rotations used below. Define\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)\n =\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}F(\\omega_L^\\ell z),\n \\quad 0\\leq j<L.\n \\label{a-spectral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rotations; the\nlocal power-series identity does not remov…","ctral-analytic-filter}\n\\end{equation}\nAbsolute convergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rotations; the\nlocal power-series identity does not remove such analytic restrictions.","vergence for $|z|<\\rho$ and the finite character sum give\n\\begin{equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rotations; the\nlocal power-series identity does not remove such analytic restrictions.","equation}\n (\\mathfrak R_{L,j}F)(z)=\\sum_{q=0}^\\infty a_{Lq+j}z^{Lq+j},\n \\quad\n \\mathfrak R_{L,j}\\mathfrak R_{L,k}\n   =\\delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rotations; the\nlocal power-series identity does not remove such analytic restrictions.","delta_{jk}\\mathfrak R_{L,j},\\quad\n \\sum_j\\mathfrak R_{L,j}=I.\n \\label{a-spectral-analytic-lanes}\n\\end{equation}\nTo prove the coefficient formula, the coefficient of $a_nz^n$ in\n\\eqref{a-spectral-analytic-filter} is\n$L^{-1}\\sum_\\ell\\omega_L^{(n-j)\\ell}$, equal to one exactly when\n$n\\equiv j\\pmod L$. The idempotent and reconstruction laws follow\nby filtering these coefficients again. For an entire $F$ the formulas\nhold everywhere. For a function continued outside its disk, the\ndomain and any chosen branches must also support all rotations; the\nlocal power-series identity does not remove such analytic restrictions.","Apply this filter to $e^z$ and put\n\\begin{equation}\n E_{L,j}(z)=\\sum_{q=0}^\\infty\\frac{z^{Lq+j}}{(Lq+j)!}.\n \\label{a-spectral-factorial-lane}\n\\end{equation}\nFor $\\alpha\\in\\C$, write\n$(\\alpha)_q=\\alpha(\\alpha+1)\\cdots(\\alpha+q-1)$ and\n$(\\alpha)_0=1$. Splitting the factorial into step-$L$ factors gives\n\\begin{align}\n \\frac{(Lq+j)!}{j!}\n &=\\prod_{r=1}^{L}\\prod_{k=0}^{q-1}(j+r+Lk)\\nonumber","Apply this filter to $e^z$ and put\n\\begin{equation}\n E_{L,j}(z)=\\sum_{q=0}^\\infty\\frac{z^{Lq+j}}{(Lq+j)!}.\n \\label{a-spectral-factorial-lane}\n\\end{equation}\nFor $\\alpha\\in\\C$, write\n$(\\alpha)_q=\\alpha(\\alpha+1)\\cdots(\\alpha+q-1)$ and\n$(\\alpha)_0=1$. Splitting the factorial into step-$L$ factors gives\n\\begin{align}\n \\frac{(Lq+j)!}{j!}\n &=\\prod_{r=1}^{L}\\prod_{k=0}^{q-1}(j+r+Lk)\\nonumber\\\\\n &=L^{Lq}\\prod_{r=1}^{L}\\left(\\frac{j+r}{L}\\right)_q.\n \\label{a-spectral-factorial-split}\n\\end{align}\nExactly one bottom parameter is one, at $r=L-j$. Define the multiset\n\\[\n \\mathcal A_{L,j}\n =\\left\\{\\frac{j+r}{L}:1\\leq r\\leq L\\right\\}\\setminus\\{1\\}.\n\\]\nEvery member is positive, so none is a forbidden nonpositive integer\nin a hypergeometric denominator. Since $(1)_q=q!$,\n\\begin{equation}\n E_{L,j}(z)=\\frac{z^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-","&=L^{Lq}\\prod_{r=1}^{L}\\left(\\frac{j+r}{L}\\right)_q.\n \\label{a-spectral-factorial-split}\n\\end{align}\nExactly one bottom parameter is one, at $r=L-j$. Define the multiset\n\\[\n \\mathcal A_{L,j}\n =\\left\\{\\frac{j+r}{L}:1\\leq r\\leq L\\right\\}\\setminus\\{1\\}.\n\\]\nEvery member is positive, so none is a forbidden nonpositive integer\nin a hypergeometric denominator. Since $(1)_q=q!$,\n\\begin{equation}\n E_{L,j}(z)=\\frac{z^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-","&=L^{Lq}\\prod_{r=1}^{L}\\left(\\frac{j+r}{L}\\right)_q.\n \\label{a-spectral-factorial-split}\n\\end{align}\nExactly one bottom parameter is one, at $r=L-j$. Define the multiset\n\\[\n \\mathcal A_{L,j}\n =\\left\\{\\frac{j+r}{L}:1\\leq r\\leq L\\right\\}\\setminus\\{1\\}.\n\\]\nEvery member is positive, so none is a forbidden nonpositive integer\nin a hypergeometric denominator. Since $(1)_q=q!$,\n\\begin{equation}\n E_{L,j}(z)=\\frac{z^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-\\\\\\mathcal A_{L,j}\\end{matrix};\n                              \\frac{z^L}{L^L}\\right).\n \\label{a-spectral-hypergeometric-lane}\n\\end{equation}\nHere ${}_0F_s(;b_1,\\ldots,b_s;w)$ is defined by\n$\\sum_{q\\geq0}w^q/[q!\\prod_r(b_r)_q]$. The ratio of successive\nterms has magnitude $|w|/[(q+1)\\prod_r|q+b_r|]$, which tends to\nzero. Its series is entire in $w$, so \\eqref{a-spectral-hypergeometric-lane}\nis an everywhere convergent identity with the factorial series.\nFor example $\\mathcal A_{3,1}=\\{2/3,4/3\\}$ and\n$E_{3,1}(z)=z\\,{}_0F_2(;2/3,4/3;z^3/27)$.","mathcal A_{L,j}\\end{matrix};\n                              \\frac{z^L}{L^L}\\right).\n \\label{a-spectral-hypergeometric-lane}\n\\end{equation}\nHere ${}_0F_s(;b_1,\\ldots,b_s;w)$ is defined by\n$\\sum_{q\\geq0}w^q/[q!\\prod_r(b_r)_q]$. The ratio of successive\nterms has magnitude $|w|/[(q+1)\\prod_r|q+b_r|]$, which tends to\nzero. Its series is entire in $w$, so \\eqref{a-spectral-hypergeometric-lane}\nis an everywhere convergent identity with the factorial series.\nFor example $\\mathcal A_{3,1}=\\{2/3,4/3\\}$ and\n$E_{3,1}(z)=z\\,{}_0F_2(;2/3,4/3;z^3/27)$.","L}{L^L}\\right).\n \\label{a-spectral-hypergeometric-lane}\n\\end{equation}\nHere ${}_0F_s(;b_1,\\ldots,b_s;w)$ is defined by\n$\\sum_{q\\geq0}w^q/[q!\\prod_r(b_r)_q]$. The ratio of successive\nterms has magnitude $|w|/[(q+1)\\prod_r|q+b_r|]$, which tends to\nzero. Its series is entire in $w$, so \\eqref{a-spectral-hypergeometric-lane}\nis an everywhere convergent identity with the factorial series.\nFor example $\\mathcal A_{3,1}=\\{2/3,4/3\\}$ and\n$E_{3,1}(z)=z\\,{}_0F_2(;2/3,4/3;z^3/27)$.","hypergeometric-lane}\n\\end{equation}\nHere ${}_0F_s(;b_1,\\ldots,b_s;w)$ is defined by\n$\\sum_{q\\geq0}w^q/[q!\\prod_r(b_r)_q]$. The ratio of successive\nterms has magnitude $|w|/[(q+1)\\prod_r|q+b_r|]$, which tends to\nzero. Its series is entire in $w$, so \\eqref{a-spectral-hypergeometric-lane}\nis an everywhere convergent identity with the factorial series.\nFor example $\\mathcal A_{3,1}=\\{2/3,4/3\\}$ and\n$E_{3,1}(z)=z\\,{}_0F_2(;2/3,4/3;z^3/27)$.","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\o","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\el","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometr","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the sa","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different num","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a root sum can","Termwise differentiation on compact subsets gives\n$E'_{L,j}=E_{L,j-1}$ for $j\\geq1$. For $j=0$, the constant term\nvanishes under differentiation and reindexing gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a root sum can contain cancellations even when the\ndefining series has positive terms…","gives\n$E'_{L,0}=E_{L,L-1}$. Let\n$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a root sum can contain cancellations even when the\ndefining series has positive terms.","$\\mathbf E_L=(E_{L,0},\\ldots,E_{L,L-1})^{\\mathsf T}$ and\ndefine the cyclic matrix by $(C_Lv)_0=v_{L-1}$ and\n$(C_Lv)_j=v_{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a root sum can contain cancellations even when the\ndefining series has positive terms.","{j-1}$ for $j\\geq1$. Then\n\\begin{equation}\n \\mathbf E_L'=C_L\\mathbf E_L,\\qquad\n \\mathbf E_L(0)=e_0,\\qquad\n C_L^L=I,\\qquad\n \\mathbf E_L(z)=e^{zC_L}e_0.\n \\label{a-spectral-companion-system}\n\\end{equation}\nOne can verify the last equality directly: $C_L^ne_0=e_{n\\bmod L}$,\nso its $j$th exponential coordinate is exactly\n\\eqref{a-spectral-factorial-lane}. Character filtering also gives\n\\begin{equation}\n E_{L,j}(z)=\\frac1L\\sum_{\\ell=0}^{L-1}\n               \\omega_L^{-j\\ell}e^{\\omega_L^\\ell z}.\n \\label{a-spectral-root-exponential}\n\\end{equation}\nThus factorial, hypergeometric, companion-state, and root-filter forms\ndescribe the same analytic function. Each form has a different numerical\nconditioning: a root sum can contain cancellations even when the\ndefining series has positive terms.","For a general finite transfer matrix $M_1$, the residue-state identity\n$v_{Lq+j}=M_1^j(M_1^L)^qv_0$ follows by adding exponents. It needs\nno hypergeometric coefficients. A factorial lane provides a particular\nanalytic coordinate system; it does not impose a unique microscopic\nHamiltonian on every such transfer problem.","\\begin{theorem}[Entire lane calculus for a bounded operator]\n\\label{a-spectral-bounded-operator-lanes}\nLet $A_0\\in\\mathcal B(\\HH)$ and $t\\in\\C$. The series\n$E_{L,j}(tA_0)$ converges absolutely in operator norm and satisfies\n\\begin{align}\n E_{L,j}(tA_0)\n &=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                           e^{\\omega_L^\\ell tA_0},\\nonumber","\\begin{theorem}[Entire lane calculus for a bounded operator]\n\\label{a-spectral-bounded-operator-lanes}\nLet $A_0\\in\\mathcal B(\\HH)$ and $t\\in\\C$. The series\n$E_{L,j}(tA_0)$ converges absolutely in operator norm and satisfies\n\\begin{align}\n E_{L,j}(tA_0)\n &=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                           e^{\\omega_L^\\ell tA_0},\\nonumber\\\\\n &=\\frac{(tA_0)^j}{j!}\\,\n {}_0F_{L-1}\\!\\left(\\begin{matrix}-\\\\\\mathcal A_{L,j}\\end{matrix};\n                      \\frac{(tA_0)^L}{L^L}\\right),\\nonumber\\\\\n \\sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.\n \\label{a-spectral-operator-lane-identities}\n\\end{align}\nIf all terms through degree $Q$ are retained in a lane, the omitted\noperator norm is at most\n\\begin{equation}\n e^a\\frac{a^{Q+1}}{(Q+1)!},\\qquad a=|t|\\norm{A_0}.\n \\label{a-spectral-lane-tail-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of norms of all exponential terms is at most $e^a$. A lane\nis a subseries of this sum, so it converges absolutely. The character\ncalculation and factorial split can therefore be applied in operator\nnorm, giving the first two identities. …","\\sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.\n \\label{a-spectral-operator-lane-identities}\n\\end{align}\nIf all terms through degree $Q$ are retained in a lane, the omitted\noperator norm is at most\n\\begin{equation}\n e^a\\frac{a^{Q+1}}{(Q+1)!},\\qquad a=|t|\\norm{A_0}.\n \\label{a-spectral-lane-tail-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of norms of all exponential terms is at most $e^a$. A lane\nis a subseries of this sum, so it converges absolutely. The character\ncalculation and factorial split can therefore be applied in operator\nnorm, giving the first two identities. Partitioning the nonnegative\ndegrees by their residues proves the third. All products are powers\nof the same operator, so there is no commuting-operator assumption.\nFor the tail, bound a lane by the full tail. Since\n$(Q+1+k)!\\geq(Q+1)!k!$,\n\\[\n \\","\\sum_{j=0}^{L-1}E_{L,j}(tA_0)&=e^{tA_0}.\n \\label{a-spectral-operator-lane-identities}\n\\end{align}\nIf all terms through degree $Q$ are retained in a lane, the omitted\noperator norm is at most\n\\begin{equation}\n e^a\\frac{a^{Q+1}}{(Q+1)!},\\qquad a=|t|\\norm{A_0}.\n \\label{a-spectral-lane-tail-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe sum of norms of all exponential terms is at most $e^a$. A lane\nis a subseries of this sum, so it converges absolutely. The character\ncalculation and factorial split can therefore be applied in operator\nnorm, giving the first two identities. Partitioning the nonnegative\ndegrees by their residues proves the third. All products are powers\nof the same operator, so there is no commuting-operator assumption.\nFor the tail, bound a lane by the full tail. Since\n$(Q+1+k)!\\geq(Q+1)!k!$,\n\\[\n \\sum_{n=Q+1}^\\infty\\frac{a^n}{n!}\n \\leq\\frac{a^{Q+1}}{(Q+1)!}\\sum_{k=0}^\\infty\\frac{a^k}{k!},\n\\]\nwhich proves \\eqref{a-spectral-lane-tail-bound}.\n\\end{proof}","lane\nis a subseries of this sum, so it converges absolutely. The character\ncalculation and factorial split can therefore be applied in operator\nnorm, giving the first two identities. Partitioning the nonnegative\ndegrees by their residues proves the third. All products are powers\nof the same operator, so there is no commuting-operator assumption.\nFor the tail, bound a lane by the full tail. Since\n$(Q+1+k)!\\geq(Q+1)!k!$,\n\\[\n \\sum_{n=Q+1}^\\infty\\frac{a^n}{n!}\n \\leq\\frac{a^{Q+1}}{(Q+1)!}\\sum_{k=0}^\\infty\\frac{a^k}{k!},\n\\]\nwhich proves \\eqref{a-spectral-lane-tail-bound}.\n\\end{proof}","is sum, so it converges absolutely. The character\ncalculation and factorial split can therefore be applied in operator\nnorm, giving the first two identities. Partitioning the nonnegative\ndegrees by their residues proves the third. All products are powers\nof the same operator, so there is no commuting-operator assumption.\nFor the tail, bound a lane by the full tail. Since\n$(Q+1+k)!\\geq(Q+1)!k!$,\n\\[\n \\sum_{n=Q+1}^\\infty\\frac{a^n}{n!}\n \\leq\\frac{a^{Q+1}}{(Q+1)!}\\sum_{k=0}^\\infty\\frac{a^k}{k!},\n\\]\nwhich proves \\eqref{a-spectral-lane-tail-bound}.\n\\end{proof}","If $A_0^L=\\alpha I$, the hypergeometric expression reduces every\nlane to a scalar function multiplying $A_0^j$. The propagator lies\nin the finite span of $I,A_0,\\ldots,A_0^{L-1}$. A more general\npolynomial identity gives the corresponding finite algebra of powers.\nThese statements follow from the supplied algebraic relation; they\ndo not assert one for an arbitrary Hamiltonian.","The coefficient maps $\\mathfrak R_{L,j}$ in\n\\eqref{a-spectral-analytic-lanes} are idempotents on analytic functions.\nThe evaluated operators $E_{L,j}(tA_0)$ generally are not.\nFor $L=2$, $E_{2,0}(tA_0)=\\cosh(tA_0)$ and\n$E_{2,1}(tA_0)=\\sinh(tA_0)$, which are neither an orthogonal\nprojector family nor a pair of independent unitary evolutions.\nThe sum reconstructs an evolution; an individual lane is an analytic\ncontribution to it.","The same separation gives the entire function\n\\[\n B(y)=\\sum_{n=0}^\\infty\\frac{(-1)^ny^n}{(n!)^2}\n     ={}_0F_1(;1;-y).\n\\]\nFor $y\\geq0$, this is $J_0(2\\sqrt y)$; the entire series itself\ndoes not need a square-root branch. Define\n$B_{L,j}(y)=\\sum_{q\\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2$.\nSquaring \\eqref{a-spectral-factorial-split} gives\n\\begin{equation}\n B_{L,j}(y)\n =\\frac{(-1)^jy^j}{(j!)^2}\\,\n {}_0F_{2L-1}\\!\\left(\n \\begin{matrix}-","The same separation gives the entire function\n\\[\n B(y)=\\sum_{n=0}^\\infty\\frac{(-1)^ny^n}{(n!)^2}\n     ={}_0F_1(;1;-y).\n\\]\nFor $y\\geq0$, this is $J_0(2\\sqrt y)$; the entire series itself\ndoes not need a square-root branch. Define\n$B_{L,j}(y)=\\sum_{q\\geq0}(-1)^{Lq+j}y^{Lq+j}/[(Lq+j)!]^2$.\nSquaring \\eqref{a-spectral-factorial-split} gives\n\\begin{equation}\n B_{L,j}(y)\n =\\frac{(-1)^jy^j}{(j!)^2}\\,\n {}_0F_{2L-1}\\!\\left(\n \\begin{matrix}-\\\\1,\\mathcal A_{L,j},\\mathcal A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$…","1,\\mathcal A_{L,j},\\mathcal A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits","1,\\mathcal A_{L,j},\\mathcal A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equati","1,\\mathcal A_{L,j},\\mathcal A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.","l A_{L,j}\\end{matrix};\n \\frac{(-1)^Ly^L}{L^{2L}}\\right).\n \\label{a-spectral-bessel-lane}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.","}\n\\end{equation}\nOne of the two $q!$ factors is the conventional hypergeometric\nfactorial and the other supplies the explicit bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.","it bottom parameter $1$.\nAll bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.","bottom parameters are positive and the ratio test again gives\nentire convergence. Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.",". Character filtering reconstructs $B=\\sum_jB_{L,j}$.\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.",".\nWith the Euler derivative $\\Theta=y\\,d/dy$, termwise differentiation\nalso gives the cyclic system\n\\begin{equation}\n \\Theta^2B_{L,j}(y)=-yB_{L,j-1\\bmod L}(y).\n \\label{a-spectral-bessel-euler}\n\\end{equation}\nFor a term of positive degree $n$, its new coefficient is\n$n^2(-1)^n/(n!)^2=-(-1)^{n-1}/[(n-1)!]^2$.\nThe degree-zero term disappears under $\\Theta^2$, which also verifies\nthe wraparound case $j=0$. The scalar Gowdy Bessel mode therefore\nadmits these analytic coordinates. Filtering that already-defined\nsolution does not add a gravitational field equation.","Let $H_\\infty$ be self-adjoint with spectral measure $E$. For real $t$\ndefine the scalar functions\n$f_j(\\lambda)=E_{L,j}(-it\\lambda)$.\nThe operator $f_j(H_\\infty)$ has domain\n\\begin{equation}\n \\Dom(f_j(H_\\infty))\n =\\left\\{\\psi:\\int_{\\R}|f_j(\\lambda)|^2\n                       \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty\\right\\}.\n \\label{a-spectral-lane-domain}\n\\end{equation}\nThe scalar identity $\\sum_jf_j(\\lambda)=e^{-it\\lambda}$ gives\n\\begin{equation}\n \\sum_jf_j(H_\\infty)\\psi=e^{-itH_\\infty}\\psi,\n \\qquad \\psi\\in\\bigcap_j\\Dom(f_j(H_\\infty)).\n \\label{a-spectral-unbounded-reconstruction}\n\\end{equation}\nThe equality holds on the stated common domain because it is a finite\nsum of spectral functions there. It d","Let $H_\\infty$ be self-adjoint with spectral measure $E$. For real $t$\ndefine the scalar functions\n$f_j(\\lambda)=E_{L,j}(-it\\lambda)$.\nThe operator $f_j(H_\\infty)$ has domain\n\\begin{equation}\n \\Dom(f_j(H_\\infty))\n =\\left\\{\\psi:\\int_{\\R}|f_j(\\lambda)|^2\n                       \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty\\right\\}.\n \\label{a-spectral-lane-domain}\n\\end{equation}\nThe scalar identity $\\sum_jf_j(\\lambda)=e^{-it\\lambda}$ gives\n\\begin{equation}\n \\sum_jf_j(H_\\infty)\\psi=e^{-itH_\\infty}\\psi,\n \\qquad \\psi\\in\\bigcap_j\\Dom(f_j(H_\\infty)).\n \\label{a-spectral-unbounded-reconstruction}\n\\end{equation}\nThe equality holds on the stated common domain because it is a finite\nsum of spectral functions there. It does not say that each lane acts\non every v","Let $H_\\infty$ be self-adjoint with spectral measure $E$. For real $t$\ndefine the scalar functions\n$f_j(\\lambda)=E_{L,j}(-it\\lambda)$.\nThe operator $f_j(H_\\infty)$ has domain\n\\begin{equation}\n \\Dom(f_j(H_\\infty))\n =\\left\\{\\psi:\\int_{\\R}|f_j(\\lambda)|^2\n                       \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty\\right\\}.\n \\label{a-spectral-lane-domain}\n\\end{equation}\nThe scalar identity $\\sum_jf_j(\\lambda)=e^{-it\\lambda}$ gives\n\\begin{equation}\n \\sum_jf_j(H_\\infty)\\psi=e^{-itH_\\infty}\\psi,\n \\qquad \\psi\\in\\bigcap_j\\Dom(f_j(H_\\infty)).\n \\label{a-spectral-unbounded-reconstruction}\n\\end{equation}\nThe equality holds on the stated common domain because it is a finite\nsum of spectral functions there. It does not say that each lane acts\non every vector.","E$. For real $t$\ndefine the scalar functions\n$f_j(\\lambda)=E_{L,j}(-it\\lambda)$.\nThe operator $f_j(H_\\infty)$ has domain\n\\begin{equation}\n \\Dom(f_j(H_\\infty))\n =\\left\\{\\psi:\\int_{\\R}|f_j(\\lambda)|^2\n                       \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty\\right\\}.\n \\label{a-spectral-lane-domain}\n\\end{equation}\nThe scalar identity $\\sum_jf_j(\\lambda)=e^{-it\\lambda}$ gives\n\\begin{equation}\n \\sum_jf_j(H_\\infty)\\psi=e^{-itH_\\infty}\\psi,\n \\qquad \\psi\\in\\bigcap_j\\Dom(f_j(H_\\infty)).\n \\label{a-spectral-unbounded-reconstruction}\n\\end{equation}\nThe equality holds on the stated common domain because it is a finite\nsum of spectral functions there. It does not say that each lane acts\non every vector.","For an explicit example, take $L=4$. The scalar formula gives\n\\[\n f_0(\\lambda)=\\tfrac12\\bigl(\\cosh(t\\lambda)+\\cos(t\\lambda)\\bigr),\n \\quad\n f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n\\]\nIf $H_\\infty$ is multiplication by $\\lambda$ on $L^2(\\R)$ and\n$t\\ne0$, these functions define unbounded operators. Their sum\ncancels the growing hyperbolic contribution. The full propagator\n$e^{-itH_\\infty}$ is nevertheless unitary on all of $L^2(\\R)$.\nFor $L=2$, by contrast, the two Schr\\\"odinger lanes are\n$\\cos(tH_\\infty)$ and $-i\\sin(tH_\\infty)$, both bounded. A larger\nlane count must not inherit this special dyadic boundedness without proof.","rmula gives\n\\[\n f_0(\\lambda)=\\tfrac12\\bigl(\\cosh(t\\lambda)+\\cos(t\\lambda)\\bigr),\n \\quad\n f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n\\]\nIf $H_\\infty$ is multiplication by $\\lambda$ on $L^2(\\R)$ and\n$t\\ne0$, these functions define unbounded operators. Their sum\ncancels the growing hyperbolic contribution. The full propagator\n$e^{-itH_\\infty}$ is nevertheless unitary on all of $L^2(\\R)$.\nFor $L=2$, by contrast, the two Schr\\\"odinger lanes are\n$\\cos(tH_\\infty)$ and $-i\\sin(tH_\\infty)$, both bounded. A larger\nlane count must not inherit this special dyadic boundedness without proof.","f_0(\\lambda)=\\tfrac12\\bigl(\\cosh(t\\lambda)+\\cos(t\\lambda)\\bigr),\n \\quad\n f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n\\]\nIf $H_\\infty$ is multiplication by $\\lambda$ on $L^2(\\R)$ and\n$t\\ne0$, these functions define unbounded operators. Their sum\ncancels the growing hyperbolic contribution. The full propagator\n$e^{-itH_\\infty}$ is nevertheless unitary on all of $L^2(\\R)$.\nFor $L=2$, by contrast, the two Schr\\\"odinger lanes are\n$\\cos(tH_\\infty)$ and $-i\\sin(tH_\\infty)$, both bounded. A larger\nlane count must not inherit this special dyadic boundedness without proof.","ambda)\\bigr),\n \\quad\n f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n\\]\nIf $H_\\infty$ is multiplication by $\\lambda$ on $L^2(\\R)$ and\n$t\\ne0$, these functions define unbounded operators. Their sum\ncancels the growing hyperbolic contribution. The full propagator\n$e^{-itH_\\infty}$ is nevertheless unitary on all of $L^2(\\R)$.\nFor $L=2$, by contrast, the two Schr\\\"odinger lanes are\n$\\cos(tH_\\infty)$ and $-i\\sin(tH_\\infty)$, both bounded. A larger\nlane count must not inherit this special dyadic boundedness without proof.","f_2(\\lambda)=\\tfrac12\\bigl(\\cos(t\\lambda)-\\cosh(t\\lambda)\\bigr).\n\\]\nIf $H_\\infty$ is multiplication by $\\lambda$ on $L^2(\\R)$ and\n$t\\ne0$, these functions define unbounded operators. Their sum\ncancels the growing hyperbolic contribution. The full propagator\n$e^{-itH_\\infty}$ is nevertheless unitary on all of $L^2(\\R)$.\nFor $L=2$, by contrast, the two Schr\\\"odinger lanes are\n$\\cos(tH_\\infty)$ and $-i\\sin(tH_\\infty)$, both bounded. A larger\nlane count must not inherit this special dyadic boundedness without proof.","A sufficient common-domain condition is\n\\begin{equation}\n \\int_{\\R} e^{2|t||\\lambda|}\n              \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty,\n \\label{a-spectral-exponential-domain}\n\\end{equation}\nsince $|f_j(\\lambda)|\\leq e^{|t||\\lambda|}$. Spectral cutoffs\n$P_R=E([-R,R])$ supply a dense set satisfying this condition. On\n$\\Ran P_R$, the restricted Hamiltonian is bounded, the norm-series\nidentities apply, and\n\\[\n \\sum_jf_j(H_\\infty)P_R\\psi=e^{-itH_\\infty}P_R\\psi\n      \\longrightarrow e^{-itH_\\infty}\\psi.\n\\]\nOn the common domain of the individual lanes, each term also converges\nto $f_j(H_\\infty)\\psi$ by its spectral-domain integrability.\nFor an arbitrary vector only convergence of the summed cutoff expression\nis asserted. The domain question for an interacting Hamiltonian therefore\nremains even when a bounded surrogate has excellent numerical residuals.","on-domain condition is\n\\begin{equation}\n \\int_{\\R} e^{2|t||\\lambda|}\n              \\,d\\langle\\psi,E(\\lambda)\\psi\\rangle<\\infty,\n \\label{a-spectral-exponential-domain}\n\\end{equation}\nsince $|f_j(\\lambda)|\\leq e^{|t||\\lambda|}$. Spectral cutoffs\n$P_R=E([-R,R])$ supply a dense set satisfying this condition. On\n$\\Ran P_R$, the restricted Hamiltonian is bounded, the norm-series\nidentities apply, and\n\\[\n \\sum_jf_j(H_\\infty)P_R\\psi=e^{-itH_\\infty}P_R\\psi\n      \\longrightarrow e^{-itH_\\infty}\\psi.\n\\]\nOn the common domain of the individual lanes, each term also converges\nto $f_j(H_\\infty)\\psi$ by its spectral-domain integrability.\nFor an arbitrary vector only convergence of the summed cutoff expression\nis asserted. The domain question for an interacting Hamiltonian therefore\nremains even when a bounded surrogate has excellent numerical residuals.","quation}\nsince $|f_j(\\lambda)|\\leq e^{|t||\\lambda|}$. Spectral cutoffs\n$P_R=E([-R,R])$ supply a dense set satisfying this condition. On\n$\\Ran P_R$, the restricted Hamiltonian is bounded, the norm-series\nidentities apply, and\n\\[\n \\sum_jf_j(H_\\infty)P_R\\psi=e^{-itH_\\infty}P_R\\psi\n      \\longrightarrow e^{-itH_\\infty}\\psi.\n\\]\nOn the common domain of the individual lanes, each term also converges\nto $f_j(H_\\infty)\\psi$ by its spectral-domain integrability.\nFor an arbitrary vector only convergence of the summed cutoff expression\nis asserted. The domain question for an interacting Hamiltonian therefore\nremains even when a bounded surrogate has excellent numerical residuals.","Let $p$ be prime, let $L=mp$ for a positive integer $m$, and take\n$0\\leq j<p$. For every $q\\geq0$,\n\\begin{equation}\n g_q=\\frac{j!(Lq)!}{(Lq+j)!}\n       =\\binom{Lq+j}{j}^{-1}\\in\\Z_p^\\times.\n \\label{a-spectral-unit-window}\n\\end{equation}\nIndeed\n$\\binom{Lq+j}{j}=\\prod_{r=1}^j(Lq+r)/r$.\nEach numerator and denominator is a $p$-adic unit because\n$Lq\\equiv0\\pmod p$ and $1\\leq r<p$. For $j=0$ the empty product\nis one. The proof also covers $p=2$ in its short window. Outside this\nwindow, a factor can cross a multiple of $p$ and the same proof gives\nno unit guarantee. This rational coefficient statement supplies no\nclaim about the $p$-adic convergence of the complex exponential series.","sitive integer $m$, and take\n$0\\leq j<p$. For every $q\\geq0$,\n\\begin{equation}\n g_q=\\frac{j!(Lq)!}{(Lq+j)!}\n       =\\binom{Lq+j}{j}^{-1}\\in\\Z_p^\\times.\n \\label{a-spectral-unit-window}\n\\end{equation}\nIndeed\n$\\binom{Lq+j}{j}=\\prod_{r=1}^j(Lq+r)/r$.\nEach numerator and denominator is a $p$-adic unit because\n$Lq\\equiv0\\pmod p$ and $1\\leq r<p$. For $j=0$ the empty product\nis one. The proof also covers $p=2$ in its short window. Outside this\nwindow, a factor can cross a multiple of $p$ and the same proof gives\nno unit guarantee. This rational coefficient statement supplies no\nclaim about the $p$-adic convergence of the complex exponential series.","_q=\\frac{j!(Lq)!}{(Lq+j)!}\n       =\\binom{Lq+j}{j}^{-1}\\in\\Z_p^\\times.\n \\label{a-spectral-unit-window}\n\\end{equation}\nIndeed\n$\\binom{Lq+j}{j}=\\prod_{r=1}^j(Lq+r)/r$.\nEach numerator and denominator is a $p$-adic unit because\n$Lq\\equiv0\\pmod p$ and $1\\leq r<p$. For $j=0$ the empty product\nis one. The proof also covers $p=2$ in its short window. Outside this\nwindow, a factor can cross a multiple of $p$ and the same proof gives\nno unit guarantee. This rational coefficient statement supplies no\nclaim about the $p$-adic convergence of the complex exponential series.","tion}\nIndeed\n$\\binom{Lq+j}{j}=\\prod_{r=1}^j(Lq+r)/r$.\nEach numerator and denominator is a $p$-adic unit because\n$Lq\\equiv0\\pmod p$ and $1\\leq r<p$. For $j=0$ the empty product\nis one. The proof also covers $p=2$ in its short window. Outside this\nwindow, a factor can cross a multiple of $p$ and the same proof gives\nno unit guarantee. This rational coefficient statement supplies no\nclaim about the $p$-adic convergence of the complex exponential series.","Analytic refinement of a factorial lane is independent of this unit\ncondition. Splitting $q=pr+a$, $0\\leq a<p$, gives the exact identity\n\\begin{equation}\n E_{L,j}(z)=\\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\\qquad 0\\leq j<L.\n \\label{a-spectral-analytic-refinement}\n\\end{equation}\nAbsolute convergence permits this partition of the summation index.\nThe labels $j+aL$ all lie in $[0,pL)$, but need not lie in the\nparent's short unit window. Exact complex analytic refinement and\nthe local $p$-adic unit assertion therefore have separate hypotheses.","Splitting $q=pr+a$, $0\\leq a<p$, gives the exact identity\n\\begin{equation}\n E_{L,j}(z)=\\sum_{a=0}^{p-1}E_{pL,j+aL}(z),\\qquad 0\\leq j<L.\n \\label{a-spectral-analytic-refinement}\n\\end{equation}\nAbsolute convergence permits this partition of the summation index.\nThe labels $j+aL$ all lie in $[0,pL)$, but need not lie in the\nparent's short unit window. Exact complex analytic refinement and\nthe local $p$-adic unit assertion therefore have separate hypotheses.","On the oscillator Fock space, let\n$\\widehat N|n\\rangle=n|n\\rangle$ and define the bounded unitary\n$V=e^{2\\pi i\\widehat N/L}$. Its spectral residues are\n\\begin{equation}\n Q_j^{(L)}=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}V^\\ell\n          =\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n\\end{equation}\nOn every number basis vector, the same finite character sum proves the\nsecond equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,\n$Q_jQ_k=\\delta_{jk}Q_j$, and $\\sum_jQ_j=I$. No exponential norm series\nfor the unbounded number operator is needed; its spectral calculus\ndefines $V$.","t N|n\\rangle=n|n\\rangle$ and define the bounded unitary\n$V=e^{2\\pi i\\widehat N/L}$. Its spectral residues are\n\\begin{equation}\n Q_j^{(L)}=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}V^\\ell\n          =\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n\\end{equation}\nOn every number basis vector, the same finite character sum proves the\nsecond equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,\n$Q_jQ_k=\\delta_{jk}Q_j$, and $\\sum_jQ_j=I$. No exponential norm series\nfor the unbounded number operator is needed; its spectral calculus\ndefines $V$.","unitary\n$V=e^{2\\pi i\\widehat N/L}$. Its spectral residues are\n\\begin{equation}\n Q_j^{(L)}=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}V^\\ell\n          =\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n\\end{equation}\nOn every number basis vector, the same finite character sum proves the\nsecond equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,\n$Q_jQ_k=\\delta_{jk}Q_j$, and $\\sum_jQ_j=I$. No exponential norm series\nfor the unbounded number operator is needed; its spectral calculus\ndefines $V$.","N/L}$. Its spectral residues are\n\\begin{equation}\n Q_j^{(L)}=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}V^\\ell\n          =\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n\\end{equation}\nOn every number basis vector, the same finite character sum proves the\nsecond equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,\n$Q_jQ_k=\\delta_{jk}Q_j$, and $\\sum_jQ_j=I$. No exponential norm series\nfor the unbounded number operator is needed; its spectral calculus\ndefines $V$.","=\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}|n\\rangle\\langle n|.\n \\label{a-spectral-number-projectors}\n\\end{equation}\nOn every number basis vector, the same finite character sum proves the\nsecond equality. These $Q_j^{(L)}$ are bounded orthogonal projectors,\n$Q_jQ_k=\\delta_{jk}Q_j$, and $\\sum_jQ_j=I$. No exponential norm series\nfor the unbounded number operator is needed; its spectral calculus\ndefines $V$.","For a coherent state\n\\[\n |\\alpha\\rangle=e^{-|\\alpha|^2/2}\n        \\sum_{n=0}^\\infty\\frac{\\alpha^n}{\\sqrt{n!}}|n\\rangle,\n \\qquad \\mu=|\\alpha|^2\\geq0,\n\\]\nthe packet probability is the Poisson residue sum\n\\begin{align}\n p_{L,j}(\\mu)\n &=\\langle\\alpha,Q_j^{(L)}\\alpha\\rangle\n   =e^{-\\mu}E_{L,j}(\\mu)\\nonumber","For a coherent state\n\\[\n |\\alpha\\rangle=e^{-|\\alpha|^2/2}\n        \\sum_{n=0}^\\infty\\frac{\\alpha^n}{\\sqrt{n!}}|n\\rangle,\n \\qquad \\mu=|\\alpha|^2\\geq0,\n\\]\nthe packet probability is the Poisson residue sum\n\\begin{align}\n p_{L,j}(\\mu)\n &=\\langle\\alpha,Q_j^{(L)}\\alpha\\rangle\n   =e^{-\\mu}E_{L,j}(\\mu)\\nonumber\\\\\n &=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                      e^{\\mu(\\omega_L^\\ell-1)}.\n \\label{a-spectral-coherent-packets}\n\\end{align}\nThe factorial series proves nonnegativity, including the limiting values\n$p_{L,0}(0)=1$ and $p_{L,j}(0)=0$ for $j>0$. Partition of the\nPoisson series proves $\\sum_jp_{L,j}=1$. Their equivalent hypergeometric\nform is $e^{-\\mu}$ times \\eqref{a-spectral-hypergeometric-lane} at $z=\\mu$.","&=\\frac1L\\sum_{\\ell=0}^{L-1}\\omega_L^{-j\\ell}\n                      e^{\\mu(\\omega_L^\\ell-1)}.\n \\label{a-spectral-coherent-packets}\n\\end{align}\nThe factorial series proves nonnegativity, including the limiting values\n$p_{L,0}(0)=1$ and $p_{L,j}(0)=0$ for $j>0$. Partition of the\nPoisson series proves $\\sum_jp_{L,j}=1$. Their equivalent hypergeometric\nform is $e^{-\\mu}$ times \\eqref{a-spectral-hypergeometric-lane} at $z=\\mu$.","=0}^{L-1}\\omega_L^{-j\\ell}\n                      e^{\\mu(\\omega_L^\\ell-1)}.\n \\label{a-spectral-coherent-packets}\n\\end{align}\nThe factorial series proves nonnegativity, including the limiting values\n$p_{L,0}(0)=1$ and $p_{L,j}(0)=0$ for $j>0$. Partition of the\nPoisson series proves $\\sum_jp_{L,j}=1$. Their equivalent hypergeometric\nform is $e^{-\\mu}$ times \\eqref{a-spectral-hypergeometric-lane} at $z=\\mu$.","\\begin{proposition}[Approach to uniform residue probability]\n\\label{a-spectral-coherent-bound}\nFor fixed $L\\geq2$ and $\\mu\\geq0$,\n\\begin{equation}\n \\left|p_{L,j}(\\mu)-\\frac1L\\right|\n \\leq\\frac1L\\sum_{\\ell=1}^{L-1}\n       e^{-\\mu(1-\\cos(2\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn \\eqref{a-spectral-coherent-packets}, the $\\ell=0$ term is $1/L$.\nThe modulus of every other term is\n$e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L$.\nThe triangle inequality gives the first bound. Among the nonzero\nresidues, the largest cosine is $\\cos(2\\pi/L)$, giving the second.\n\\end{proof}","iform residue probability]\n\\label{a-spectral-coherent-bound}\nFor fixed $L\\geq2$ and $\\mu\\geq0$,\n\\begin{equation}\n \\left|p_{L,j}(\\mu)-\\frac1L\\right|\n \\leq\\frac1L\\sum_{\\ell=1}^{L-1}\n       e^{-\\mu(1-\\cos(2\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn \\eqref{a-spectral-coherent-packets}, the $\\ell=0$ term is $1/L$.\nThe modulus of every other term is\n$e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L$.\nThe triangle inequality gives the first bound. Among the nonzero\nresidues, the largest cosine is $\\cos(2\\pi/L)$, giving the second.\n\\end{proof}","bability]\n\\label{a-spectral-coherent-bound}\nFor fixed $L\\geq2$ and $\\mu\\geq0$,\n\\begin{equation}\n \\left|p_{L,j}(\\mu)-\\frac1L\\right|\n \\leq\\frac1L\\sum_{\\ell=1}^{L-1}\n       e^{-\\mu(1-\\cos(2\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn \\eqref{a-spectral-coherent-packets}, the $\\ell=0$ term is $1/L$.\nThe modulus of every other term is\n$e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L$.\nThe triangle inequality gives the first bound. Among the nonzero\nresidues, the largest cosine is $\\cos(2\\pi/L)$, giving the second.\n\\end{proof}","}\nFor fixed $L\\geq2$ and $\\mu\\geq0$,\n\\begin{equation}\n \\left|p_{L,j}(\\mu)-\\frac1L\\right|\n \\leq\\frac1L\\sum_{\\ell=1}^{L-1}\n       e^{-\\mu(1-\\cos(2\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn \\eqref{a-spectral-coherent-packets}, the $\\ell=0$ term is $1/L$.\nThe modulus of every other term is\n$e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L$.\nThe triangle inequality gives the first bound. Among the nonzero\nresidues, the largest cosine is $\\cos(2\\pi/L)$, giving the second.\n\\end{proof}","\\pi\\ell/L))}\n \\leq\\frac{L-1}{L}\n       e^{-\\mu(1-\\cos(2\\pi/L))}.\n \\label{a-spectral-packet-mixing-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn \\eqref{a-spectral-coherent-packets}, the $\\ell=0$ term is $1/L$.\nThe modulus of every other term is\n$e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}/L$.\nThe triangle inequality gives the first bound. Among the nonzero\nresidues, the largest cosine is $\\cos(2\\pi/L)$, giving the second.\n\\end{proof}","The exponent is positive for each fixed $L$, but decreases as\n$2\\pi^2/L^2$ when $L$ grows. The conclusion is therefore uniform in\nthe finite packet label $j$ at fixed $L$; it is not a depth-uniform\nestimate at arbitrarily large residue modulus.","Finally the annihilation operator routes adjacent residues:\n\\begin{equation}\n Q_r^{(L)}aQ_s^{(L)}=0\\quad\\text{unless }r\\equiv s-1\\pmod L,\n \\qquad\n Q_r^{(L)}a^\\dagger Q_s^{(L)}=0\\quad\\text{unless }r\\equiv s+1\\pmod L.\n \\label{a-spectral-ladder-routing}\n\\end{equation}\nOn a finite number-state combination this follows from\n$a|n\\rangle=\\sqrt n\\,|n-1\\rangle$ and\n$a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle$.\nEach residue projector preserves $\\Dom(a)$ and $\\Dom(a^\\dagger)$,\nbecause it only removes terms from their weighted number norm.\nTruncation of a vector in either domain converges in the corresponding\ngraph norm, so the identities extend to those domains.\nFor the isolated oscillator Hamiltonian, which is a function of\n$\\widehat N$, the packet projectors commute with the evolution and\nthe weights stay constant. A driving term containing ladder operators\ncan transfer weight according to the routing rule. Selecting such a\ndrive is further model","Finally the annihilation operator routes adjacent residues:\n\\begin{equation}\n Q_r^{(L)}aQ_s^{(L)}=0\\quad\\text{unless }r\\equiv s-1\\pmod L,\n \\qquad\n Q_r^{(L)}a^\\dagger Q_s^{(L)}=0\\quad\\text{unless }r\\equiv s+1\\pmod L.\n \\label{a-spectral-ladder-routing}\n\\end{equation}\nOn a finite number-state combination this follows from\n$a|n\\rangle=\\sqrt n\\,|n-1\\rangle$ and\n$a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle$.\nEach residue projector preserves $\\Dom(a)$ and $\\Dom(a^\\dagger)$,\nbecause it only removes terms from their weighted number norm.\nTruncation of a vector in either domain converges in the corresponding\ngraph norm, so the identities extend to those domains.\nFor the isolated oscillator Hamiltonian, which is a function of\n$\\widehat N$, the packet projectors commute with the evolution and\nthe weights stay constant. A driving term containing ladder operators\ncan transfer weight according to the routing rule. Selecting such a\ndrive is further model data.","s adjacent residues:\n\\begin{equation}\n Q_r^{(L)}aQ_s^{(L)}=0\\quad\\text{unless }r\\equiv s-1\\pmod L,\n \\qquad\n Q_r^{(L)}a^\\dagger Q_s^{(L)}=0\\quad\\text{unless }r\\equiv s+1\\pmod L.\n \\label{a-spectral-ladder-routing}\n\\end{equation}\nOn a finite number-state combination this follows from\n$a|n\\rangle=\\sqrt n\\,|n-1\\rangle$ and\n$a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle$.\nEach residue projector preserves $\\Dom(a)$ and $\\Dom(a^\\dagger)$,\nbecause it only removes terms from their weighted number norm.\nTruncation of a vector in either domain converges in the corresponding\ngraph norm, so the identities extend to those domains.\nFor the isolated oscillator Hamiltonian, which is a function of\n$\\widehat N$, the packet projectors commute with the evolution and\nthe weights stay constant. A driving term containing ladder operators\ncan transfer weight according to the routing rule. Selecting such a\ndrive is further model data.","idues:\n\\begin{equation}\n Q_r^{(L)}aQ_s^{(L)}=0\\quad\\text{unless }r\\equiv s-1\\pmod L,\n \\qquad\n Q_r^{(L)}a^\\dagger Q_s^{(L)}=0\\quad\\text{unless }r\\equiv s+1\\pmod L.\n \\label{a-spectral-ladder-routing}\n\\end{equation}\nOn a finite number-state combination this follows from\n$a|n\\rangle=\\sqrt n\\,|n-1\\rangle$ and\n$a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle$.\nEach residue projector preserves $\\Dom(a)$ and $\\Dom(a^\\dagger)$,\nbecause it only removes terms from their weighted number norm.\nTruncation of a vector in either domain converges in the corresponding\ngraph norm, so the identities extend to those domains.\nFor the isolated oscillator Hamiltonian, which is a function of\n$\\widehat N$, the packet projectors commute with the evolution and\nthe weights stay constant. A driving term containing ladder operators\ncan transfer weight according to the routing rule. Selecting such a\ndrive is further model data.","\\rangle=\\sqrt n\\,|n-1\\rangle$ and\n$a^\\dagger|n\\rangle=\\sqrt{n+1}\\,|n+1\\rangle$.\nEach residue projector preserves $\\Dom(a)$ and $\\Dom(a^\\dagger)$,\nbecause it only removes terms from their weighted number norm.\nTruncation of a vector in either domain converges in the corresponding\ngraph norm, so the identities extend to those domains.\nFor the isolated oscillator Hamiltonian, which is a function of\n$\\widehat N$, the packet projectors commute with the evolution and\nthe weights stay constant. A driving term containing ladder operators\ncan transfer weight according to the routing rule. Selecting such a\ndrive is further model data.","\\section{Connecting the group to its generator}\nThe next chapter proves that the constructed group is the spectral evolution of this Hamiltonian:\n\\[\n\\mathcal U(t)=e^{-itH_m^{\\mathrm{phys}}}.\n\\]\nThe proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal generator has exactly that domain. The equality is therefore a conclusion about the complete Hilbert-space group, not an inference from agreement on finitely many modes.","\\clearpage\n\\section*{Completion and continuum evolution}\n\\section{The complete coefficient Hilbert space}\nLet\n\\[\n\\KK=\\ell^2(\\Z^3;\\C^4)\n=\\left\\{a=(a_n):\\sum_{n\\in\\Z^3}\\norm{a_n}^2<\\infty\\right\\}.\n\\]\nIts inner product is $\\inner{a}{b}=\\sum_n\\inner{a_n}{b_n}$. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of amplitudes embeds by assigning zero to every absent mode.","For a finite set $S\\subset\\Z^3$, define the truncation $P_Sa$ by retaining $a_n$ when $n\\in S$ and assigning zero otherwise. Then\n\\[\n\\norm{a-P_Sa}^2=\\sum_{n\\notin S}\\norm{a_n}^2.\n\\]\nThe sum on the right can be made arbitrarily small because the full nonnegative series converges. Thus finite packets are dense in $\\KK$. The statement concerns the norm of the omitted amplitudes and does not depend on a particular ordering of the integer lattice.","\\begin{theorem}[Completion of a modewise isometry]\nSuppose that for every mode $n$, $V_n:\\C^4\\to\\C^4$ is a complex linear isometry. The coordinate rule $(Va)_n=V_na_n$ defines a complex linear isometry on $\\KK$. If each $V_n$ has an inverse isometry $W_n$, the induced $V$ and $W$ are inverse maps on $\\KK$.\n\\end{theorem}\n\\begin{proof}\nFor each $n$, $\\norm{V_na_n}^2=\\norm{a_n}^2$. Summability of the latter terms gives membership of $Va$ in $\\KK$, and summing the equalities gives $\\norm{Va}=\\norm{a}$. Linearity holds coordinatewise. The two inverse identities hold on each coordinate and therefore on the coefficient fields themselves.\n\\end{proof}","ewise isometry]\nSuppose that for every mode $n$, $V_n:\\C^4\\to\\C^4$ is a complex linear isometry. The coordinate rule $(Va)_n=V_na_n$ defines a complex linear isometry on $\\KK$. If each $V_n$ has an inverse isometry $W_n$, the induced $V$ and $W$ are inverse maps on $\\KK$.\n\\end{theorem}\n\\begin{proof}\nFor each $n$, $\\norm{V_na_n}^2=\\norm{a_n}^2$. Summability of the latter terms gives membership of $Va$ in $\\KK$, and summing the equalities gives $\\norm{Va}=\\norm{a}$. Linearity holds coordinatewise. The two inverse identities hold on each coordinate and therefore on the coefficient fields themselves.\n\\end{proof}","ometry]\nSuppose that for every mode $n$, $V_n:\\C^4\\to\\C^4$ is a complex linear isometry. The coordinate rule $(Va)_n=V_na_n$ defines a complex linear isometry on $\\KK$. If each $V_n$ has an inverse isometry $W_n$, the induced $V$ and $W$ are inverse maps on $\\KK$.\n\\end{theorem}\n\\begin{proof}\nFor each $n$, $\\norm{V_na_n}^2=\\norm{a_n}^2$. Summability of the latter terms gives membership of $Va$ in $\\KK$, and summing the equalities gives $\\norm{Va}=\\norm{a}$. Linearity holds coordinatewise. The two inverse identities hold on each coordinate and therefore on the coefficient fields themselves.\n\\end{proof}","$V_n:\\C^4\\to\\C^4$ is a complex linear isometry. The coordinate rule $(Va)_n=V_na_n$ defines a complex linear isometry on $\\KK$. If each $V_n$ has an inverse isometry $W_n$, the induced $V$ and $W$ are inverse maps on $\\KK$.\n\\end{theorem}\n\\begin{proof}\nFor each $n$, $\\norm{V_na_n}^2=\\norm{a_n}^2$. Summability of the latter terms gives membership of $Va$ in $\\KK$, and summing the equalities gives $\\norm{Va}=\\norm{a}$. Linearity holds coordinatewise. The two inverse identities hold on each coordinate and therefore on the coefficient fields themselves.\n\\end{proof}","Apply this construction to\n\\[\nV_n(t)=e^{-itH_n},\\qquad V_{h,n}(t)=e^{-itH_{h,n}}.\n\\]\nThe finite-dimensional norm preservation already proved supplies its hypothesis. Define\n\\[\n(U(t)a)_n=V_n(t)a_n,\\qquad (U_h(t)a)_n=V_{h,n}(t)a_n.\n\\]\nBoth maps are everywhere-defined unitary maps on $\\KK$. Their inverses are obtained at time $-t$. The modewise time-group laws imply\n\\[\nU(s+t)=U(s)U(t),\\qquad U_h(s+t)=U_h(s)U_h(t).\n\\]\nThis construction uses the bounded unitary map on each mode. It does not apply an unbounded Hamiltonian to an arbitrary coefficient field. In particular, the evolution is defined for more initial states than the Hamiltonian action constructed below.","\\subsection{The invariant finite packets}\nBoth families commute with every finite truncation:\n\\[\nU(t)P_S=P_SU(t),\\qquad U_h(t)P_S=P_SU_h(t).\n\\]\nThe equality follows directly from the diagonal mode action. It says that the full evolution restricts to the finite-packet evolution already constructed. It also says that the missing tail evolves without feeding new amplitude into the retained mode set in this free model.","\\begin{lemma}[Two tails and a finite core]\nLet $V$ and $W$ be linear isometries on a normed space. For any $a$ and any approximation $b$,\n\\[\n\\norm{Va-Wa}\\le 2\\norm{a-b}+\\norm{Vb-Wb}.\n\\]\n\\end{lemma}\n\\begin{proof}\nWrite the residual as $V(a-b)+(Vb-Wb)+W(b-a)$. The triangle inequality bounds its norm by the sum of the three norms. Each outer term has norm $\\norm{a-b}$ because its operator is an isometry.\n\\end{proof}","Take $V=U_h(t)$, $W=U(t)$, and $b=P_Sa$. For a fixed finite $S$, the compact-time packet estimate gives\n\\[\n\\norm{U_h(t)P_Sa-U(t)P_Sa}\\le K_{S,a,T}h^2\n\\]\nfor all $|t|\\le T$ and sufficiently small positive $h$. The constant is allowed to depend on the support, the amplitudes, and the time window. Combining the estimates gives\n\\[\n\\norm{U_h(t)a-U(t)a}\\le\n2\\norm{a-P_Sa}+K_{S,a,T}h^2.\n\\]","\\begin{theorem}[Compact-time strong convergence]\nFor every $a\\in\\KK$, every $T\\ge0$, and every $\\varepsilon>0$, there exists $\\delta>0$ such that\n\\[\n0<h<\\delta,\\quad |t|\\le T\n\\quad\\Longrightarrow\\quad\n\\norm{U_h(t)a-U(t)a}<\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nChoose a finite $S$ such that $\\norm{a-P_Sa}<\\varepsilon/4$. The finite-packet theorem gives a sufficiently small positive $\\delta$ for which the core residual is less than $\\varepsilon/2$ throughout the time interval. The two-tail inequality then makes the full residual less than $\\varepsilon$. The choice of $S$ precedes the choice of $h$ and works for every time in the same interval.\n\\end{proof}","ergence]\nFor every $a\\in\\KK$, every $T\\ge0$, and every $\\varepsilon>0$, there exists $\\delta>0$ such that\n\\[\n0<h<\\delta,\\quad |t|\\le T\n\\quad\\Longrightarrow\\quad\n\\norm{U_h(t)a-U(t)a}<\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nChoose a finite $S$ such that $\\norm{a-P_Sa}<\\varepsilon/4$. The finite-packet theorem gives a sufficiently small positive $\\delta$ for which the core residual is less than $\\varepsilon/2$ throughout the time interval. The two-tail inequality then makes the full residual less than $\\varepsilon$. The choice of $S$ precedes the choice of $h$ and works for every time in the same interval.\n\\end{proof}","0<h<\\delta,\\quad |t|\\le T\n\\quad\\Longrightarrow\\quad\n\\norm{U_h(t)a-U(t)a}<\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nChoose a finite $S$ such that $\\norm{a-P_Sa}<\\varepsilon/4$. The finite-packet theorem gives a sufficiently small positive $\\delta$ for which the core residual is less than $\\varepsilon/2$ throughout the time interval. The two-tail inequality then makes the full residual less than $\\varepsilon$. The choice of $S$ precedes the choice of $h$ and works for every time in the same interval.\n\\end{proof}","ad\\Longrightarrow\\quad\n\\norm{U_h(t)a-U(t)a}<\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nChoose a finite $S$ such that $\\norm{a-P_Sa}<\\varepsilon/4$. The finite-packet theorem gives a sufficiently small positive $\\delta$ for which the core residual is less than $\\varepsilon/2$ throughout the time interval. The two-tail inequality then makes the full residual less than $\\varepsilon$. The choice of $S$ precedes the choice of $h$ and works for every time in the same interval.\n\\end{proof}","m{U_h(t)a-U(t)a}<\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nChoose a finite $S$ such that $\\norm{a-P_Sa}<\\varepsilon/4$. The finite-packet theorem gives a sufficiently small positive $\\delta$ for which the core residual is less than $\\varepsilon/2$ throughout the time interval. The two-tail inequality then makes the full residual less than $\\varepsilon$. The choice of $S$ precedes the choice of $h$ and works for every time in the same interval.\n\\end{proof}","The quantifiers are important. One initial field and one compact time interval are fixed before the mesh scale is chosen. The theorem does not assert convergence in operator norm on the entire unit ball. It also does not assign a universal $O(h^2)$ rate to every $a\\in\\KK$. For arbitrary square-summable data, a small Fourier tail may require a large cutoff, and the finite-core constant may grow with that cutoff.","\\section{Fourier synthesis on the physical torus}\nThe scalar Fourier characters $e_n(x)=e^{2\\pi i n\\cdot x}$ are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is\n\\[\n\\F:\\HH\\longrightarrow\\KK,\n\\qquad \\HH=L^2(\\T^3;\\C^4).\n\\]\nThe four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation,\n\\[\n\\norm{\\F f}_{\\KK}=\\norm{f}_{\\HH},\\qquad\n\\F^{-1}\\F f=f,\\qquad \\F\\F^{-1}a=a.\n\\]\nThe inverse is understood as convergence in $L^2$. A general field need not have a pointwise-convergent Fourier series, and no such convergence is required here.","n(x)=e^{2\\pi i n\\cdot x}$ are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is\n\\[\n\\F:\\HH\\longrightarrow\\KK,\n\\qquad \\HH=L^2(\\T^3;\\C^4).\n\\]\nThe four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation,\n\\[\n\\norm{\\F f}_{\\KK}=\\norm{f}_{\\HH},\\qquad\n\\F^{-1}\\F f=f,\\qquad \\F\\F^{-1}a=a.\n\\]\nThe inverse is understood as convergence in $L^2$. A general field need not have a pointwise-convergent Fourier series, and no such convergence is required here.","ar Fourier coefficient map to each of its four coordinates. The resulting map is\n\\[\n\\F:\\HH\\longrightarrow\\KK,\n\\qquad \\HH=L^2(\\T^3;\\C^4).\n\\]\nThe four-spinor Parseval identity and surjectivity of synthesis produce a complex linear isometric equivalence. In ordinary notation,\n\\[\n\\norm{\\F f}_{\\KK}=\\norm{f}_{\\HH},\\qquad\n\\F^{-1}\\F f=f,\\qquad \\F\\F^{-1}a=a.\n\\]\nThe inverse is understood as convergence in $L^2$. A general field need not have a pointwise-convergent Fourier series, and no such convergence is required here.","For a finite-support coefficient field, the inverse is the familiar finite sum\n\\[\n\\F^{-1}a=\\sum_n e_na_n.\n\\]\nThe complete-space equivalence extends that reconstruction while retaining its exact norm. Its surjectivity is the step that rules out having built only a proper closed subspace of the physical field space.","\\begin{definition}[Physical propagators]\nDefine\n\\[\n\\mathcal U(t)=\\F^{-1}U(t)\\F,\n\\qquad\n\\mathcal U_h(t)=\\F^{-1}U_h(t)\\F.\n\\]\n\\end{definition}\nBoth preserve the physical $L^2$ norm and inner product. Their group laws follow by inserting $\\F\\F^{-1}=I$ between consecutive factors. Their residual satisfies the exact equality\n\\[\n\\norm{\\mathcal U_h(t)f-\\mathcal U(t)f}_{\\HH}\n=\\norm{U_h(t)\\F f-U(t)\\F f}_{\\KK}.\n\\]\nThe compact-time convergence theorem therefore transfers to every $f\\in\\HH$ without an additional constant.","t)\\F,\n\\qquad\n\\mathcal U_h(t)=\\F^{-1}U_h(t)\\F.\n\\]\n\\end{definition}\nBoth preserve the physical $L^2$ norm and inner product. Their group laws follow by inserting $\\F\\F^{-1}=I$ between consecutive factors. Their residual satisfies the exact equality\n\\[\n\\norm{\\mathcal U_h(t)f-\\mathcal U(t)f}_{\\HH}\n=\\norm{U_h(t)\\F f-U(t)\\F f}_{\\KK}.\n\\]\nThe compact-time convergence theorem therefore transfers to every $f\\in\\HH$ without an additional constant.","\\clearpage\n\\section*{The unbounded Hamiltonian}\n\\section{A domain defined by the operator output}\nThe continuum mode energy grows with $|n|$. A square-summable sequence $a_n$ need not remain square summable after multiplication by $H_n$. This is why the full Hamiltonian is represented as a partial linear map with a specified domain.","Define\n\\[\n\\Dom(\\widehat H_m)\n=\\left\\{a\\in\\KK:\\sum_n\\norm{H_na_n}^2<\\infty\\right\\},\n\\qquad\n(\\widehat H_ma)_n=H_na_n.\n\\]\nThe domain is maximal for this coefficient action: it contains precisely those coefficient fields for which the stated output belongs to the ambient Hilbert space. It is a vector subspace. For example, $\\norm{x+y}^2\\le2\\norm{x}^2+2\\norm{y}^2$ and linearity of each $H_n$ prove closure under addition. Scalar multiplication preserves summability as well.","Every finite-support field lies in this domain, since the output again has finite support. Density of finite-support fields therefore proves density of $\\Dom(\\widehat H_m)$ in $\\KK$. Density matters because it gives uniqueness of an adjoint value when such a value exists.","\\section{The adjoint domain and self-adjointness}\nModewise Hermiticity gives symmetry on the maximal domain. If $a,b$ belong to that domain, the two series in\n\\[\n\\inner{\\widehat H_ma}{b}\n=\\sum_n\\inner{H_na_n}{b_n}\n=\\sum_n\\inner{a_n}{H_nb_n}\n=\\inner{a}{\\widehat H_mb}\n\\]\nconverge absolutely by Cauchy--Schwarz. This establishes the inclusion $\\widehat H_m\\subseteq\\widehat H_m^*$, including the corresponding domain inclusion.","\\begin{theorem}[Self-adjoint maximal multiplier]\nThe densely defined operator $\\widehat H_m$ equals its Hilbert adjoint.\n\\end{theorem}\n\\begin{proof}\nLet $b\\in\\Dom(\\widehat H_m^*)$ and write $c=\\widehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $","\\begin{theorem}[Self-adjoint maximal multiplier]\nThe densely defined operator $\\widehat H_m$ equals its Hilbert adjoint.\n\\end{theorem}\n\\begin{proof}\nLet $b\\in\\Dom(\\widehat H_m^*)$ and write $c=\\widehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{pro","\\begin{theorem}[Self-adjoint maximal multiplier]\nThe densely defined operator $\\widehat H_m$ equals its Hilbert adjoint.\n\\end{theorem}\n\\begin{proof}\nLet $b\\in\\Dom(\\widehat H_m^*)$ and write $c=\\widehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","equals its Hilbert adjoint.\n\\end{theorem}\n\\begin{proof}\nLet $b\\in\\Dom(\\widehat H_m^*)$ and write $c=\\widehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","et $b\\in\\Dom(\\widehat H_m^*)$ and write $c=\\widehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","idehat H_m^*b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","b$. By the adjoint relation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","lation,\n\\[\n\\inner{\\widehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","ehat H_ma}{b}=\\inner{a}{c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","c}\n\\qquad(a\\in\\Dom(\\widehat H_m)).\n\\]\nFix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","Fix a mode $n$ and choose $a$ with the single nonzero coefficient $a_n=v$, where $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","e $v\\in\\C^4$ is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes\n\\[\n\\inner{H_nv}{b_n}=\\inner{v}{c_n}.\n\\]\nHermiticity gives $\\inner{v}{H_nb_n}=\\inner{v}{c_n}$ for every $v$. Taking $v=H_nb_n-c_n$ shows $c_n=H_nb_n$. Because $c\\in\\KK$, the sequence $(H_nb_n)_n$ is square summable. Thus $b\\in\\Dom(\\widehat H_m)$ and the two operator actions agree there. This proves $\\widehat H_m^*\\subseteq\\widehat H_m$. Symmetry supplies the reverse inclusion.\n\\end{proof}","Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If $a^{(r)}\\to a$ and $\\widehat H_ma^{(r)}\\to b$ in $\\KK$, then convergence holds at each coordinate. Since the finite matrix $H_n$ is continuous,\n\\[\nb_n=\\lim_rH_na_n^{(r)}=H_na_n.\n\\]\nThe output sequence equals $b\\in\\KK$, so $a$ belongs to the maximal domain and $\\widehat H_ma=b$. This direct argument is included as an ordinary explanatory deduction.","\\section{Finite packets approximate the graph}\nFor a domain vector $a$, truncation commutes with the diagonal multiplier:\n\\[\n\\widehat H_mP_Sa=P_S\\widehat H_ma.\n\\]\nBoth $a$ and $\\widehat H_ma$ are square summable. Given $\\varepsilon>0$, choose finite sets that make each of their tails small and take their union. The resulting one finite set $S$ satisfies\n\\[\n\\norm{a-P_Sa}<\\varepsilon,\n\\qquad\n\\norm{\\widehat H_ma-\\widehat H_mP_Sa}<\\varepsilon.\n\\]\nThis graph-approximation statement is represented in the formal source. The finite Fourier fields are dense for the graph norm\n\\[\n\\norm{a}_{\\mathrm{graph}}^2\n=\\norm{a}^2+\\norm{\\widehat H_ma}^2.\n\\]\nThe term \\emph{graph core} describes this mathematical conclusion. The source proves the simultaneous approximation theorem w","\\section{Finite packets approximate the graph}\nFor a domain vector $a$, truncation commutes with the diagonal multiplier:\n\\[\n\\widehat H_mP_Sa=P_S\\widehat H_ma.\n\\]\nBoth $a$ and $\\widehat H_ma$ are square summable. Given $\\varepsilon>0$, choose finite sets that make each of their tails small and take their union. The resulting one finite set $S$ satisfies\n\\[\n\\norm{a-P_Sa}<\\varepsilon,\n\\qquad\n\\norm{\\widehat H_ma-\\widehat H_mP_Sa}<\\varepsilon.\n\\]\nThis graph-approximation statement is represented in the formal source. The finite Fourier fields are dense for the graph norm\n\\[\n\\norm{a}_{\\mathrm{graph}}^2\n=\\norm{a}^2+\\norm{\\widehat H_ma}^2.\n\\]\nThe term \\emph{graph core} describes this mathematical conclusion. The source proves the simultaneous approximation theorem without introducing a separate \\path{IsCore} wrapper.","$\\widehat H_ma$ are square summable. Given $\\varepsilon>0$, choose finite sets that make each of their tails small and take their union. The resulting one finite set $S$ satisfies\n\\[\n\\norm{a-P_Sa}<\\varepsilon,\n\\qquad\n\\norm{\\widehat H_ma-\\widehat H_mP_Sa}<\\varepsilon.\n\\]\nThis graph-approximation statement is represented in the formal source. The finite Fourier fields are dense for the graph norm\n\\[\n\\norm{a}_{\\mathrm{graph}}^2\n=\\norm{a}^2+\\norm{\\widehat H_ma}^2.\n\\]\nThe term \\emph{graph core} describes this mathematical conclusion. The source proves the simultaneous approximation theorem without introducing a separate \\path{IsCore} wrapper.","\\section{The exact first-order Fourier domain}\nThe Clifford square identity sharpens the definition of the maximal domain. Since $H_n^*=H_n$ and $H_n^2=(m^2+|p(n)|^2)I_4$,\n\\[\n\\norm{H_nv}^2\n=\\inner{v}{H_n^*H_nv}\n=(m^2+|p(n)|^2)\\norm{v}^2.\n\\]\nFor the coefficient field this gives the exact equivalence\n\\[\na\\in\\Dom(\\widehat H_m)\n\\quad\\Longleftrightarrow\\quad\n\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2<\\infty.\n\\]\nThe mass term is already summable because $a\\in\\KK$. Membership is therefore equivalent to summability of $\\sum_n|p(n)|^2\\norm{a_n}^2$.","in}\nThe Clifford square identity sharpens the definition of the maximal domain. Since $H_n^*=H_n$ and $H_n^2=(m^2+|p(n)|^2)I_4$,\n\\[\n\\norm{H_nv}^2\n=\\inner{v}{H_n^*H_nv}\n=(m^2+|p(n)|^2)\\norm{v}^2.\n\\]\nFor the coefficient field this gives the exact equivalence\n\\[\na\\in\\Dom(\\widehat H_m)\n\\quad\\Longleftrightarrow\\quad\n\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2<\\infty.\n\\]\nThe mass term is already summable because $a\\in\\KK$. Membership is therefore equivalent to summability of $\\sum_n|p(n)|^2\\norm{a_n}^2$.","maximal domain. Since $H_n^*=H_n$ and $H_n^2=(m^2+|p(n)|^2)I_4$,\n\\[\n\\norm{H_nv}^2\n=\\inner{v}{H_n^*H_nv}\n=(m^2+|p(n)|^2)\\norm{v}^2.\n\\]\nFor the coefficient field this gives the exact equivalence\n\\[\na\\in\\Dom(\\widehat H_m)\n\\quad\\Longleftrightarrow\\quad\n\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2<\\infty.\n\\]\nThe mass term is already summable because $a\\in\\KK$. Membership is therefore equivalent to summability of $\\sum_n|p(n)|^2\\norm{a_n}^2$.","\\begin{theorem}[Weighted domain and mass independence]\nFor every real $m$,\n\\[\n\\Dom(\\widehat H_m)\n=\\left\\{a\\in\\KK:\n\\sum_n(1+|p(n)|^2)\\norm{a_n}^2<\\infty\\right\\}.\n\\]\nConsequently $\\Dom(\\widehat H_{m_1})=\\Dom(\\widehat H_{m_2})$ for every two real masses.\n\\end{theorem}\n\\begin{proof}\nThe weighted energy series is the sum of the nonnegative momentum series and $m^2\\sum_n\\norm{a_n}^2$. The second series is finite. Thus the energy series is finite exactly when the momentum series is finite. Adding the already finite unweighted series gives precisely the displayed inhomogeneous first-order condition. This condition contains no mass parameter.\n\\end{proof}","The inhomogeneous weight retains the zero mode even when $m=0$. There is no loss of the ambient square-summability requirement in the massless case. The phrase first-order Fourier domain refers to this exact condition. It agrees with the conventional Fourier definition of periodic $H^1$ regularity; this manuscript does not claim a new checked equivalence with a separate weak-derivative implementation.","\\subsection{A concrete domain boundary}\nAs an ordinary example, put nonzero coefficients only on the positive axis modes $n=(k,0,0)$, $k\\ge1$, and set $a_n=k^{-s}v$ for a fixed unit spinor $v$. All remaining coefficients are zero. Then\n\\[\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n\\]\nwhich is finite exactly when $s>1/2$. The additional momentum series is\n\\[\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n\\]\nwhich is finite exactly when $s>3/2$. Thus the choice $s=1$ gives an $L^2$ field outside the Hamiltonian domain, while $s=2$ gives one inside it. These thresholds use axis-supported data. A radially populated three-dimensional family has a different counting factor.","rete domain boundary}\nAs an ordinary example, put nonzero coefficients only on the positive axis modes $n=(k,0,0)$, $k\\ge1$, and set $a_n=k^{-s}v$ for a fixed unit spinor $v$. All remaining coefficients are zero. Then\n\\[\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n\\]\nwhich is finite exactly when $s>1/2$. The additional momentum series is\n\\[\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n\\]\nwhich is finite exactly when $s>3/2$. Thus the choice $s=1$ gives an $L^2$ field outside the Hamiltonian domain, while $s=2$ gives one inside it. These thresholds use axis-supported data. A radially populated three-dimensional family has a different counting factor.","an ordinary example, put nonzero coefficients only on the positive axis modes $n=(k,0,0)$, $k\\ge1$, and set $a_n=k^{-s}v$ for a fixed unit spinor $v$. All remaining coefficients are zero. Then\n\\[\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n\\]\nwhich is finite exactly when $s>1/2$. The additional momentum series is\n\\[\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n\\]\nwhich is finite exactly when $s>3/2$. Thus the choice $s=1$ gives an $L^2$ field outside the Hamiltonian domain, while $s=2$ gives one inside it. These thresholds use axis-supported data. A radially populated three-dimensional family has a different counting factor.","mple, put nonzero coefficients only on the positive axis modes $n=(k,0,0)$, $k\\ge1$, and set $a_n=k^{-s}v$ for a fixed unit spinor $v$. All remaining coefficients are zero. Then\n\\[\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n\\]\nwhich is finite exactly when $s>1/2$. The additional momentum series is\n\\[\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n\\]\nwhich is finite exactly when $s>3/2$. Thus the choice $s=1$ gives an $L^2$ field outside the Hamiltonian domain, while $s=2$ gives one inside it. These thresholds use axis-supported data. A radially populated three-dimensional family has a different counting factor.","e axis modes $n=(k,0,0)$, $k\\ge1$, and set $a_n=k^{-s}v$ for a fixed unit spinor $v$. All remaining coefficients are zero. Then\n\\[\n\\sum_n\\norm{a_n}^2=\\sum_{k\\ge1}k^{-2s},\n\\]\nwhich is finite exactly when $s>1/2$. The additional momentum series is\n\\[\n4\\pi^2\\sum_{k\\ge1}k^{2-2s},\n\\]\nwhich is finite exactly when $s>3/2$. Thus the choice $s=1$ gives an $L^2$ field outside the Hamiltonian domain, while $s=2$ gives one inside it. These thresholds use axis-supported data. A radially populated three-dimensional family has a different counting factor.","\\section{Exact operator energy}\nFor $a\\in\\Dom(\\widehat H_m)$, summing the modal norm identity gives\n\\[\n\\norm{\\widehat H_ma}^2\n=\\sum_n(m^2+|p(n)|^2)\\norm{a_n}^2.\n\\]\nThis is the squared norm of the Hamiltonian output. It is nonnegative. It must be distinguished from the expectation $\\inner{a}{\\widehat H_ma}$, which may have either sign. The finite-packet discussion of positive and negative energy branches already exhibits that distinction.","An immediate ordinary consequence is the explicit graph norm\n\\[\n\\norm{a}_{\\mathrm{graph}}^2\n=\\sum_n(1+m^2+|p(n)|^2)\\norm{a_n}^2.\n\\]\nThe graph norm is therefore equivalent to the norm defined by the first-order Fourier weight. Writing $w(n)=1+|p(n)|^2$, one has\n\\[\nw(n)\\le1+m^2+|p(n)|^2\\le(1+m^2)w(n).\n\\]\nThe domains coincide for all masses even though these graph norms and the operator actions depend on the mass.","\\section{The Hamiltonian on physical spinor fields}\nUse the checked Fourier equivalence to define\n\\[\n\\Dom(H_m^{\\mathrm{phys}})=\\F^{-1}\\Dom(\\widehat H_m),\n\\qquad\nH_m^{\\mathrm{phys}}f=\\F^{-1}\\widehat H_m\\F f.\n\\]\nThe second formula is used only for $f$ in the displayed domain. Its Fourier intertwining identity is\n\\[\n\\F(H_m^{\\mathrm{phys}}f)=\\widehat H_m(\\F f).\n\\]","\\begin{theorem}[Physical torus Hamiltonian]\nThe operator $H_m^{\\mathrm{phys}}$ is densely defined, symmetric, self-adjoint, and closed on $\\HH=L^2(\\T^3;\\C^4)$. Its domain consists exactly of fields satisfying\n\\[\n\\sum_{n\\in\\Z^3}(1+|2\\pi n|^2)\\norm{\\widehat f(n)}^2<\\infty,\n\\]\nand for every domain vector,\n\\[\n\\norm{H_m^{\\mathrm{phys}}f}_{\\HH}^2\n=\\sum_n(m^2+|2\\pi n|^2)\\norm{\\widehat f(n)}^2.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe domain characterization is the coefficient characterization applied to $\\F f$. Density transports through the isometric equivalence. For two domain vectors, transporting the inner product and using coefficient-space symmetry proves physical-space symmetry.",", symmetric, self-adjoint, and closed on $\\HH=L^2(\\T^3;\\C^4)$. Its domain consists exactly of fields satisfying\n\\[\n\\sum_{n\\in\\Z^3}(1+|2\\pi n|^2)\\norm{\\widehat f(n)}^2<\\infty,\n\\]\nand for every domain vector,\n\\[\n\\norm{H_m^{\\mathrm{phys}}f}_{\\HH}^2\n=\\sum_n(m^2+|2\\pi n|^2)\\norm{\\widehat f(n)}^2.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe domain characterization is the coefficient characterization applied to $\\F f$. Density transports through the isometric equivalence. For two domain vectors, transporting the inner product and using coefficient-space symmetry proves physical-space symmetry.","To prove the reverse adjoint-domain inclusion, let $g$ have physical adjoint value $z$. For every coefficient domain vector $a$, test the physical adjoint identity at $\\F^{-1}a$. Fourier isometry gives\n\\[\n\\inner{\\widehat H_ma}{\\F g}=\\inner{a}{\\F z}.\n\\]\nThus $\\F g$ belongs to the coefficient adjoint domain with adjoint value $\\F z$. Coefficient self-adjointness puts $\\F g$ in the maximal coefficient domain, hence puts $g$ in the physical domain. The actions agree. Symmetry supplies the opposite inclusion, giving equality with the adjoint. Closedness follows, and Fourier isometry transports the exact energy sum.\n\\end{proof}","djoint-domain inclusion, let $g$ have physical adjoint value $z$. For every coefficient domain vector $a$, test the physical adjoint identity at $\\F^{-1}a$. Fourier isometry gives\n\\[\n\\inner{\\widehat H_ma}{\\F g}=\\inner{a}{\\F z}.\n\\]\nThus $\\F g$ belongs to the coefficient adjoint domain with adjoint value $\\F z$. Coefficient self-adjointness puts $\\F g$ in the maximal coefficient domain, hence puts $g$ in the physical domain. The actions agree. Symmetry supplies the opposite inclusion, giving equality with the adjoint. Closedness follows, and Fourier isometry transports the exact energy sum.\n\\end{proof}","On finite Fourier polynomials, the operator agrees with the matrix differential expression $-i\\sum_j\\alpha_j\\partial_j+m\\beta$ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every $L^2$ representative. Equality with a separately constructed weak differential operator is a further interface statement if that alternate realization is introduced.","We use natural units \\(\\hbar=c=1\\). Space is the flat unit torus","\\[\n\\mathbb T^3=(\\mathbb R/\\mathbb Z)^3,\n\\]","equipped with normalized Haar measure \\(dx\\), so\n\\(\\int_{\\mathbb T^3}1\\,dx=1\\). Integration of a periodic representative\nmay equivalently be taken over \\([0,1)^3\\) with Lebesgue measure. A\nFourier mode is \\(n=(n_1,n_2,n_3)\\in\\mathbb Z^3\\), its momentum is\n\\(p(n)=2\\pi n\\), and its scalar plane wave is","\\[\ne_n(x)=e^{2\\pi i n\\cdot x}=e^{i p(n)\\cdot x}.\n\\]","Components of a spinor are numbered \\(0,1,2,3\\), in the exact order of\nthe source representation. For \\(v,w\\in\\mathbb C^4\\),","\\[\n\\langle v,w\\rangle=\\sum_{r=0}^{3}\\overline{v_r}w_r,\n\\qquad\n\\|v\\|_2^2=\\sum_{r=0}^{3}|v_r|^2.\n\\]","\\[\n\\langle f,g\\rangle_{L^2}\n=\\int_{\\mathbb T^3}\\langle f(x),g(x)\\rangle\\,dx\n\\]","on \\(L^2(\\mathbb T^3;\\mathbb C^4)\\). Fix a finite set\n\\(S\\subset\\mathbb Z^3\\). A packet with amplitudes \\(a_n\\in\\mathbb C^4\\)\nis","\\[\nf_a(x)=\\sum_{n\\in S}e_n(x)a_n.\n\\]","\\[\n\\mathcal K_S=\\bigoplus_{n\\in S}\\mathbb C^4,\n\\qquad\n\\|a\\|_{\\mathcal K_S}^2=\\sum_{n\\in S}\\|a_n\\|_2^2.\n\\]","The packet space \\(\\mathcal F_S=\\{f_a:a\\in\\mathcal K_S\\}\\) will be\nidentified isometrically with \\(\\mathcal K_S\\). Its elements are smooth\nperiodic functions, so the differential and centered-translation\ncalculations below raise no regularity question. Empty support is\nallowed: both spaces then contain only zero and all packet identities\nreduce to \\(0=0\\).","The parameters \\(m,h,t\\) are arbitrary real numbers. The mathematics\ndoes not require \\(m\\ge0\\); the usual physical rest-mass interpretation\nuses that additional convention. The shell value \\(E\\) is always a\nnonnegative square root, even when \\(m<0\\). Mesh limits in the checked\nstatements are \\(h\\to0^+\\). Some explicit estimates proved here hold for\nevery real \\(h\\), since the centered symbol is even in \\(h\\).","eters \\(m,h,t\\) are arbitrary real numbers. The mathematics\ndoes not require \\(m\\ge0\\); the usual physical rest-mass interpretation\nuses that additional convention. The shell value \\(E\\) is always a\nnonnegative square root, even when \\(m<0\\). Mesh limits in the checked\nstatements are \\(h\\to0^+\\). Some explicit estimates proved here hold for\nevery real \\(h\\), since the centered symbol is even in \\(h\\).","\\[\n\\alpha_x=\n\\begin{pmatrix}\n0&0&0&1\\\\\n0&0&1&0\\\\\n0&1&0&0\\\\\n1&0&0&0\n\\end{pmatrix},\n\\qquad\n\\alpha_y=\n\\begin{pmatrix}\n0&0&0&-i\\\\\n0&0&i&0\\\\\n0&-i&0&0\\\\\ni&0&0&0\n\\end{pmatrix},\n\\]","\\[\n\\alpha_z=\n\\begin{pmatrix}\n0&0&1&0\\\\\n0&0&0&-1\\\\\n1&0&0&0\\\\\n0&-1&0&0\n\\end{pmatrix},\n\\qquad\n\\beta=\n\\begin{pmatrix}\n1&0&0&0\\\\\n0&1&0&0\\\\\n0&0&-1&0\\\\\n0&0&0&-1\n\\end{pmatrix}.\n\\]","In particular, \\(\\alpha_yv=(-iv_3,iv_2,-iv_1,iv_0)\\); this fixes a sign\nthat cannot be changed silently in a worked example. The matrices act on\ncolumn spinors. Grouping the first two and last two components gives\ntheir block form without any coordinate permutation:","\\[\n\\alpha_j=\n\\begin{pmatrix}0&\\sigma_j\\\\\\sigma_j&0\\end{pmatrix},\n\\qquad\n\\beta=\\begin{pmatrix}I_2&0\\\\0&-I_2\\end{pmatrix},\n\\]","\\[\n\\sigma_x=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\n\\quad\n\\sigma_y=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix},\n\\quad\n\\sigma_z=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}.\n\\]","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH01 (Hermitian Clifford relations).} The four\ndisplayed matrices are Hermitian involutions. Distinct spatial\ngenerators anticommute, and each spatial generator anticommutes with\n\\(\\beta\\):","\\[\n\\alpha_j^*=\\alpha_j,\\quad \\beta^*=\\beta,\\quad\n\\alpha_j^2=\\beta^2=I_4,\n\\]","\\[\n\\alpha_j\\alpha_k+\\alpha_k\\alpha_j=0\\quad(j\\ne k),\n\\qquad\n\\alpha_j\\beta+\\beta\\alpha_j=0.\n\\]","\\emph{Proof.} The real diagonal entries of \\(\\beta\\) are\n\\(1,1,-1,-1\\), so its adjoint and square have the stated forms. Each\n\\(\\sigma_j\\) is Hermitian and squares to \\(I_2\\), by direct\nmultiplication. Their products are","\\[\n\\sigma_x\\sigma_y=i\\sigma_z,\\qquad\n\\sigma_y\\sigma_z=i\\sigma_x,\\qquad\n\\sigma_z\\sigma_x=i\\sigma_y,\n\\]","\\[\n\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\n\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}\n=\\begin{pmatrix}i&0\\\\0&-i\\end{pmatrix}=i\\sigma_z.\n\\]","\\[\n\\alpha_j^*=\n\\begin{pmatrix}0&\\sigma_j^*\\\\\\sigma_j^*&0\\end{pmatrix}\n=\\alpha_j,\\qquad\n\\alpha_j\\alpha_k=\n\\begin{pmatrix}\\sigma_j\\sigma_k&0\\\\0&\\sigma_j\\sigma_k\\end{pmatrix}.\n\\]","Thus \\(\\alpha_j^2=I_4\\) and the spatial anticommutators vanish. Finally,","\\[\n\\alpha_j\\beta=\n\\begin{pmatrix}0&-\\sigma_j\\\\\\sigma_j&0\\end{pmatrix},\n\\qquad\n\\beta\\alpha_j=\n\\begin{pmatrix}0&\\sigma_j\\\\-\\sigma_j&0\\end{pmatrix},\n\\]","whose sum is zero. Every identity is an identity of the exact source\nmatrices. \\(\\square\\)","For \\(p\\in\\mathbb R^3\\), define","\\[\nH(m,p)=m\\beta+p_x\\alpha_x+p_y\\alpha_y+p_z\\alpha_z.\n\\]","\\[\nH(m,p)=\n\\begin{pmatrix}\nm&0&p_z&p_x-ip_y\\\\\n0&m&p_x+ip_y&-p_z\\\\\np_z&p_x-ip_y&-m&0\\\\\np_x+ip_y&-p_z&0&-m\n\\end{pmatrix}.\n\\]","On mode \\(n\\), the continuum Hamiltonian is \\(H_n=H(m,p(n))\\). It is the\nsymbol of the differential expression","\\[\n-i\\sum_{j=x,y,z}\\alpha_j\\partial_j+m\\beta\n\\]","on the finite packet space: \\(\\partial_je_n=ip_j(n)e_n\\), so applying\nthe expression to \\(e_na_n\\) gives \\(e_nH_na_n\\). For real spacing\n\\(h\\), set","\\[\nq_j(h,p)=\n\\begin{cases}\n\\sin(hp_j)/h,&h\\ne0,\\\\\np_j,&h=0.\n\\end{cases}\n\\]","Write \\(q_h(n)=q(h,p(n))\\) and \\(H_{h,n}=H(m,q_h(n))\\). The function\n\\(q_j\\) is even in \\(h\\), and its value at zero is justified by\n\\(\\sin z/z\\to1\\). For \\(h\\ne0\\), a centered translation difference on a\nperiodic smooth function is","\\[\n(D_{h,j}f)(x)=\\frac{f(x+he_j)-f(x-he_j)}{2h}.\n\\]","Here \\(e_j\\) is the j-th spatial unit vector, distinct from the Fourier\ncharacters \\(e_n\\) and spinor vectors. Its action on a plane wave is\nexactly","\\[\nD_{h,j}e_n\n=e_n\\frac{e^{ihp_j(n)}-e^{-ihp_j(n)}}{2h}\n=i\\frac{\\sin(hp_j(n))}{h}e_n.\n\\]","The centered differential expression \\(-i\\sum_j\\alpha_jD_{h,j}+m\\beta\\)\ntherefore has symbol \\(H_{h,n}\\). Its signs follow from \\(-i\\cdot i=1\\).\nAt \\(h=0\\), the symbol equals \\(H_n\\) by definition. These formulas take\nplace on the continuum torus and use no sampled spatial data.","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH02 (Hermiticity and the mass shell).} For every\nreal \\(m\\) and \\(p\\in\\mathbb R^3\\),","\\[\nH(m,p)^*=H(m,p),\\qquad\nH(m,p)^2=(m^2+|p|^2)I_4.\n\\]","In particular, \\(H_n,H_{h,n}\\) are Hermitian for all real \\(m,h\\), and","\\[\nH_n^2=E_n^2I_4,\\quad E_n=\\sqrt{m^2+|p(n)|^2},\n\\]","\\[\nH_{h,n}^2=E_{h,n}^2I_4,\\quad\nE_{h,n}=\\sqrt{m^2+|q_h(n)|^2}.\n\\]","\\emph{Proof.} Adjoint is additive, and a real multiple of a Hermitian\nmatrix is Hermitian. PH01 therefore proves the first identity. Expanding\nthe square gives the square terms \\(m^2\\beta^2+\\sum_jp_j^2\\alpha_j^2\\),\nthe terms \\(mp_j(\\beta\\alpha_j+\\alpha_j\\beta)\\), and the terms\n\\(p_jp_k(\\alpha_j\\alpha_k+\\alpha_k\\alpha_j)\\) for \\(j<k\\). The\nanticommutators vanish and all four squares are \\(I_4\\), giving\n\\((m^2+\\sum_jp_j^2)I_4\\). Substitution proves both mode statements.\n\\(\\square\\)","roof.} Adjoint is additive, and a real multiple of a Hermitian\nmatrix is Hermitian. PH01 therefore proves the first identity. Expanding\nthe square gives the square terms \\(m^2\\beta^2+\\sum_jp_j^2\\alpha_j^2\\),\nthe terms \\(mp_j(\\beta\\alpha_j+\\alpha_j\\beta)\\), and the terms\n\\(p_jp_k(\\alpha_j\\alpha_k+\\alpha_k\\alpha_j)\\) for \\(j<k\\). The\nanticommutators vanish and all four squares are \\(I_4\\), giving\n\\((m^2+\\sum_jp_j^2)I_4\\). Substitution proves both mode statements.\n\\(\\square\\)","x is Hermitian. PH01 therefore proves the first identity. Expanding\nthe square gives the square terms \\(m^2\\beta^2+\\sum_jp_j^2\\alpha_j^2\\),\nthe terms \\(mp_j(\\beta\\alpha_j+\\alpha_j\\beta)\\), and the terms\n\\(p_jp_k(\\alpha_j\\alpha_k+\\alpha_k\\alpha_j)\\) for \\(j<k\\). The\nanticommutators vanish and all four squares are \\(I_4\\), giving\n\\((m^2+\\sum_jp_j^2)I_4\\). Substitution proves both mode statements.\n\\(\\square\\)","Hermiticity here is a statement about matrices acting on all of\n\\(\\mathbb C^4\\). Every linear map on this finite space is bounded and\neverywhere defined. This is not a construction of the adjoint or domain\nof an unbounded differential operator on the full field space.","Let \\(H=H(m,p)\\) and \\(E=\\sqrt{m^2+|p|^2}\\). Define the exact propagator","\\[\nU_H(t)=e^{-itH}=\\sum_{k=0}^{\\infty}\\frac{(-itH)^k}{k!}.\n\\]","The series converges absolutely in operator norm, being dominated by\n\\(\\sum_k(|t|\\|H\\|)^k/k!\\). This is exact continuous-time evolution, with\nno time step or series truncation.","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH03 (spectral splitting and closed propagator).} If\n\\(E>0\\), define","\\[\nP_+=\\frac12\\left(I_4+\\frac HE\\right),\\qquad\nP_-=\\frac12\\left(I_4-\\frac HE\\right).\n\\]","Then \\(P_\\pm^*=P_\\pm\\), \\(P_\\pm^2=P_\\pm\\), \\(P_+P_-=P_-P_+=0\\),\n\\(P_++P_-=I_4\\), and \\(HP_\\pm=\\pm EP_\\pm\\). Both projectors have rank\ntwo, and","\\[\nU_H(t)=e^{-itE}P_++e^{itE}P_-\n=\\cos(tE)I_4-i\\frac{\\sin(tE)}{E}H.\n\\]","If \\(E=0\\), then \\(H=0\\) and \\(U_H(t)=I_4\\). Equivalently, the closed\nexpression is valid with the continuous convention \\(\\sin(tE)/E=t\\) at\n\\(E=0\\).","\\emph{Proof.} For \\(E>0\\), the Hermitian matrix \\(K=H/E\\) satisfies\n\\(K^2=I_4\\). Thus","\\[\nP_\\pm^2=\\tfrac14(I_4\\pm2K+K^2)=\\tfrac12(I_4\\pm K)=P_\\pm,\n\\]","and \\(P_+P_-=\\tfrac14(I_4-K^2)=0\\); the reversed product is the same.\nTheir adjoints equal themselves and their sum is \\(I_4\\). Also\n\\(K(I_4\\pm K)=K\\pm I_4=\\pm(I_4\\pm K)\\), proving \\(HP_\\pm=\\pm EP_\\pm\\).\nEach displayed Clifford matrix has trace zero. Hence\n\\(\\operatorname{tr}H=0\\) and \\(\\operatorname{tr}P_\\pm=2\\). An idempotent\nsplits the space into its range and kernel, acting as identity and zero\nrespectively. Its trace is therefore its rank, proving rank two.","The identity \\(I_4=P_++P_-\\) and the two eigenvalue identities give, by\ninduction, \\(H^k=E^kP_++(-E)^kP_-\\) for every \\(k\\ge0\\). Substitute into\nthe absolutely convergent series to obtain the spectral expression.\nExpanding its projectors gives the cosine and sine expression.","When \\(E=0\\), the sum \\(m^2+|p|^2\\) of nonnegative squares is zero.\nTherefore \\(m=0,p=0\\), and \\(H=0\\). Its exponential is \\(I_4\\). The\nquotient \\(\\sin(tE)/E\\) tends to \\(t\\) at zero; multiplying its filled\nvalue by \\(H=0\\) gives the same identity. No division by zero is used.\n\\(\\square\\)","The proof applies unchanged to \\(H_{h,n}\\), with energy \\(E_{h,n}\\). A\npositive-energy continuum mode can have \\(E_{h,n}=0\\) at a nonzero\nspacing when \\(m=0\\) and all centered components vanish. That mode uses\nthe zero-energy branch. Positivity of the continuum energy does not\nlicense division by the centered energy at every spacing.","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH04 (group law, unitarity, and stability).} For\nreal \\(s,t\\),","\\[\nU_H(0)=I_4,\\qquad U_H(s+t)=U_H(s)U_H(t),\n\\]","\\[\nU_H(-t)=U_H(t)^{-1}=U_H(t)^*,\n\\qquad U_H(t)^*U_H(t)=U_H(t)U_H(t)^*=I_4.\n\\]","\\[\n\\langle U_H(t)v,U_H(t)w\\rangle=\\langle v,w\\rangle,\\qquad\n\\|U_H(t)v\\|_2=\\|v\\|_2\n\\]","for all spinors. These statements hold for both mode Hamiltonians at\nevery real \\(m,h,t\\).","\\emph{Proof.} When \\(E>0\\), the spectral form and orthogonality of\n\\(P_\\pm\\) give","\\[\nU_H(s)U_H(t)\n=e^{-i(s+t)E}P_++e^{i(s+t)E}P_-=U_H(s+t).\n\\]","At zero time the sum is \\(I_4\\). Hermiticity of the projectors and\nreality of \\(tE\\) give \\(U_H(t)^*=e^{itE}P_++e^{-itE}P_-=U_H(-t)\\).\nSetting \\(s=-t\\) proves the inverse identities. When \\(E=0\\), every\nassertion follows from \\(U_H(t)=I_4\\).","For any matrix \\(A\\), expansion and interchange of two finite sums gives\n\\(\\langle Av,w\\rangle=\\langle v,A^*w\\rangle\\). Thus","\\[\n\\langle U_H(t)v,U_H(t)w\\rangle\n=\\langle v,U_H(t)^*U_H(t)w\\rangle=\\langle v,w\\rangle.\n\\]","Take \\(w=v\\) and then nonnegative square roots to obtain norm\npreservation. PH02 supplies the mode cases. \\(\\square\\)","An equivalent bounded-operator argument is represented in the formal source:\n\\(-itH\\) is skew-adjoint, and its exponential is unitary. Our spectral\nproof supplies additional explicit projectors and formulas.","Let \\(J_n:\\mathbb C^4\\to L^2(\\mathbb T^3;\\mathbb C^4)\\) be\n\\(J_nv=e_nv\\).","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH05 (mode isometry and finite Parseval identity).}\nFor modes \\(n,k\\) and spinors \\(v,w\\),","\\[\n\\langle J_nv,J_kw\\rangle_{L^2}\n=\\begin{cases}\\langle v,w\\rangle,&n=k,\\\\0,&n\\ne k.\\end{cases}\n\\]","In particular, \\(J_n\\) is a linear isometry and","\\[\n\\left\\|\\sum_{n\\in S}J_na_n\\right\\|_{L^2}^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2.\n\\]","\\[\n\\langle J_nv,J_kw\\rangle_{L^2}\n=\\langle v,w\\rangle\\int_{\\mathbb T^3}e^{2\\pi i(k-n)\\cdot x}\\,dx.\n\\]","The integral is the product of three one-dimensional integrals. For\ninteger \\(r\\), the integral of \\(e^{2\\pi irx}\\) over \\([0,1)\\) is \\(1\\)\nwhen \\(r=0\\); for \\(r\\ne0\\), it is \\((e^{2\\pi ir}-1)/(2\\pi ir)=0\\). The\nproduct is therefore \\(1\\) exactly when \\(n=k\\). Linearity is immediate,\nand setting \\(k=n,w=v\\) proves isometry. Expansion of the squared norm\nof the finite sum gives a double sum of pairings. The off-diagonal\npairings vanish and the diagonal pairings are \\(\\|a_n\\|_2^2\\).\n\\(\\square\\)","eger \\(r\\), the integral of \\(e^{2\\pi irx}\\) over \\([0,1)\\) is \\(1\\)\nwhen \\(r=0\\); for \\(r\\ne0\\), it is \\((e^{2\\pi ir}-1)/(2\\pi ir)=0\\). The\nproduct is therefore \\(1\\) exactly when \\(n=k\\). Linearity is immediate,\nand setting \\(k=n,w=v\\) proves isometry. Expansion of the squared norm\nof the finite sum gives a double sum of pairings. The off-diagonal\npairings vanish and the diagonal pairings are \\(\\|a_n\\|_2^2\\).\n\\(\\square\\)","ntegral of \\(e^{2\\pi irx}\\) over \\([0,1)\\) is \\(1\\)\nwhen \\(r=0\\); for \\(r\\ne0\\), it is \\((e^{2\\pi ir}-1)/(2\\pi ir)=0\\). The\nproduct is therefore \\(1\\) exactly when \\(n=k\\). Linearity is immediate,\nand setting \\(k=n,w=v\\) proves isometry. Expansion of the squared norm\nof the finite sum gives a double sum of pairings. The off-diagonal\npairings vanish and the diagonal pairings are \\(\\|a_n\\|_2^2\\).\n\\(\\square\\)","\\[\nf(t,x)=\\sum_{n\\in S}e_n(x)U_{H_n}(t)a_n,\\qquad\nf_h(t,x)=\\sum_{n\\in S}e_n(x)U_{H_{h,n}}(t)a_n.\n\\]","Both remain in \\(\\mathcal F_S\\). Their operators \\(U_S(t),U_{h,S}(t)\\)\nhave the finite block diagonal coefficient actions with blocks\n\\(U_{H_n}(t),U_{H_{h,n}}(t)\\).","\\needspace{4\\baselineskip}\n\\textbf{Theorem PH06 (exact finite-packet norm conservation).} For every\nreal \\(m,h,t\\), finite \\(S\\), and coefficient family \\(a\\),","\\[\n\\|f(t)\\|_{L^2}^2=\\|f_h(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2=\\|f_a\\|_{L^2}^2.\n\\]","Both packet evolutions preserve the \\(L^2\\) inner product between two\npackets in \\(\\mathcal F_S\\), and each forms a one-parameter unitary\ngroup there.","\\[\n\\|f_h(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|U_{H_{h,n}}(t)a_n\\|_2^2\n=\\sum_{n\\in S}\\|a_n\\|_2^2.\n\\]","The continuum argument is identical. For coefficient families \\(a,b\\),\nthe same two propositions give","\\[\n\\langle f_{h,a}(t),f_{h,b}(t)\\rangle_{L^2}\n=\\sum_{n\\in S}\n\\langle U_{H_{h,n}}(t)a_n,U_{H_{h,n}}(t)b_n\\rangle\n=\\sum_{n\\in S}\\langle a_n,b_n\\rangle.\n\\]","Composition of coefficient blocks proves the packet group law, and\ninverse time proves surjectivity. This establishes unitarity on\n\\(\\mathcal F_S\\) using only finite sums. \\(\\square\\)","The quantity conserved is squared Hilbert norm, or total probability for\na normalized state. The parameter \\(m\\), the positive shell value\n\\(E_n\\), and Hamiltonian expectation are different quantities. Calling\nsquared norm a packet mass does not identify it with the rest mass or\nphysical Hamiltonian energy.","\\[\n\\Delta_{h,n}:=H_{h,n}-H_n\n=\\sum_{j=x,y,z}\\delta_j(h,n)\\alpha_j,\n\\qquad\n\\delta_j(h,n)=q_j(h,p(n))-p_j(n).\n\\]","\\[\nC(n)=\\frac16\\left(\\sum_{j=x,y,z}|p_j(n)|^6\\right)^{1/2}.\n\\]","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH07 (quadratic generator consistency).} For every\nreal \\(h\\) and fixed mode \\(n\\),","\\[\n\\|\\Delta_{h,n}\\|=|\\delta(h,n)|\\le h^2C(n).\n\\]","\\[\n\\left\\|\\sum_{n\\in S}e_n\\Delta_{h,n}a_n\\right\\|_{L^2}^2\n=\\sum_{n\\in S}|\\delta(h,n)|^2\\|a_n\\|_2^2,\n\\]","\\[\n\\left\\|\\sum_{n\\in S}e_n\\Delta_{h,n}a_n\\right\\|_{L^2}\n\\le h^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}.\n\\]","\\emph{Proof.} First prove the scalar estimate without a sign\nrestriction on \\(h\\). The fundamental theorem of calculus gives\n\\(|\\sin u|\\le|u|\\), since \\(\\sin u=\\int_0^u\\cos v\\,dv\\) and\n\\(|\\cos v|\\le1\\). Therefore","\\[\n|1-\\cos u|=2\\sin^2(u/2)\\le u^2/2.\n\\]","For \\(z\\ge0\\), integrate this bound in\n\\(\\sin z-z=\\int_0^z(\\cos u-1)\\,du\\) to obtain \\(|\\sin z-z|\\le z^3/6\\).\nOddness of the residual proves \\(|\\sin z-z|\\le|z|^3/6\\) for negative\n\\(z\\). For \\(h\\ne0\\), substitute \\(z=hp_j(n)\\) and divide by \\(|h|\\):","\\[\n|\\delta_j(h,n)|\\le \\frac{h^2|p_j(n)|^3}{6}.\n\\]","For \\(h=0\\), the residual is zero by definition and the bound still\nholds. Squaring and adding proves \\(|\\delta(h,n)|\\le h^2C(n)\\).","The Hermitian matrix \\(\\Delta_{h,n}\\) satisfies\n\\(\\Delta_{h,n}^2=|\\delta(h,n)|^2I_4\\) by PH01. Hence","\\[\n\\|\\Delta_{h,n}v\\|_2^2\n=\\langle v,\\Delta_{h,n}^*\\Delta_{h,n}v\\rangle\n=|\\delta(h,n)|^2\\|v\\|_2^2.\n\\]","Its induced operator norm is exactly \\(|\\delta(h,n)|\\): the upper bound\nholds on every vector and equality holds on every unit vector. Apply\nPH05 to the residual coefficients for the packet equality, substitute\nthe scalar bounds, and take nonnegative square roots. \\(\\square\\)","The checked generator results state corresponding \\(O(h^2)\\) estimates\nas \\(h\\to0^+\\). The explicit constant and all-real inequality above are\ntraditional deductions. Precisely, \\(r(h)=O(h^2)\\) means that some\n\\(M,\\varepsilon>0\\) satisfy \\(\\|r(h)\\|\\le Mh^2\\) for\n\\(0<h<\\varepsilon\\). Fixed support and amplitudes permit finite\nsummation of such bounds; no estimate uniform over all integer modes\nfollows.","The reverse triangle inequality in \\(\\mathbb R^4\\), applied to\n\\((m,q_h(n))\\) and \\((m,p(n))\\), gives","\\[\n|E_{h,n}-E_n|\\le|q_h(n)-p(n)|\\le h^2C(n).\n\\]","Moreover, \\(|\\sin(hp_j)/h|\\le|p_j|\\) for \\(h\\ne0\\), and the filled value\nis equal at zero. Thus \\(0\\le E_{h,n}\\le E_n\\). For a fixed mode,\nTaylor's formula gives","\\[\nq_j(h,p(n))=p_j(n)-\\frac{h^2p_j(n)^3}{6}+O(h^4),\n\\]","\\[\nE_{h,n}^2=E_n^2-\\frac{h^2}{3}\\sum_jp_j(n)^4+O(h^4).\n\\]","For clarity, Taylor's theorem through degree four has zero even\ncoefficients for sine, and its fifth-derivative remainder is at most\n\\(|hp_j|^5/120\\). Division by \\(|h|\\) gives \\(O(h^4)\\); squaring the\nthree expansions and summing preserves that order for a fixed mode. If\n\\(E_n>0\\), the square root is smooth near \\(E_n^2\\), so","\\[\nE_{h,n}=E_n-\\frac{h^2}{6E_n}\\sum_jp_j(n)^4+O(h^4).\n\\]","If \\(E_n=0\\), then \\(m=0,n=0\\) and \\(E_{h,n}=0\\) exactly for every\n\\(h\\). The expansion containing \\(1/E_n\\) is not used.","\\[\n|e^{\\pm itE_{h,n}}-e^{\\pm itE_n}|\n\\le |t|\\,|E_{h,n}-E_n|,\n\\]","by integrating the derivative of \\(s\\mapsto e^{\\pm its}\\) between the\ntwo real energies. Phase comparison alone does not compare full spinor\nevolution: the eigenspaces also depend on the kinetic vector. The\noperator argument below controls both effects.","\\needspace{10\\baselineskip}\n\\atlasfigure{04_dispersion}{The continuum and centered shell branches along one momentum axis, with $m=1$, $h=1/12$. Marked points are integer torus modes; intermediate points use the real-symbol extension.}","\\needspace{4\\baselineskip}\n\\textbf{Theorem PH08 (fixed-mode and fixed-packet convergence).} For\nfixed real \\(m,t\\) and fixed \\(n\\),","\\[\nU_{H_{h,n}}(t)-U_{H_n}(t)=O(h^2)\\quad(h\\to0^+).\n\\]","For fixed finite \\(S\\) and amplitudes \\(a\\),","\\[\nf_h(t)-f(t)=O(h^2)\n\\quad\\hbox{in }L^2(\\mathbb T^3;\\mathbb C^4),\n\\]","and in particular \\(\\|f_h(t)-f(t)\\|_{L^2}\\to0\\). The exact residual\nidentity is","\\[\n\\|f_h(t)-f(t)\\|_{L^2}^2\n=\\sum_{n\\in S}\\|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\\|_2^2.\n\\]","\\emph{Proof.} Give a fixed-time proof independent of the later\ncompact-time extension. For arbitrary matrices \\(X,Y\\), telescoping\nproducts give","\\[\nX^k-Y^k=\\sum_{r=0}^{k-1}X^{k-1-r}(X-Y)Y^r\n\\quad(k\\ge1).\n\\]","No commutation is assumed. Expanding the summands gives successive\ndifferences \\(X^{k-r}Y^r-X^{k-1-r}Y^{r+1}\\), which cancel. If\n\\(\\|X\\|,\\|Y\\|\\le R\\), then","\\[\n\\|X^k-Y^k\\|\\le kR^{k-1}\\|X-Y\\|.\n\\]","\\[\n\\|e^X-e^Y\\|\n\\le\\|X-Y\\|\\sum_{k=1}^{\\infty}\\frac{kR^{k-1}}{k!}\n=e^R\\|X-Y\\|.\n\\]","PH02 gives \\(\\|H_n\\|=E_n,\\|H_{h,n}\\|=E_{h,n}\\), by the norm-squared\nargument of PH07. Since \\(E_{h,n}\\le E_n\\), take \\(X=-itH_{h,n}\\),\n\\(Y=-itH_n\\), and \\(R=|t|E_n\\). The result is","\\[\n\\|U_{H_{h,n}}(t)-U_{H_n}(t)\\|\n\\le |t|e^{|t|E_n}h^2C(n).\n\\]","This proves the modewise order, including \\(t=0\\). The difference of\npackets is the finite sum of the mode residuals, so PH05 gives its exact\nresidual identity. Thus","\\[\n\\|f_h(t)-f(t)\\|_{L^2}\n\\le h^2\\left(\n\\sum_{n\\in S}|t|^2e^{2|t|E_n}C(n)^2\\|a_n\\|_2^2\n\\right)^{1/2}.\n\\]","The constant is finite for fixed \\(t,S,a\\), proving the order and\nconvergence. \\(\\square\\)","The formal algebraic-coordinate proof obtains this order from continuity and local\ndifferentiability of the Banach-algebra exponential. Its operator norm\nbelongs to the algebraic coordinate spinor type. The coefficient\nconversion in its packet definitions is continuous, so its checked\ntorus-\\(L^2\\) convergence is genuine. Our Euclidean PH08 also has the\ndirect proof above. The following bridge explains why its coordinates\nagree with the separately defined Hilbert propagators.","Let \\(\\mathcal A\\) be the coordinate space \\(\\{0,1,2,3\\}\\to\\mathbb C\\)\nwith maximum norm \\(\\|v\\|_\\infty=\\max_r|v_r|\\), and let \\(\\mathcal H\\)\nbe the same coordinates with the Euclidean inner product. The map","\\[\nT:\\mathcal A\\longrightarrow\\mathcal H,\\qquad (Tv)_r=v_r\n\\]","\\[\n\\|v\\|_\\infty\\le\\|Tv\\|_2\\le2\\|v\\|_\\infty\n\\]","show that both directions are continuous. The first follows because each\ncoordinate square is bounded by the sum; the second because each of four\nsquares is at most \\(\\|v\\|_\\infty^2\\).","\\needspace{4\\baselineskip}\n\\textbf{Proposition PH09 (coordinate conjugacy of generators and\nexponentials).} Let \\(A\\) be a complex-linear endomorphism of\n\\(\\mathcal A\\), and let \\(A^{\\mathcal H}=TAT^{-1}\\). For every complex\n\\(z\\),","\\[\ne^{zA^{\\mathcal H}}=Te^{zA}T^{-1}.\n\\]","For the source Clifford actions, the continuum and centered Hilbert\ngenerators are exactly these conjugates of their algebraic generators.\nTheir exact propagators therefore have the same coordinates after\napplying \\(T\\). The algebraic-coordinate packet, transported coefficientwise by\n\\(T\\), agrees with the Hilbert packet for the corresponding\namplitudes.","\\emph{Proof.} Induction gives \\((TAT^{-1})^k=TA^kT^{-1}\\). For\n\\(k=0\\), use \\(TT^{-1}=I\\); for the induction step, cancel the adjacent\n\\(T^{-1}T\\). Every exponential partial sum therefore satisfies","\\[\n\\sum_{k=0}^{N}\\frac{z^k(A^{\\mathcal H})^k}{k!}\n=T\\left(\\sum_{k=0}^{N}\\frac{z^kA^k}{k!}\\right)T^{-1}.\n\\]","\\[\n\\|TBT^{-1}\\|_2\\le\\|T\\|\\,\\|B\\|_\\infty\\,\\|T^{-1}\\|\n\\le2\\|B\\|_\\infty.\n\\]","The source's transported Clifford action is defined by\n\\(v\\mapsto T(A(T^{-1}v))\\). The same is true for the displayed diagonal\nbeta action. Conjugation preserves real coefficients and finite sums,\nproving the Hamiltonian equality. Apply the exponential identity with\n\\(z=-it\\). Substituting each coefficient equality into the finite\nFourier sum then proves equality of the packets. \\(\\square\\)","\\needspace{4\\baselineskip}\n\\textbf{Example PH10 (axis and diagonal modes at \\(h=1/12\\)).} Set\n\\(m=1\\), \\(h=1/12\\), and","\\[\nS=\\{n_0,n_A,n_D\\},\\qquad\nn_0=(0,0,0),\\quad n_A=(1,0,0),\\quad n_D=(1,1,1).\n\\]","Let \\(e_0=(1,0,0,0)^T\\) denote the first spinor basis vector, and choose","\\[\na_{n_0}=\\frac1{\\sqrt2}e_0,\\qquad\na_{n_A}=\\frac12e_0,\\qquad\na_{n_D}=\\frac12e_0.\n\\]","Their squared norms are \\(1/2,1/4,1/4\\), so the packet is normalized.\nIts continuum and centered momenta are","\\needspace{8\\baselineskip}\n\\begin{center}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nMode & \\(p(n)\\) & \\(q_h(n)\\)","\\(n_0\\) & \\((0,0,0)\\) & \\((0,0,0)\\)","\\(n_A\\) & \\((2\\pi,0,0)\\) & \\((6,0,0)\\)","\\(n_D\\) & \\((2\\pi,2\\pi,2\\pi)\\) & \\((6,6,6)\\)","Each nonzero component has \\(hp_j=\\pi/6\\), so\n\\(\\sin(hp_j)/h=(1/2)/(1/12)=6\\). Thus","\\needspace{8\\baselineskip}\n\\begin{center}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nMode & \\(E_n\\) & \\(E_{h,n}\\)","\\(n_0\\) & \\(1\\) & \\(1\\)","\\(n_A\\) & \\(\\sqrt{1+4\\pi^2}\\) & \\(\\sqrt{37}\\)","\\(n_D\\) & \\(\\sqrt{1+12\\pi^2}\\) & \\(\\sqrt{109}\\)","These are exact symbolic equalities. The centered component \\(6\\) is\ndistinct from the integer mode label \\(1\\) and continuum momentum\n\\(2\\pi\\).","For any real kinetic vector \\(r\\), the exact matrix gives","\\[\nH(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.\n\\]","Set \\(E_r=\\sqrt{1+|r|^2}\\), \\(c_r(t)=\\cos(E_rt)\\), and\n\\(s_r(t)=\\sin(E_rt)/E_r\\). PH03 gives","\\[\nu_r(t)=U_{H(1,r)}(t)e_0=\n\\begin{pmatrix}\nc_r(t)-is_r(t)\\\\\n0\\\\\n-ir_zs_r(t)\\\\\n(r_y-ir_x)s_r(t)\n\\end{pmatrix}.\n\\]","In particular, \\(u_0(t)=e^{-it}e_0\\). For the centered axis and diagonal\nmodes,","\\[\nu_A(t)=\n\\begin{pmatrix}\n\\cos(\\sqrt{37}t)-i\\sin(\\sqrt{37}t)/\\sqrt{37}\\\\\n0\\\\\n0\\\\\n-6i\\sin(\\sqrt{37}t)/\\sqrt{37}\n\\end{pmatrix},\n\\]","\\[\nu_D(t)=\n\\begin{pmatrix}\n\\cos(\\sqrt{109}t)-i\\sin(\\sqrt{109}t)/\\sqrt{109}\\\\\n0\\\\\n-6i\\sin(\\sqrt{109}t)/\\sqrt{109}\\\\\n6(1-i)\\sin(\\sqrt{109}t)/\\sqrt{109}\n\\end{pmatrix}.\n\\]","\\[\nf_h(t,x)=\\frac{e^{-it}}{\\sqrt2}e_0\n+\\frac12e^{2\\pi ix_1}u_A(t)\n+\\frac12e^{2\\pi i(x_1+x_2+x_3)}u_D(t).\n\\]","The continuum packet has the same form with \\(u_A,u_D\\) replaced by\n\\(u_{(2\\pi,0,0)}\\) and \\(u_{(2\\pi,2\\pi,2\\pi)}\\). At \\(t=0\\), both equal","\\[\nf_a(x)=\\left(\\frac1{\\sqrt2}+\\frac12e^{2\\pi ix_1}\n+\\frac12e^{2\\pi i(x_1+x_2+x_3)}\\right)e_0.\n\\]","\\[\n\\|u_r(t)\\|_2^2\n=c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2)\n=\\cos^2(E_rt)+\\sin^2(E_rt)=1.\n\\]","Here \\(|c-is|^2=c^2+s^2\\) and \\(|r_y-ir_x|^2=r_x^2+r_y^2\\), since the\ncoefficients are real. Fourier orthogonality gives","\\[\n\\|f_h(t)\\|_{L^2}^2=\\tfrac12+\\tfrac14+\\tfrac14=1,\\qquad\n\\|f(t)\\|_{L^2}^2=1.\n\\]","This is an integrated statement. The initial pointwise density is not\nconstant: at \\(x=0\\), it is \\((1+1/\\sqrt2)^2=3/2+\\sqrt2\\). Cross terms\ncan change local density while their torus integrals vanish.\nNormalization does not mean that each point has density one.","The error also has an exact expression. For fixed mode, let \\(r=p(n)\\),\n\\(q=q_h(n)\\), with the corresponding \\(c_r,s_r,c_q,s_q\\). Coordinate\nsubtraction gives","\\[\n\\|u_q(t)-u_r(t)\\|_2^2\n=(c_q-c_r)^2+(s_q-s_r)^2\n+\\sum_{j=x,y,z}(q_js_q-r_js_r)^2.\n\\]","\\[\n\\begin{aligned}\n\\|f_h(t)-f(t)\\|_{L^2}^2\n={}&\\frac14\\|u_{(6,0,0)}(t)-u_{(2\\pi,0,0)}(t)\\|_2^2\\\\\n&+\\frac14\\|u_{(6,6,6)}(t)-u_{(2\\pi,2\\pi,2\\pi)}(t)\\|_2^2.\n\\end{aligned}\n\\]","\\needspace{10\\baselineskip}\n\\atlasfigure{06_packet_evolution}{The centered three-mode packet at $m=1$, $h=1/12$ changes local density and spinor composition while keeping total squared norm one. The image evaluates the closed formula at finitely many times.}","\\needspace{4\\baselineskip}\n\\textbf{Lemma (finite Hermitian Duhamel estimate).} Let \\(A,B\\) be\nHermitian matrices on a nonzero finite-dimensional complex Hilbert\nspace. For every real \\(t\\),","\\[\n\\|e^{-itA}-e^{-itB}\\|\\le |t|\\,\\|A-B\\|.\n\\]","\\emph{Proof.} First justify differentiation. On \\(|s|\\le R\\), the\nseries for \\(e^{-isB}\\) converges uniformly, and its derivative series\nis norm-dominated by","\\[\n\\sum_{k=1}^{\\infty}\\frac{kR^{k-1}\\|B\\|^k}{k!}\n=\\|B\\|e^{R\\|B\\|}.\n\\]","\\[\n\\frac{d}{ds}e^{-isB}=-iB e^{-isB}=-ie^{-isB}B.\n\\]","The same holds for \\(A\\). Since \\(A\\) commutes with every power of\nitself, it commutes with \\(e^{-iuA}\\). Fix \\(t\\) and define\n\\(F(s)=e^{-i(t-s)A}e^{-isB}\\). The product rule gives","\\[\nF'(s)=i e^{-i(t-s)A}(A-B)e^{-isB}.\n\\]","The endpoints are \\(F(0)=e^{-itA}\\) and \\(F(t)=e^{-itB}\\). Integrating\nentrywise in the finite matrix space gives the oriented-integral\nidentity","\\[\ne^{-itB}-e^{-itA}\n=i\\int_0^t e^{-i(t-s)A}(A-B)e^{-isB}\\,ds.\n\\]","For Hermitian matrices the two exponentials in the integrand are\nunitary. This does not rely on the special Clifford algebra: the series\ngives \\((e^{-iuA})^*=e^{iuA}\\), and the Cauchy product of the two\nabsolutely convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","ifford algebra: the series\ngives \\((e^{-iuA})^*=e^{iuA}\\), and the Cauchy product of the two\nabsolutely convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","he series\ngives \\((e^{-iuA})^*=e^{iuA}\\), and the Cauchy product of the two\nabsolutely convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","*=e^{iuA}\\), and the Cauchy product of the two\nabsolutely convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","he Cauchy product of the two\nabsolutely convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","y convergent series gives \\(e^{iuA}e^{-iuA}=I\\). Indeed, the\ncoefficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","efficient of \\(A^k\\) for \\(k>0\\) is\n\\((iu)^k\\sum_{r=0}^{k}\\binom{k}{r}(-1)^r/k!=0\\), and the constant\ncoefficient is \\(I\\). A unitary matrix on a nonzero space has norm one.\nSubmultiplicativity therefore bounds the integrand norm by \\(\\|A-B\\|\\).\nFor \\(t\\ge0\\), integration gives \\(t\\|A-B\\|\\); for \\(t<0\\), reversing\nthe interval gives \\((-t)\\|A-B\\|\\). This proves the estimate for every\nreal time. \\(\\square\\)","\\needspace{4\\baselineskip}\n\\textbf{Theorem PH11 (time-explicit finite-packet bound).} For every\nreal \\(m,h,t\\), fixed mode \\(n\\), and fixed finite packet,","\\[\n\\|U_{H_{h,n}}(t)-U_{H_n}(t)\\|\n\\le |t|\\,|\\delta(h,n)|\\le |t|h^2C(n),\n\\]","\\[\n\\begin{aligned}\n\\|f_h(t)-f(t)\\|_{L^2}\n&\\le |t|\\left(\\sum_{n\\in S}|\\delta(h,n)|^2\\|a_n\\|_2^2\\right)^{1/2}\\\\\n&\\le |t|h^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}.\n\\end{aligned}\n\\]","Consequently, for every \\(T\\ge0\\),","\\[\n\\sup_{|t|\\le T}\\|f_h(t)-f(t)\\|_{L^2}\n\\le Th^2\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2},\n\\]","\\emph{Proof.} Apply the Duhamel lemma with \\(A=H_{h,n}\\), \\(B=H_n\\).\nBoth are Hermitian by PH02, and PH07 gives the residual norm and bound.\nFor each amplitude,","\\[\n\\|(U_{H_{h,n}}(t)-U_{H_n}(t))a_n\\|_2\n\\le |t|\\,|\\delta(h,n)|\\,\\|a_n\\|_2\n\\le |t|h^2C(n)\\|a_n\\|_2.\n\\]","Square and sum over \\(S\\), apply the exact residual Parseval identity of\nPH08, and take the nonnegative square root. The middle coefficient is\nexactly the initial generator-residual norm in PH07 because these\nClifford differences satisfy\n\\(\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4\\). This identification\nis specific to the displayed symbol differences; the general Duhamel\nlemma is an operator-norm estimate. Replacing \\(|t|\\) by \\(T\\) gives the\ncompact-time bound. Its right side tends to zero as \\(h\\to0\\).\n\\(\\square\\)","apply the exact residual Parseval identity of\nPH08, and take the nonnegative square root. The middle coefficient is\nexactly the initial generator-residual norm in PH07 because these\nClifford differences satisfy\n\\(\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4\\). This identification\nis specific to the displayed symbol differences; the general Duhamel\nlemma is an operator-norm estimate. Replacing \\(|t|\\) by \\(T\\) gives the\ncompact-time bound. Its right side tends to zero as \\(h\\to0\\).\n\\(\\square\\)","exact residual Parseval identity of\nPH08, and take the nonnegative square root. The middle coefficient is\nexactly the initial generator-residual norm in PH07 because these\nClifford differences satisfy\n\\(\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4\\). This identification\nis specific to the displayed symbol differences; the general Duhamel\nlemma is an operator-norm estimate. Replacing \\(|t|\\) by \\(T\\) gives the\ncompact-time bound. Its right side tends to zero as \\(h\\to0\\).\n\\(\\square\\)","quare root. The middle coefficient is\nexactly the initial generator-residual norm in PH07 because these\nClifford differences satisfy\n\\(\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4\\). This identification\nis specific to the displayed symbol differences; the general Duhamel\nlemma is an operator-norm estimate. Replacing \\(|t|\\) by \\(T\\) gives the\ncompact-time bound. Its right side tends to zero as \\(h\\to0\\).\n\\(\\square\\)","The middle coefficient is\nexactly the initial generator-residual norm in PH07 because these\nClifford differences satisfy\n\\(\\Delta_{h,n}^*\\Delta_{h,n}=|\\delta(h,n)|^2I_4\\). This identification\nis specific to the displayed symbol differences; the general Duhamel\nlemma is an operator-norm estimate. Replacing \\(|t|\\) by \\(T\\) gives the\ncompact-time bound. Its right side tends to zero as \\(h\\to0\\).\n\\(\\square\\)","For PH10, \\(C(n_0)=0\\), \\(C(n_A)=(2\\pi)^3/6\\), and\n\\(C(n_D)=\\sqrt3(2\\pi)^3/6\\), so","\\[\n\\left(\\sum_{n\\in S}C(n)^2\\|a_n\\|_2^2\\right)^{1/2}\n=\\frac{(2\\pi)^3}{6}=\\frac{4\\pi^3}{3}.\n\\]","At its spacing, put \\(d=2\\pi-6>0\\). The axis residual has magnitude\n\\(d\\), the diagonal residual has magnitude \\(\\sqrt3d\\), and their\nsquared amplitude weights are \\(1/4,1/4\\). Hence the exact middle\ncoefficient is","\\[\n\\left(\\tfrac14d^2+\\tfrac14\\,3d^2\\right)^{1/2}=d,\n\\]","\\[\n\\|f_{1/12}(t)-f(t)\\|_{L^2}\n\\le(2\\pi-6)|t|\\le\\frac{\\pi^3}{108}|t|.\n\\]","The triangle inequality also gives \\(\\|f_h(t)-f(t)\\|_{L^2}\\le2\\) for\nthis normalized packet, so one may use the smaller bound. Neither says\nthat a fixed nonzero spacing has small error for arbitrarily large time.\nNor does the time bound remove the cutoff: its coefficient contains high\npowers of mode momenta. For arbitrary \\(L^2\\) data, the Fourier-tail argument in the following part\nprovides convergence without this uniform quadratic coefficient.","Let \\(H_S,H_{h,S}\\) denote the finite block diagonal Hamiltonians. In\ncoefficient coordinates, squared norm and continuum Hamiltonian\nexpectation are","\\[\nN(a)=\\sum_{n\\in S}\\|a_n\\|_2^2,\\qquad\n\\mathcal E(a)=\\sum_{n\\in S}\\langle a_n,H_na_n\\rangle,\n\\]","with centered expectation \\(\\mathcal E_h(a)\\) obtained by replacing\n\\(H_n\\) with \\(H_{h,n}\\). Hermiticity makes each expectation real: its\ncomplex conjugate is\n\\(\\langle H_na_n,a_n\\rangle=\\langle a_n,H_na_n\\rangle\\).","\\[\n\\mathcal E(U_S(t)a)=\\mathcal E(a),\\qquad\n\\mathcal E_h(U_{h,S}(t)a)=\\mathcal E_h(a).\n\\]","Here the packet operators act on coefficients through the isometry\n\\(\\mathcal K_S\\cong\\mathcal F_S\\). Squared-norm conservation does not\nidentify either expectation with \\(N(a)\\). At common initial data,","\\[\n|\\mathcal E_h(a)-\\mathcal E(a)|\n\\le h^2\\sum_{n\\in S}C(n)\\|a_n\\|_2^2.\n\\]","\\[\n\\langle U_H(t)a,H U_H(t)a\\rangle\n=\\langle a,U_H(t)^*HU_H(t)a\\rangle\n=\\langle a,Ha\\rangle.\n\\]","\\[\n|\\langle a_n,(H_{h,n}-H_n)a_n\\rangle|\n\\le\\|a_n\\|_2\\|\\Delta_{h,n}a_n\\|_2\n\\le h^2C(n)\\|a_n\\|_2^2.\n\\]","The finite-sum triangle inequality proves the comparison. \\(\\square\\)","A unit vector in the negative-energy range of \\(P_-\\) has squared norm\none but expectation \\(-E\\); a unit vector in the positive-energy range\nhas expectation \\(E\\). Norm and energy are therefore different even for\none mode. For \\(e_0\\), the matrix gives\n\\(\\langle e_0,H(m,p)e_0\\rangle=m\\), since the kinetic terms have zero\ndiagonal entry in that component. PH10 has expectation \\(1\\) for both\nHamiltonians. Its values \\(\\sqrt{37},\\sqrt{109}\\) are shell magnitudes,\nand its spinors contain both energy signs. Their expectations need not\nequal a positive shell value.","negative-energy range of \\(P_-\\) has squared norm\none but expectation \\(-E\\); a unit vector in the positive-energy range\nhas expectation \\(E\\). Norm and energy are therefore different even for\none mode. For \\(e_0\\), the matrix gives\n\\(\\langle e_0,H(m,p)e_0\\rangle=m\\), since the kinetic terms have zero\ndiagonal entry in that component. PH10 has expectation \\(1\\) for both\nHamiltonians. Its values \\(\\sqrt{37},\\sqrt{109}\\) are shell magnitudes,\nand its spinors contain both energy signs. Their expectations need not\nequal a positive shell value.","Centered approximation & The multiplier with kinetic symbol $\\sin(hp_j)/h$ on the same integer mode index set.","\\clearpage\n\\section*{Resolution and finite dynamics}\n\\section{Packets and transfer blocks}\nThe useful starting point is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","section*{Resolution and finite dynamics}\n\\section{Packets and transfer blocks}\nThe useful starting point is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","ection{Packets and transfer blocks}\nThe useful starting point is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","and transfer blocks}\nThe useful starting point is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","nsfer blocks}\nThe useful starting point is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","is a resolved operator rather than a preferred coordinate matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","e matrix. Let $V$ be a complex vector space with a finite family of projectors $(P_a)_{a\\in A}$ satisfying\n\\[\nP_aP_b=\\delta_{ab}P_a,\\qquad \\sum_{a\\in A}P_a=I.\n\\]\nThe component $P_av$ is the packet of $v$ selected by $a$. For a linear map $L:V\\to V$, its transfer block from packet $a$ to packet $b$ is\n\\[\nL_{ba}=P_bLP_a.\n\\]\nCompleteness gives $L=\\sum_{a,b}L_{ba}$. Both indices matter. Keeping only $P_aLP_a$ would remove every transfer between distinct packets.","For two operators $L$ and $M$, insert the resolution between them:\n\\[\nP_cMLP_a=\\sum_{b\\in A}(P_cMP_b)(P_bLP_a).\n\\]\nThis is block multiplication written without a choice of basis inside each packet. The intermediate index records the interface through which a composition passes. It is also the reason that a change of resolution must specify how fine packets assemble into coarse packets.","Suppose $P_A=\\sum_{a\\in C(A)}P_a$ and $P_B=\\sum_{b\\in C(B)}P_b$. Then\n\\[\nP_BLP_A=\\sum_{b\\in C(B)}\\sum_{a\\in C(A)}P_bLP_a.\n\\]\nThe identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.","\\subsection{What changes at completion}\nThe finite identity $\\sum_aP_a=I$ becomes an approximation statement for an infinite orthogonal family. Finite partial sums must converge in the chosen topology. On the coefficient Hilbert space, canonical finite truncations converge in norm. For an unbounded Hamiltonian, approximating the state alone is insufficient to control its image. The graph approximation later in the volume controls both quantities at once.","Throughout this section, $\\HH$ is a complex Hilbert space and $I$ is a finite index set. The inner product is conjugate-linear in its first argument and linear in its second. Every operator initially belongs to $\\mathcal B(\\HH)$, the bounded operators defined on all of $\\HH$. The dimension of $\\HH$ may be infinite. Finiteness refers to the number of packets. Countable resolutions and unbounded generators require the additional hypotheses developed later.","\\begin{definition}[Finite orthogonal atlas]\nA finite orthogonal atlas is a family $(P_a)_{a\\in I}$ satisfying\n\\begin{equation}\\label{f:atlas-relations}\n P_a^*=P_a,\\qquad P_aP_b=\\delta_{ab}P_a,\n \\qquad \\sum_{a\\in I}P_a=\\id.\n\\end{equation}\nIts packet of a vector $\\psi$ is $\\psi_a=P_a\\psi$. Its packet weight is $p_a=\\norm{\\psi_a}^2$. Zero projectors are allowed, so a label can describe an empty sector.\n\\end{definition}","The word weight avoids assigning physical units before an application supplies them. If $\\norm{\\psi}=1$, these weights form a probability distribution. If the state carries another normalization, they sum to its squared norm. They are not automatically the energy expectation of a Hamiltonian. For example, a normalized energy eigenvector always has total packet weight one, regardless of its eigenvalue.","\\begin{proposition}[Reconstruction and norm separation]\\label{f:parseval}\nFor every $\\psi,\\phi\\in\\HH$,\n\\begin{equation}\\label{f:reconstruct}\n \\psi=\\sum_a\\psi_a,\\qquad\n \\inner{\\psi}{\\phi}=\\sum_a\\inner{P_a\\psi}{P_a\\phi},\\qquad\n \\norm{\\psi}^2=\\sum_a\\norm{\\psi_a}^2.\n\\end{equation}\nPackets with different indices are orthogonal, including packets obtained from different vectors.\n\\end{proposition}","\\begin{proof}\nEvaluate $\\sum_aP_a=\\id$ on $\\psi$. For $a\\ne b$, self-adjointness and orthogonality give\n\\[\n \\inner{P_a\\psi}{P_b\\phi}\n =\\inner{\\psi}{P_aP_b\\phi}=0.\n\\]\nExpand both entries of $\\inner{\\psi}{\\phi}$ using the finite reconstruction. All terms with unequal indices vanish. Taking $\\phi=\\psi$ proves the norm identity.\n\\end{proof}","\\begin{example}[Two packets inside four coordinates]\\label{f:four-coordinate}\nLet $\\HH=\\C^4$ with its usual inner product, and set\n\\[\n \\psi=\\frac1{\\sqrt7}(1,i,2,-i)^{\\mathsf T},\\qquad\n P_A=\\operatorname{diag}(1,1,0,0),\\quad\n P_B=\\operatorname{diag}(0,0,1,1).\n\\]\nThen $\\psi_A=(1,i,0,0)^{\\mathsf T}/\\sqrt7$ and\n$\\psi_B=(0,0,2,-i)^{\\mathsf T}/\\sqrt7$. The weights are $p_A=2/7$ and $p_B=5/7$. Refining into the four coordinate projectors gives weights $1/7,1/7,4/7,1/7$. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights would discard their internal phases and amplitudes.\n\\end{example}","s]\\label{f:four-coordinate}\nLet $\\HH=\\C^4$ with its usual inner product, and set\n\\[\n \\psi=\\frac1{\\sqrt7}(1,i,2,-i)^{\\mathsf T},\\qquad\n P_A=\\operatorname{diag}(1,1,0,0),\\quad\n P_B=\\operatorname{diag}(0,0,1,1).\n\\]\nThen $\\psi_A=(1,i,0,0)^{\\mathsf T}/\\sqrt7$ and\n$\\psi_B=(0,0,2,-i)^{\\mathsf T}/\\sqrt7$. The weights are $p_A=2/7$ and $p_B=5/7$. Refining into the four coordinate projectors gives weights $1/7,1/7,4/7,1/7$. The two descriptions resolve the same vector at different levels. The coarse description still contains the two-dimensional packet vectors. Retaining only the two scalar weights would discard their internal phases and amplitudes.\n\\end{example}","The same atlas resolves an operator on both its input and output sides. For $X\\in\\mathcal B(\\HH)$, define\n\\begin{equation}\\label{f:corner-def}\n X_{ba}=P_bXP_a.\n\\end{equation}\nThe first index names the destination and the second names the source. As a map between packet spaces, this is $X_{ba}:\\Ran P_a\\to\\Ran P_b$. As a bounded operator on $\\HH$, it is extended by zero on the orthogonal complement of $\\Ran P_a$. Keeping these two descriptions aligned prevents a block from being composed with an incompatible input space.","same atlas resolves an operator on both its input and output sides. For $X\\in\\mathcal B(\\HH)$, define\n\\begin{equation}\\label{f:corner-def}\n X_{ba}=P_bXP_a.\n\\end{equation}\nThe first index names the destination and the second names the source. As a map between packet spaces, this is $X_{ba}:\\Ran P_a\\to\\Ran P_b$. As a bounded operator on $\\HH$, it is extended by zero on the orthogonal complement of $\\Ran P_a$. Keeping these two descriptions aligned prevents a block from being composed with an incompatible input space.","\\begin{proposition}[Block calculus]\\label{f:block-calculus}\nFor bounded $X,Y$,\n\\begin{align}\n X&=\\sum_{b,a}X_{ba}, & P_bX\\psi&=\\sum_aX_{ba}\\psi_a,\\label{f:block-reconstruction}","\\begin{proposition}[Block calculus]\\label{f:block-calculus}\nFor bounded $X,Y$,\n\\begin{align}\n X&=\\sum_{b,a}X_{ba}, & P_bX\\psi&=\\sum_aX_{ba}\\psi_a,\\label{f:block-reconstruction}\\\\\n (XY)_{ca}&=\\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.\n \\label{f:block-product}\n\\end{align}\nThe identity has diagonal blocks $P_a$ and zero off-diagonal blocks. If $X=X^*$, its opposite blocks are adjoints.\n\\end{proposition}","(XY)_{ca}&=\\sum_bX_{cb}Y_{ba}, & (X^*)_{ab}&=(X_{ba})^*.\n \\label{f:block-product}\n\\end{align}\nThe identity has diagonal blocks $P_a$ and zero off-diagonal blocks. If $X=X^*$, its opposite blocks are adjoints.\n\\end{proposition}","\\begin{proof}\nInsert $\\sum_bP_b=\\id$ to the left of $X$ and $\\sum_aP_a=\\id$ to its right. This gives the first formula. Evaluate the right resolution on $\\psi$ for the second. For the product,\n\\[\n P_cXYP_a=P_cX\\Bigl(\\sum_bP_b\\Bigr)YP_a\n =\\sum_b(P_cXP_b)(P_bYP_a),\n\\]\nwhere $P_b^2=P_b$ supplies the repeated middle projector. Finally,\n$(P_bXP_a)^*=P_aX^*P_b$.\n\\end{proof}","\\begin{example}[A coarse block retains internal channels]\\label{f:corner-example}\nUse the two-packet split of Example~\\ref{f:four-coordinate} and the Hermitian matrix\n\\[\n H=\\begin{pmatrix}\n 0&1&0&2\\\\1&0&-1&0\\\\0&-1&0&1\\\\2&0&1&0\n \\end{pmatrix}.\n\\]\nRelative to $\\C^4=\\C^2_A\\oplus\\C^2_B$, the four blocks are\n\\[\n H_{AA}=H_{BB}=\\begin{pmatrix}0&1","2&0&1&0\n \\end{pmatrix}.\n\\]\nRelative to $\\C^4=\\C^2_A\\oplus\\C^2_B$, the four blocks are\n\\[\n H_{AA}=H_{BB}=\\begin{pmatrix}0&1","2&0&1&0\n \\end{pmatrix}.\n\\]\nRelative to $\\C^4=\\C^2_A\\oplus\\C^2_B$, the four blocks are\n\\[\n H_{AA}=H_{BB}=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n H_{BA}=\\begin{pmatrix}0&-1\\\\2&0\\end{pmatrix},\\qquad\n H_{AB}=H_{BA}^*.\n\\]\nThe block from $A$ to $B$ contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.\n\\end{example}","2&0\\end{pmatrix},\\qquad\n H_{AB}=H_{BA}^*.\n\\]\nThe block from $A$ to $B$ contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depend on the state occupying its source and destination.\n\\end{example}","Deleting off-diagonal blocks produces another operator, $\\sum_aP_aXP_a$. It agrees with $X$ exactly when all omitted blocks vanish. Thus an atlas resolution is lossless, while keeping only its diagonal is an additional operation. Similarly, a finite-dimensional coordinate choice inside the packet ranges turns the blocks into numerical submatrices, but that coordinate representation is not needed for reconstruction itself.","Let $\\psi(t)$ be a differentiable trajectory and suppose that\n\\begin{equation}\\label{f:flow}\n \\dot\\psi(t)=K(t)\\psi(t),\\qquad K(t)^*=-K(t).\n\\end{equation}\nEach $K(t)$ is bounded. The argument uses the stated derivative of the trajectory at the time under consideration. It does not manufacture a solution of an arbitrary differential equation. For a fixed bounded generator, the exponential series provides a separate construction of the evolution. A state-dependent vector field requires its own existence and uniqueness hypotheses.","Define the receiving-first current by\n\\begin{equation}\\label{f:current}\n J_{b\\leftarrow a}(t)\n =2\\operatorname{Re}\\inner{\\psi_b(t)}{K(t)\\psi_a(t)}\n =2\\operatorname{Re}\\inner{\\psi_b(t)}{K_{ba}(t)\\psi_a(t)}.\n\\end{equation}\nThe second equality follows because $P_b\\psi_b=\\psi_b$. A positive value means that the contribution assigned to the pair increases the weight in $b$. The current is a real number, and its sign convention is fixed by the derivative calculation below.","\\begin{theorem}[Finite packet transfer]\\label{f:transfer-theorem}\nFor the trajectory in \\eqref{f:flow},\n\\begin{align}\n \\dot p_b&=\\sum_aJ_{b\\leftarrow a},\\label{f:continuity}\\\\\n J_{b\\leftarrow a}&=-J_{a\\leftarrow b},\\qquad\n J_{a\\leftarrow a}=0,\\label{f:antisymmetry}\\\\\n \\frac{d}{dt}\\sum_ap_a&=0.\n\\end{align}\nFor every subset $C\\subseteq I$, its total weight obeys\n\\begin{equation}\\label{f:boundary-current}\n \\frac{d}{dt}\\sum_{b\\in C}p_b\n =\\sum_{b\\in C}\\sum_{a\\notin C}J_{b\\leftarrow a}.\n\\end{equation}\n\\end{theorem}","\\frac{d}{dt}\\sum_ap_a&=0.\n\\end{align}\nFor every subset $C\\subseteq I$, its total weight obeys\n\\begin{equation}\\label{f:boundary-current}\n \\frac{d}{dt}\\sum_{b\\in C}p_b\n =\\sum_{b\\in C}\\sum_{a\\notin C}J_{b\\leftarrow a}.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe projectors are fixed in time, so the product rule for the inner product gives\n\\begin{align*}\n \\dot p_b\n &=2\\operatorname{Re}\\inner{P_b\\psi}{P_b\\dot\\psi}\n =2\\operatorname{Re}\\inner{\\psi_b}{K\\psi}\\\\\n &=\\sum_a2\\operatorname{Re}\\inner{\\psi_b}{K\\psi_a}.\n\\end{align*}\nThe reconstruction of $\\psi$ justifies the last finite sum. Skew-adjointness gives\n\\[\n \\inner{\\psi_a}{K\\psi_b}\n =-\\inner{K\\psi_a}{\\psi_b}\n =-\\overline{\\inner{\\psi_b}{K\\psi_a}}.\n\\]\nTaking real parts proves antisymmetry. Putting $a=b$ gives zero diagonal current. Every internal ordered pair in $C\\times C$ cancels with its reverse. Splitting the sum over all sources into $C$ and its complement proves \\eqref{f:boundary-current}. Taking $C=I$ proves total conservation.\n\\end{proof}","&=\\sum_a2\\operatorname{Re}\\inner{\\psi_b}{K\\psi_a}.\n\\end{align*}\nThe reconstruction of $\\psi$ justifies the last finite sum. Skew-adjointness gives\n\\[\n \\inner{\\psi_a}{K\\psi_b}\n =-\\inner{K\\psi_a}{\\psi_b}\n =-\\overline{\\inner{\\psi_b}{K\\psi_a}}.\n\\]\nTaking real parts proves antisymmetry. Putting $a=b$ gives zero diagonal current. Every internal ordered pair in $C\\times C$ cancels with its reverse. Splitting the sum over all sources into $C$ and its complement proves \\eqref{f:boundary-current}. Taking $C=I$ proves total conservation.\n\\end{proof}","i_b}{K\\psi_a}.\n\\end{align*}\nThe reconstruction of $\\psi$ justifies the last finite sum. Skew-adjointness gives\n\\[\n \\inner{\\psi_a}{K\\psi_b}\n =-\\inner{K\\psi_a}{\\psi_b}\n =-\\overline{\\inner{\\psi_b}{K\\psi_a}}.\n\\]\nTaking real parts proves antisymmetry. Putting $a=b$ gives zero diagonal current. Every internal ordered pair in $C\\times C$ cancels with its reverse. Splitting the sum over all sources into $C$ and its complement proves \\eqref{f:boundary-current}. Taking $C=I$ proves total conservation.\n\\end{proof}","s the last finite sum. Skew-adjointness gives\n\\[\n \\inner{\\psi_a}{K\\psi_b}\n =-\\inner{K\\psi_a}{\\psi_b}\n =-\\overline{\\inner{\\psi_b}{K\\psi_a}}.\n\\]\nTaking real parts proves antisymmetry. Putting $a=b$ gives zero diagonal current. Every internal ordered pair in $C\\times C$ cancels with its reverse. Splitting the sum over all sources into $C$ and its complement proves \\eqref{f:boundary-current}. Taking $C=I$ proves total conservation.\n\\end{proof}","In the Hamiltonian convention $K=-iH$ with $H=H^*$ and $\\hbar=1$, linearity in the second slot yields\n\\begin{equation}\\label{f:hamiltonian-current}\n J_{b\\leftarrow a}\n =2\\operatorname{Im}\\inner{\\psi_b}{H\\psi_a}.\n\\end{equation}\nIndeed, $\\operatorname{Re}(-iz)=\\operatorname{Im}z$. The plus sign belongs to this inner-product convention and to the choice that the first index receives the current. It should not be changed independently of either convention.","\\begin{example}[An exactly solved exchange]\\label{f:rabi}\nOn $\\C^2$, take the coordinate atlas,\n\\[\n H=g\\begin{pmatrix}0&1","\\begin{example}[An exactly solved exchange]\\label{f:rabi}\nOn $\\C^2$, take the coordinate atlas,\n\\[\n H=g\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\quad g>0,\n \\qquad \\psi(0)=\\begin{pmatrix}1\\\\0\\end{pmatrix}.\n\\]\nDirect differentiation verifies the solution\n\\[\n \\psi(t)=\\begin{pmatrix}\\cos(gt)","0\\end{pmatrix}.\n\\]\nDirect differentiation verifies the solution\n\\[\n \\psi(t)=\\begin{pmatrix}\\cos(gt)\\\\-i\\sin(gt)\\end{pmatrix},\n \\quad p_1=\\cos^2(gt),\\quad p_2=\\sin^2(gt).\n\\]\nThe current is\n\\[\n J_{2\\leftarrow1}\n =2\\operatorname{Im}\\bigl(i g\\sin(gt)\\cos(gt)\\bigr)\n =g\\sin(2gt)=\\dot p_2.\n\\]\nIt reverses sign during the cycle. At $t=0$ the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.\n\\end{example}","-i\\sin(gt)\\end{pmatrix},\n \\quad p_1=\\cos^2(gt),\\quad p_2=\\sin^2(gt).\n\\]\nThe current is\n\\[\n J_{2\\leftarrow1}\n =2\\operatorname{Im}\\bigl(i g\\sin(gt)\\cos(gt)\\bigr)\n =g\\sin(2gt)=\\dot p_2.\n\\]\nIt reverses sign during the cycle. At $t=0$ the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instantaneous flow must be kept separate.\n\\end{example}","The same proof applies to $\\dot\\psi=K(\\psi,t)\\psi$ when the operator evaluated along the supplied trajectory is skew-adjoint. No derivative of the function $K(\\psi,t)$ appears in the first derivative of $\\norm{P_b\\psi}^2$. Such derivatives would enter a second derivative or a linearized perturbation equation. Pointwise conservation therefore survives many nonlinear choices of generator, while well-posedness remains a separate question.","Let $\\mathcal A$ be a finite set of coarse labels and let $f:I\\to\\mathcal A$ assign every fine packet to exactly one coarse label. Write $C_A=f^{-1}(A)$ and define\n\\begin{equation}\\label{f:coarse-projector}\n Q_A=\\sum_{a\\in C_A}P_a.\n\\end{equation}\nThe fibers are disjoint and cover $I$, so $(Q_A)$ is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a different physical system.","\\begin{theorem}[Exact coarse transfer]\\label{f:coarse-theorem}\nThe coarse weight and current are\n\\begin{align}\n p_A^{\\mathrm c}&=\\norm{Q_A\\psi}^2=\\sum_{a\\in C_A}p_a,\\\\\n J_{B\\leftarrow A}^{\\mathrm c}\n &=2\\operatorname{Re}\\inner{Q_B\\psi}{KQ_A\\psi}\n =\\sum_{b\\in C_B}\\sum_{a\\in C_A}J_{b\\leftarrow a}.\n\\end{align}\nThey satisfy $\\dot p_B^{\\mathrm c}=\\sum_AJ_{B\\leftarrow A}^{\\mathrm c}$ and\n$J_{B\\leftarrow A}^{\\mathrm c}=-J_{A\\leftarrow B}^{\\mathrm c}$. In particular, the sum of fine currents internal to a coarse cell is zero.\n\\end{theorem}","J_{B\\leftarrow A}^{\\mathrm c}\n &=2\\operatorname{Re}\\inner{Q_B\\psi}{KQ_A\\psi}\n =\\sum_{b\\in C_B}\\sum_{a\\in C_A}J_{b\\leftarrow a}.\n\\end{align}\nThey satisfy $\\dot p_B^{\\mathrm c}=\\sum_AJ_{B\\leftarrow A}^{\\mathrm c}$ and\n$J_{B\\leftarrow A}^{\\mathrm c}=-J_{A\\leftarrow B}^{\\mathrm c}$. In particular, the sum of fine currents internal to a coarse cell is zero.\n\\end{theorem}","For an incidence matrix $R_{Aa}=1$ when $f(a)=A$ and zero otherwise, these formulas read\n\\begin{equation}\\label{f:incidence}\n p^{\\mathrm c}=Rp,\\qquad J^{\\mathrm c}=RJR^{\\mathsf T},\\qquad\n R^{\\mathsf T}\\boldsymbol1_{\\mathrm c}=\\boldsymbol1_{\\mathrm f}.\n\\end{equation}\nThe last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If $S$ performs a second grouping, then\n\\[\n S(Rp)=(SR)p,\\qquad S(RJR^{\\mathsf T})S^{\\mathsf T}\n =(SR)J(SR)^{\\mathsf T}.\n\\]\nThus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.","Aa}=1$ when $f(a)=A$ and zero otherwise, these formulas read\n\\begin{equation}\\label{f:incidence}\n p^{\\mathrm c}=Rp,\\qquad J^{\\mathrm c}=RJR^{\\mathsf T},\\qquad\n R^{\\mathsf T}\\boldsymbol1_{\\mathrm c}=\\boldsymbol1_{\\mathrm f}.\n\\end{equation}\nThe last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If $S$ performs a second grouping, then\n\\[\n S(Rp)=(SR)p,\\qquad S(RJR^{\\mathsf T})S^{\\mathsf T}\n =(SR)J(SR)^{\\mathsf T}.\n\\]\nThus a tower of exact groupings is associative. There is no accumulation of approximation error in these identities.","\\begin{example}[Internal exchange disappears from the coarse boundary]\nConsider four labels with nonzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","ternal exchange disappears from the coarse boundary]\nConsider four labels with nonzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","ears from the coarse boundary]\nConsider four labels with nonzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","arse boundary]\nConsider four labels with nonzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","with nonzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","nzero currents\n\\[\n J_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","_{2\\leftarrow1}=3,\\quad J_{3\\leftarrow2}=2,\\quad\n J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","J_{4\\leftarrow3}=1,\\quad J_{1\\leftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","eftarrow4}=\\tfrac12,\n\\]\ntheir negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are\n$(-5/2,1,1,1/2)$. For $A=\\{1,2\\}$ and $B=\\{3,4\\}$, the internal currents $3$ and $1$ cancel, leaving\n$J^{\\mathrm c}_{B\\leftarrow A}=2-1/2=3/2$.\nHence $\\dot p_A^{\\mathrm c}=-3/2$ and $\\dot p_B^{\\mathrm c}=3/2$.\n\\end{example}","Exact balance should not be confused with closure of a reduced equation. The scalar vector $p^{\\mathrm c}$ generally does not determine the state-dependent current $J^{\\mathrm c}$. Internal phase or correlation data may still be needed. Even at the fine level, a continuity equation alone does not uniquely determine its current. On three labels, the antisymmetric matrix\n\\[\n \\begin{pmatrix}0&c&-c","Exact balance should not be confused with closure of a reduced equation. The scalar vector $p^{\\mathrm c}$ generally does not determine the state-dependent current $J^{\\mathrm c}$. Internal phase or correlation data may still be needed. Even at the fine level, a continuity equation alone does not uniquely determine its current. On three labels, the antisymmetric matrix\n\\[\n \\begin{pmatrix}0&c&-c\\\\-c&0&c\\\\c&-c&0\\end{pmatrix}\n\\]\nhas zero row sums for any real $c$. Adding it changes circulation while preserving all weight derivatives. The operator and state specify a particular current through \\eqref{f:current}. Observing only the derivatives cannot recover that additional information.","c&-c&0\\end{pmatrix}\n\\]\nhas zero row sums for any real $c$. Adding it changes circulation while preserving all weight derivatives. The operator and state specify a particular current through \\eqref{f:current}. Observing only the derivatives cannot recover that additional information.","\\foundationfigure{04_coarse_balance}{An instantaneous antisymmetric current array and its two-cell aggregation. The internal arrows cancel, leaving net transfer $3/2$ across the coarse boundary. This balance example does not assume a constructed Hamiltonian trajectory.}","A resolution can change with time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let $U(t)$ be differentiable in operator norm and satisfy both $U^*U=\\id$ and $UU^*=\\id$. Set\n\\begin{equation}\\label{f:moving-defs}\n P_a(t)=U(t)P_a^{(0)}U(t)^*,\\qquad\n \\phi(t)=U(t)^*\\psi(t),\\qquad \\Gamma(t)=U(t)^*\\dot U(t).\n\\end{equation}\nThe coordinates $\\phi$ see fixed projectors $P_a^{(0)}$. The physical weight in the moving packet is the same scalar:\n\\[\n \\norm{P_a(t)\\psi(t)}^2\n =\\norm{P_a^{(0)}\\phi(t)}^2.\n\\]","th time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let $U(t)$ be differentiable in operator norm and satisfy both $U^*U=\\id$ and $UU^*=\\id$. Set\n\\begin{equation}\\label{f:moving-defs}\n P_a(t)=U(t)P_a^{(0)}U(t)^*,\\qquad\n \\phi(t)=U(t)^*\\psi(t),\\qquad \\Gamma(t)=U(t)^*\\dot U(t).\n\\end{equation}\nThe coordinates $\\phi$ see fixed projectors $P_a^{(0)}$. The physical weight in the moving packet is the same scalar:\n\\[\n \\norm{P_a(t)\\psi(t)}^2\n =\\norm{P_a^{(0)}\\phi(t)}^2.\n\\]","rt the entire atlas by a unitary family. Let $U(t)$ be differentiable in operator norm and satisfy both $U^*U=\\id$ and $UU^*=\\id$. Set\n\\begin{equation}\\label{f:moving-defs}\n P_a(t)=U(t)P_a^{(0)}U(t)^*,\\qquad\n \\phi(t)=U(t)^*\\psi(t),\\qquad \\Gamma(t)=U(t)^*\\dot U(t).\n\\end{equation}\nThe coordinates $\\phi$ see fixed projectors $P_a^{(0)}$. The physical weight in the moving packet is the same scalar:\n\\[\n \\norm{P_a(t)\\psi(t)}^2\n =\\norm{P_a^{(0)}\\phi(t)}^2.\n\\]","\\begin{proposition}[The transformed generator]\\label{f:moving-theorem}\nIf $\\dot\\psi=K\\psi$ and $K^*=-K$, then\n\\begin{equation}\\label{f:effective-generator}\n \\dot\\phi=K_{\\mathrm{eff}}\\phi,\\qquad\n K_{\\mathrm{eff}}=U^*KU-\\Gamma,\\qquad\n \\Gamma^*=-\\Gamma,\\quad K_{\\mathrm{eff}}^*=-K_{\\mathrm{eff}}.\n\\end{equation}\nConsequently the fixed-atlas current formula, applied to $\\phi$ and $K_{\\mathrm{eff}}$, gives the derivative of the moving physical packet weights.\n\\end{proposition}","\\begin{proof}\nDifferentiate $U^*U=\\id$ to obtain\n$\\dot U^*U+U^*\\dot U=0$. This proves $\\Gamma^*=-\\Gamma$.\nMultiplication on the right by $U^*$ gives $\\dot U^*=-\\Gamma U^*$. Therefore\n\\[\n \\dot\\phi=\\dot U^*\\psi+U^*\\dot\\psi\n =-\\Gamma\\phi+U^*KU\\phi.\n\\]\nUnitary conjugation preserves skew-adjointness. The difference of the two skew-adjoint operators in \\eqref{f:effective-generator} is skew-adjoint, so the transfer theorem applies.\n\\end{proof}","The connection $\\Gamma$ records the velocity of the frame. Omitting it would interpret a change of packet coordinates as if it were entirely a change of the physical state. In Hamiltonian notation the corresponding effective Hamiltonian is\n$H_{\\mathrm{eff}}=U^*HU-i\\Gamma$, which is self-adjoint because $\\Gamma$ is skew-adjoint.","\\begin{example}[A stationary vector viewed through a rotating split]\\label{f:rotating-frame}\nLet $\\psi(t)=e_1$, $K=0$, and\n\\[\n U(t)=\\begin{pmatrix}\\cos\\theta(t)&-\\sin\\theta(t)","\\begin{example}[A stationary vector viewed through a rotating split]\\label{f:rotating-frame}\nLet $\\psi(t)=e_1$, $K=0$, and\n\\[\n U(t)=\\begin{pmatrix}\\cos\\theta(t)&-\\sin\\theta(t)\\\\\n \\sin\\theta(t)&\\cos\\theta(t)\\end{pmatrix}.\n\\]\nWith the coordinate projectors as $P_a^{(0)}$,\n\\[\n \\phi=\\begin{pmatrix}\\cos\\theta","\\sin\\theta(t)&\\cos\\theta(t)\\end{pmatrix}.\n\\]\nWith the coordinate projectors as $P_a^{(0)}$,\n\\[\n \\phi=\\begin{pmatrix}\\cos\\theta","\\sin\\theta(t)&\\cos\\theta(t)\\end{pmatrix}.\n\\]\nWith the coordinate projectors as $P_a^{(0)}$,\n\\[\n \\phi=\\begin{pmatrix}\\cos\\theta\\\\-\\sin\\theta\\end{pmatrix},\\quad\n \\Gamma=\\dot\\theta\\begin{pmatrix}0&-1\\\\1&0\\end{pmatrix},\\quad\n K_{\\mathrm{eff}}=-\\Gamma.\n\\]\nThe moving weights are $p_1=\\cos^2\\theta$ and $p_2=\\sin^2\\theta$, with\n$J_{2\\leftarrow1}=\\dot\\theta\\sin(2\\theta)=\\dot p_2$.\nThe state has not moved in the original Hilbert space. The measured sectors have moved around it. The connection term accounts for the entire change.\n\\end{example}","1&0\\end{pmatrix},\\quad\n K_{\\mathrm{eff}}=-\\Gamma.\n\\]\nThe moving weights are $p_1=\\cos^2\\theta$ and $p_2=\\sin^2\\theta$, with\n$J_{2\\leftarrow1}=\\dot\\theta\\sin(2\\theta)=\\dot p_2$.\nThe state has not moved in the original Hilbert space. The measured sectors have moved around it. The connection term accounts for the entire change.\n\\end{example}","The same principle governs a time-dependent operator block. Write $V=\\dot UU^*$ for the frame velocity in laboratory coordinates. Then\n\\[\n \\dot P_a=[V,P_a],\\qquad\n \\frac{d}{dt}(P_bXP_a)\n =[V,P_b]XP_a+P_b\\dot XP_a+P_bX[V,P_a].\n\\]\nBoth packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and completeness, but it need not preserve self-adjointness or orthogonality. The unitary hypothesis is what keeps these transported objects within the present orthogonal atlas setting.","\\foundationfigure{05_moving_frame}{A stationary laboratory vector observed in a rotating atlas. The varying packet weights are generated by the connection term $-U^*\\dot U$ in moving coordinates.}","For this section take $\\HH=\\C^n$. A density matrix is a positive semidefinite Hermitian matrix $\\rho$ with $\\operatorname{Tr}\\rho=1$. Its probability in a sharp packet is\n\\begin{equation}\\label{f:mixed-probability}\n p_a=\\operatorname{Tr}(P_a\\rho).\n\\end{equation}\nHermiticity makes this trace real. Positivity makes it nonnegative, since cyclicity and $P_a^2=P_a$ give\n$\\operatorname{Tr}(P_a\\rho)=\\operatorname{Tr}(P_a\\rho P_a)\\ge0$.\nCompleteness gives $\\sum_ap_a=\\operatorname{Tr}\\rho=1$.\nThese are separate uses of the density-matrix assumptions.","a positive semidefinite Hermitian matrix $\\rho$ with $\\operatorname{Tr}\\rho=1$. Its probability in a sharp packet is\n\\begin{equation}\\label{f:mixed-probability}\n p_a=\\operatorname{Tr}(P_a\\rho).\n\\end{equation}\nHermiticity makes this trace real. Positivity makes it nonnegative, since cyclicity and $P_a^2=P_a$ give\n$\\operatorname{Tr}(P_a\\rho)=\\operatorname{Tr}(P_a\\rho P_a)\\ge0$.\nCompleteness gives $\\sum_ap_a=\\operatorname{Tr}\\rho=1$.\nThese are separate uses of the density-matrix assumptions.","Suppose a differentiable matrix trajectory obeys the von Neumann equation\n\\begin{equation}\\label{f:von-neumann}\n \\dot\\rho=-i[H,\\rho],\\qquad H=H^*.\n\\end{equation}\nFor its coherent transfer define\n\\begin{equation}\\label{f:mixed-current}\n J^{(\\rho)}_{b\\leftarrow a}\n =2\\operatorname{Im}\\operatorname{Tr}(P_bHP_a\\rho).\n\\end{equation}","\\begin{proposition}[Trace-current balance]\\label{f:mixed-theorem}\nFor a Hermitian $\\rho$ satisfying \\eqref{f:von-neumann},\n\\[\n \\dot p_b=\\sum_aJ^{(\\rho)}_{b\\leftarrow a},\\qquad\n J^{(\\rho)}_{b\\leftarrow a}=-J^{(\\rho)}_{a\\leftarrow b}.\n\\]\nThe subset boundary formula and exact coarse-current summation remain valid.\n\\end{proposition}","\\begin{proof}\nLet $z_{ba}=\\operatorname{Tr}(P_bHP_a\\rho)$. Taking the conjugate trace and cyclically moving the first factor to the end yields\n\\[\n \\overline{z_{ba}}=\\operatorname{Tr}(\\rho P_aHP_b)\n =\\operatorname{Tr}(P_aHP_b\\rho)=z_{ab}.\n\\]\nCompleteness and cyclicity resolve the derivative as\n\\[\n \\dot p_b=-i\\sum_a(z_{ba}-z_{ab})\n =\\sum_a2\\operatorname{Im}z_{ba}.\n\\]\nThe conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exactly as before.\n\\end{proof}","For a rank-one state $\\rho=|\\psi\\rangle\\langle\\psi|$, where the associated operator sends $x$ to $\\psi\\inner{\\psi}{x}$, one has\n$\\operatorname{Tr}(M\\rho)=\\inner{\\psi}{M\\psi}$. Substituting $M=P_bHP_a$ proves that \\eqref{f:mixed-current} is precisely the pure-state formula. No normalization is needed for that identity itself.","In the coordinate atlas, the current is $2\\operatorname{Im}(H_{ba}\\rho_{ab})$. It depends on off-diagonal coherence. A useful test is to compare\n\\[\n \\rho_+=\\frac12\\begin{pmatrix}1&i","In the coordinate atlas, the current is $2\\operatorname{Im}(H_{ba}\\rho_{ab})$. It depends on off-diagonal coherence. A useful test is to compare\n\\[\n \\rho_+=\\frac12\\begin{pmatrix}1&i\\\\-i&1\\end{pmatrix},\\qquad\n \\rho_-=\\frac12\\begin{pmatrix}1&-i\\\\i&1\\end{pmatrix},\\qquad\n \\rho_0=\\frac12\\begin{pmatrix}1&0\\\\0&1\\end{pmatrix}.\n\\]\nAll three are density matrices with packet probabilities $(1/2,1/2)$. For $H=g\\left(\\begin{smallmatrix}0&1","0&1\\end{pmatrix}.\n\\]\nAll three are density matrices with packet probabilities $(1/2,1/2)$. For $H=g\\left(\\begin{smallmatrix}0&1","0&1\\end{pmatrix}.\n\\]\nAll three are density matrices with packet probabilities $(1/2,1/2)$. For $H=g\\left(\\begin{smallmatrix}0&1\\\\1&0\\end{smallmatrix}\\right)$, however, their currents $J_{2\\leftarrow1}$ are respectively $g,-g,0$. The same instantaneous populations can therefore have different derivatives under the same Hamiltonian. A closed equation involving only those two populations cannot represent all three coherent states.","1&0\\end{smallmatrix}\\right)$, however, their currents $J_{2\\leftarrow1}$ are respectively $g,-g,0$. The same instantaneous populations can therefore have different derivatives under the same Hamiltonian. A closed equation involving only those two populations cannot represent all three coherent states.","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the par","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace.","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating","Let $\\HH_A$ and $\\HH_B$ be finite-dimensional Hilbert spaces with finite orthogonal atlases $(P_a)$ and $(Q_\\mu)$. Their tensor atlas is\n\\[\n R_{a\\mu}=P_a\\otimes Q_\\mu.\n\\]\nIt is orthogonal because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative form…","al because tensor multiplication gives\n$R_{a\\mu}R_{b\\nu}=\\delta_{ab}\\delta_{\\mu\\nu}R_{a\\mu}$.\nIts sum is $\\id_A\\otimes\\id_B$. For a density matrix $\\rho_{AB}$, define\n\\begin{equation}\\label{f:joint-marginal}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","l}\n p_{a\\mu}=\\operatorname{Tr}(R_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","_{a\\mu}\\rho_{AB}),\\qquad\n p_a^A=\\sum_\\mu p_{a\\mu},\\quad p_\\mu^B=\\sum_a p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","p_{a\\mu}.\n\\end{equation}\nThese marginalizations are exact atlas coarsenings. For example,\n\\[\n p_a^A=\\operatorname{Tr}\\bigl((P_a\\otimes\\id_B)\\rho_{AB}\\bigr)\n =\\operatorname{Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","Tr}(P_a\\rho_A),\\qquad \\rho_A=\\operatorname{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","e{Tr}_B\\rho_{AB}.\n\\]\nChoose full orthonormal bases of the two Hilbert spaces, with $(e_k)_k$ indexing the basis of $\\HH_B$. In the resulting product basis the partial trace has entries\n$(\\rho_A)_{ij}=\\sum_k(\\rho_{AB})_{(i,k),(j,k)}$.\nThis explicit formula shows that it preserves Hermiticity and trace. Positivity follows by evaluating the quadratic form on $x$ and summing the nonnegative forms of $x\\otimes e_k$. Thus the reduced matrix is itself a density matrix.","For a product state $\\rho_{AB}=\\sigma_A\\otimes\\tau_B$, the trace factorizes and $p_{a\\mu}=p_a^Ap_\\mu^B$. General states need not have that property. The normalized vector\n$\\Psi=(e_0\\otimes e_0+e_1\\otimes e_1)/\\sqrt2$ has joint probabilities\n\\[\n (p_{a\\mu})=\\begin{pmatrix}1/2&0","For a product state $\\rho_{AB}=\\sigma_A\\otimes\\tau_B$, the trace factorizes and $p_{a\\mu}=p_a^Ap_\\mu^B$. General states need not have that property. The normalized vector\n$\\Psi=(e_0\\otimes e_0+e_1\\otimes e_1)/\\sqrt2$ has joint probabilities\n\\[\n (p_{a\\mu})=\\begin{pmatrix}1/2&0\\\\0&1/2\\end{pmatrix},\n \\qquad p^A=p^B=(1/2,1/2).\n\\]\nThe product density matrix $(\\id_A/2)\\otimes(\\id_B/2)$ has the same marginals but every joint entry is $1/4$. Marginals therefore fail to reconstruct even the joint measurement distribution. Moreover, the classical mixture of $e_0\\otimes e_0$ and $e_1\\otimes e_1$ has the same joint table as $\\Psi$ but lacks its off-diagonal coherence. A single atlas table does not reconstruct the full density matrix or certify entanglement on its own.","0&1/2\\end{pmatrix},\n \\qquad p^A=p^B=(1/2,1/2).\n\\]\nThe product density matrix $(\\id_A/2)\\otimes(\\id_B/2)$ has the same marginals but every joint entry is $1/4$. Marginals therefore fail to reconstruct even the joint measurement distribution. Moreover, the classical mixture of $e_0\\otimes e_0$ and $e_1\\otimes e_1$ has the same joint table as $\\Psi$ but lacks its off-diagonal coherence. A single atlas table does not reconstruct the full density matrix or certify entanglement on its own.","Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis $(e_a)_{a\\in I}$ of $\\C^n$, use its rank-one projectors, and choose rates $\\gamma_{ba}\\ge0$ for $b\\ne a$. Set diagonal rates to zero for convenience and define\n\\begin{equation}\\label{f:jump-operators}\n L_{ba}=\\sqrt{\\gamma_{ba}}\\,|e_b\\rangle\\langle e_a|,\n \\qquad\n \\mathcal D_L(\\rho)=L\\rho L^*-\tfrac12(L^*L\\rho+\\rho L^*L).\n\\end{equation}\nConsider a supplied differentiable matrix trajectory satisfying\n\\begin{equation}\\label{f:lindblad}\n \\dot\\rho=\\sum_{b\\ne a}\\mathcal D_{L_{ba}}(\\rho).\n\\end{equation}\nThis choice of jumps is tied to the selected basis. A general dissipative operator or a higher-rank atlas does not automatically produce the same closed rate equation.","of $\\C^n$, use its rank-one projectors, and choose rates $\\gamma_{ba}\\ge0$ for $b\\ne a$. Set diagonal rates to zero for convenience and define\n\\begin{equation}\\label{f:jump-operators}\n L_{ba}=\\sqrt{\\gamma_{ba}}\\,|e_b\\rangle\\langle e_a|,\n \\qquad\n \\mathcal D_L(\\rho)=L\\rho L^*-\tfrac12(L^*L\\rho+\\rho L^*L).\n\\end{equation}\nConsider a supplied differentiable matrix trajectory satisfying\n\\begin{equation}\\label{f:lindblad}\n \\dot\\rho=\\sum_{b\\ne a}\\mathcal D_{L_{ba}}(\\rho).\n\\end{equation}\nThis choice of jumps is tied to the selected basis. A general dissipative operator or a higher-rank atlas does not automatically produce the same closed rate equation.","\\begin{proposition}[Population gain and loss]\\label{f:jump-theorem}\nThe diagonal populations $p_b=\\rho_{bb}$ of a Hermitian solution satisfy\n\\begin{equation}\\label{f:jump-current}\n \\dot p_b=\\sum_aF_{b\\leftarrow a},\\qquad\n F_{b\\leftarrow a}=\\gamma_{ba}p_a-\\gamma_{ab}p_b,\n \\qquad F_{b\\leftarrow a}=-F_{a\\leftarrow b}.\n\\end{equation}\nThe total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.\n\\end{proposition}","\\begin{proof}\nFor the one transition $a\\to b$,\n\\[\n L_{ba}\\rho L_{ba}^*=\\gamma_{ba}p_aP_b,\\qquad\n L_{ba}^*L_{ba}=\\gamma_{ba}P_a.\n\\]\nIts diagonal gain is $\\gamma_{ba}p_a$ at $b$, and the two halves of the anticommutator remove that same amount at $a$. No other diagonal entry changes from this one jump. Sum over ordered transitions to obtain \\eqref{f:jump-current}. Antisymmetry is immediate, and the finite boundary cancellation applies.\n\\end{proof}","In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For $i\\ne j$, let $r_i=\\sum_{b\\ne i}\\gamma_{bi}$ be the total rate leaving $i$. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently\n\\begin{equation}\\label{f:coherence-decay}\n \\dot\\rho_{ij}=-\\tfrac12(r_i+r_j)\\rho_{ij}.\n\\end{equation}\nFor fixed rates, $\\rho_{ij}(t)=e^{-(r_i+r_j)t/2}\\rho_{ij}(0)$. This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable ways.","on closes without knowing the coherences. The coherences nevertheless have their own dynamics. For $i\\ne j$, let $r_i=\\sum_{b\\ne i}\\gamma_{bi}$ be the total rate leaving $i$. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently\n\\begin{equation}\\label{f:coherence-decay}\n \\dot\\rho_{ij}=-\\tfrac12(r_i+r_j)\\rho_{ij}.\n\\end{equation}\nFor fixed rates, $\\rho_{ij}(t)=e^{-(r_i+r_j)t/2}\\rho_{ij}(0)$. This decay law belongs to the stated rank-one jump family. Additional Hamiltonian or dephasing terms alter it in identifiable ways.","\\begin{example}[Two-state relaxation]\nTake $\\gamma_{21}=\\gamma_{12}=k>0$ and $p_1+p_2=1$. Then\n\\[\n \\dot p_1=k(1-2p_1),\\qquad\n p_1(t)=\\tfrac12+\\bigl(p_1(0)-\\tfrac12\\bigr)e^{-2kt},\n \\qquad \\rho_{12}(t)=e^{-kt}\\rho_{12}(0).\n\\]\nThe populations relax toward equal weights. In the coherent exchange of Example~\\ref{f:rabi}, the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding $-i[H,\\rho]$ to \\eqref{f:lindblad} adds the coherent current to the jump current. The resulting population equation then generally depends on coherence again.\n\\end{example}","\\dot p_1=k(1-2p_1),\\qquad\n p_1(t)=\\tfrac12+\\bigl(p_1(0)-\\tfrac12\\bigr)e^{-2kt},\n \\qquad \\rho_{12}(t)=e^{-kt}\\rho_{12}(0).\n\\]\nThe populations relax toward equal weights. In the coherent exchange of Example~\\ref{f:rabi}, the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but that common bookkeeping does not make their mechanisms identical. Adding $-i[H,\\rho]$ to \\eqref{f:lindblad} adds the coherent current to the jump current. The resulting population equation then generally depends on coherence again.\n\\end{example}","The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells $A,B$, suppose\n\\[\n \\sum_{b\\in C_B}\\gamma_{ba}=q_{BA}\n \\quad\\text{for every }a\\in C_A.\n\\]\nThen summing the fine gains gives $q_{BA}p_A^{\\mathrm c}$, so\n$\\dot p_B^{\\mathrm c}=\\sum_{A\\ne B}(q_{BA}p_A^{\\mathrm c}-q_{AB}p_B^{\\mathrm c})$.\nWithout this equality of outgoing rates to each destination cell, the fine population distribution inside a source cell matters. This direct consequence of the gain sum shows precisely what an autonomous coarse model asks in addition to current conservation.","\\foundationfigure{06_density_and_jumps}{Coherent exchange and the specified symmetric rank-one jump dynamics have different population laws. The equal-population coherent states $\\rho_+$ and $\\rho_-$ give opposite initial derivatives of $p_2$, while the classical jump equilibrium has zero derivative.}","An antisymmetric current also organizes derivatives of scalar functionals of the packet distribution. Let $p_a(t)>0$ be differentiable, normalized, and obey $\\dot p_b=\\sum_aJ_{b\\leftarrow a}$ with $J$ antisymmetric. For a differentiable real function $f$ on the positive half-line,\n\\begin{equation}\\label{f:functional-current}\n \\frac{d}{dt}\\sum_b f(p_b)\n =\\frac12\\sum_{b,a}J_{b\\leftarrow a}\n       \\bigl(f'(p_b)-f'(p_a)\\bigr).\n\\end{equation}\nTo prove it, apply the ordinary chain rule and substitute the balance equation. Exchange the two indices in a second copy of the sum and use antisymmetry. Averaging the two expressions yields \\eqref{f:functional-current}.","For Shannon entropy $S(p)=-\\sum_ap_a\\log p_a$, this gives\n\\begin{equation}\\label{f:entropy-current}\n \\dot S=-\\frac12\\sum_{b,a}J_{b\\leftarrow a}\\log\\frac{p_b}{p_a}.\n\\end{equation}\nThe constant derivative term cancels because the total current is zero. For a real $q$, the power sum $M_q=\\sum_ap_a^q$ has derivative\n\\[\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n\\]\nFor $q\\ne1$, the R\\'enyi entropy $H_q=\\log M_q/(1-q)$ therefore satisfies\n$\\dot H_q=\\dot M_q/((1-q)M_q)$. Strict positivity makes the powers and logarithmic derivatives valid, and $M_q>0$ when the finite index set is nonempty. Boundary points with zero populations require their own limiting argument and are not covered by these derivative statements as","For Shannon entropy $S(p)=-\\sum_ap_a\\log p_a$, this gives\n\\begin{equation}\\label{f:entropy-current}\n \\dot S=-\\frac12\\sum_{b,a}J_{b\\leftarrow a}\\log\\frac{p_b}{p_a}.\n\\end{equation}\nThe constant derivative term cancels because the total current is zero. For a real $q$, the power sum $M_q=\\sum_ap_a^q$ has derivative\n\\[\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n\\]\nFor $q\\ne1$, the R\\'enyi entropy $H_q=\\log M_q/(1-q)$ therefore satisfies\n$\\dot H_q=\\dot M_q/((1-q)M_q)$. Strict positivity makes the powers and logarithmic derivatives valid, and $M_q>0$ when the finite index set is nonempty. Boundary points with zero populations require their own limiting argument and are not covered by these derivative statements as written.",")=-\\sum_ap_a\\log p_a$, this gives\n\\begin{equation}\\label{f:entropy-current}\n \\dot S=-\\frac12\\sum_{b,a}J_{b\\leftarrow a}\\log\\frac{p_b}{p_a}.\n\\end{equation}\nThe constant derivative term cancels because the total current is zero. For a real $q$, the power sum $M_q=\\sum_ap_a^q$ has derivative\n\\[\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n\\]\nFor $q\\ne1$, the R\\'enyi entropy $H_q=\\log M_q/(1-q)$ therefore satisfies\n$\\dot H_q=\\dot M_q/((1-q)M_q)$. Strict positivity makes the powers and logarithmic derivatives valid, and $M_q>0$ when the finite index set is nonempty. Boundary points with zero populations require their own limiting argument and are not covered by these derivative statements as written.","equation}\\label{f:entropy-current}\n \\dot S=-\\frac12\\sum_{b,a}J_{b\\leftarrow a}\\log\\frac{p_b}{p_a}.\n\\end{equation}\nThe constant derivative term cancels because the total current is zero. For a real $q$, the power sum $M_q=\\sum_ap_a^q$ has derivative\n\\[\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n\\]\nFor $q\\ne1$, the R\\'enyi entropy $H_q=\\log M_q/(1-q)$ therefore satisfies\n$\\dot H_q=\\dot M_q/((1-q)M_q)$. Strict positivity makes the powers and logarithmic derivatives valid, and $M_q>0$ when the finite index set is nonempty. Boundary points with zero populations require their own limiting argument and are not covered by these derivative statements as written.","uation}\nThe constant derivative term cancels because the total current is zero. For a real $q$, the power sum $M_q=\\sum_ap_a^q$ has derivative\n\\[\n \\dot M_q=\\frac q2\\sum_{b,a}J_{b\\leftarrow a}\n (p_b^{q-1}-p_a^{q-1}).\n\\]\nFor $q\\ne1$, the R\\'enyi entropy $H_q=\\log M_q/(1-q)$ therefore satisfies\n$\\dot H_q=\\dot M_q/((1-q)M_q)$. Strict positivity makes the powers and logarithmic derivatives valid, and $M_q>0$ when the finite index set is nonempty. Boundary points with zero populations require their own limiting argument and are not covered by these derivative statements as written.","For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let $\\pi_a>0$ be a fixed probability distribution, and assume\n\\begin{equation}\\label{f:detailed-balance}\n \\gamma_{ba}\\pi_a=\\gamma_{ab}\\pi_b.\n\\end{equation}\nDefine relative densities $u_a=p_a/\\pi_a$ and conductances $c_{ba}=\\gamma_{ba}\\pi_a$. Then $c_{ba}=c_{ab}\\ge0$ and\n$F_{b\\leftarrow a}=c_{ba}(u_a-u_b)$.","\\Needspace{13\\baselineskip}\n\\begin{theorem}[Relative-entropy dissipation]\\label{f:relative-entropy}\nFor a strictly positive solution of the finite jump balance and a fixed positive reference satisfying \\eqref{f:detailed-balance},\n\\begin{align}\\label{f:dissipation}\n D(p\\Vert\\pi)&=\\sum_ap_a\\log\\frac{p_a}{\\pi_a},\\\\\n \\frac{d}{dt}D(p\\Vert\\pi)\n &=-\\frac12\\sum_{b,a}c_{ba}(u_b-u_a)(\\log u_b-\\log u_a)\\le0.\n \\notag\n\\end{align}\n\\end{theorem}","\\begin{proof}\nDifferentiate with $\\pi$ fixed. The terms without logarithms sum to zero. Pairing opposite currents leaves\n$\\tfrac12\\sum_{b,a}F_{b\\leftarrow a}(\\log u_b-\\log u_a)$.\nInsert the conductance expression for $F$. The logarithm is increasing on $(0,\\infty)$, so each product\n$(u_b-u_a)(\\log u_b-\\log u_a)$ is nonnegative. Every conductance is nonnegative, proving the inequality.\n\\end{proof}","The conductance form includes inactive edges without evaluating a logarithm of a zero flux. When $c_{ba}=0$, its contribution is simply zero. When $c_{ba}>0$, a summand vanishes exactly when $u_b=u_a$. Thus zero instantaneous dissipation means that the relative density is constant on each connected component of the positive-conductance graph. If that graph is connected, normalization forces the constant to be one and $p=\\pi$. This identifies the equality case. The derivative calculation by itself does not provide a convergence rate to equilibrium.","ncludes inactive edges without evaluating a logarithm of a zero flux. When $c_{ba}=0$, its contribution is simply zero. When $c_{ba}>0$, a summand vanishes exactly when $u_b=u_a$. Thus zero instantaneous dissipation means that the relative density is constant on each connected component of the positive-conductance graph. If that graph is connected, normalization forces the constant to be one and $p=\\pi$. This identifies the equality case. The derivative calculation by itself does not provide a convergence rate to equilibrium.","Throughout this chapter, inner products are conjugate linear in the first variable. A bounded state generator is denoted by $K$, with $K^*=-K$. When a Hamiltonian is present, $K=-iH$. The receiving label is written first in $J_{ba}$.","\\begin{definition}[Finite effect atlas]\nLet $\\HH$ be a complex Hilbert space and let $A$ be a finite label set. An effect atlas is a family of bounded operators satisfying\n\\[\n F_a=F_a^*\\geq0,\\qquad \\sum_{a\\in A}F_a=I.\n\\]\nIts weight in a state $\\psi$ is $p_a=\\inner{\\psi}{F_a\\psi}$. The partition identity implies $0\\leq F_a\\leq I$, so $p_a\\geq0$ and $\\sum_a p_a=\\norm\\psi^2$.\n\\end{definition}","For a sharp atlas, $F_a=P_a$ and $p_a=\\norm{P_a\\psi}^2$. In general, $p_a$ is not $\\norm{F_a\\psi}^2$. The latter quantity is $\\inner{\\psi}{F_a^2\\psi}$ and therefore loses precisely the positive defect $F_a-F_a^2$.","\\begin{example}[Overlapping two-channel weights]\nOn $\\C^2$, take\n\\[\n F_+=\\begin{pmatrix}4/5&0","\\begin{example}[Overlapping two-channel weights]\nOn $\\C^2$, take\n\\[\n F_+=\\begin{pmatrix}4/5&0\\\\0&1/5\\end{pmatrix},\\qquad\n F_-=\\begin{pmatrix}1/5&0\\\\0&4/5\\end{pmatrix},\\qquad \\psi=\\binom10.\n\\]\nThe true weights are $4/5$ and $1/5$. Squared effect-vector norms would instead give $16/25$ and $1/25$, whose sum is $17/25$. The state still satisfies $F_+\\psi+F_-\\psi=\\psi$, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are different assertions.\n\\end{example}","0&4/5\\end{pmatrix},\\qquad \\psi=\\binom10.\n\\]\nThe true weights are $4/5$ and $1/5$. Squared effect-vector norms would instead give $16/25$ and $1/25$, whose sum is $17/25$. The state still satisfies $F_+\\psi+F_-\\psi=\\psi$, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are different assertions.\n\\end{example}","Suppose that a differentiable state curve satisfies $\\dot\\psi=K\\psi$. Define\n\\begin{equation}\n J^{F}_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}\n             -\\operatorname{Re}\\inner{F_a\\psi}{KF_b\\psi}.\n \\label{eq:foundation-effect-current}\n\\end{equation}\nThis current is antisymmetric directly from its definition. Skew-adjointness also gives\n\\[\n J^F_{ba}=2\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}.\n\\]\nThe same bilinear expression occurs for sharp packets, but its interpretation now concerns expectation weights.","\\begin{proposition}[Effect continuity]\nUnder these hypotheses,\n\\[\n \\dot p_b=\\sum_a J^F_{ba},\\qquad \\sum_b\\dot p_b=0.\n\\]\nIf $F_b=F_b(t)$ is differentiable in operator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Ant","\\begin{proposition}[Effect continuity]\nUnder these hypotheses,\n\\[\n \\dot p_b=\\sum_a J^F_{ba},\\qquad \\sum_b\\dot p_b=0.\n\\]\nIf $F_b=F_b(t)$ is differentiable in operator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels","\\begin{proposition}[Effect continuity]\nUnder these hypotheses,\n\\[\n \\dot p_b=\\sum_a J^F_{ba},\\qquad \\sum_b\\dot p_b=0.\n\\]\nIf $F_b=F_b(t)$ is differentiable in operator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","$F_b=F_b(t)$ is differentiable in operator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","s differentiable in operator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","perator norm, the first identity becomes\n\\[\n \\dot p_b=s_b+\\sum_a J^F_{ba},\\qquad\n s_b=\\inner{\\psi}{\\dot F_b\\psi},\\qquad \\sum_b s_b=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nSelf-adjointness of $F_b$ and differentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","rentiation give\n\\[\n \\dot p_b=2\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","eratorname{Re}\\inner{F_b\\psi}{K\\psi}\n\\]\nfor a fixed effect. Summing the two terms of \\eqref{eq:foundation-effect-current} over $a$ and using $\\sum_aF_a=I$ gives\n\\[\n \\sum_aJ^F_{ba}=\\operatorname{Re}\\inner{F_b\\psi}{K\\psi}\n       -\\operatorname{Re}\\inner{\\psi}{KF_b\\psi}.\n\\]\nThe second real part is the negative of the first. The moving case adds the product-rule term $s_b$. Its total is zero because $\\sum_b\\dot F_b=0$. Antisymmetry cancels the finite double sum of currents.\n\\end{proof}","If a map $r:A\\to B$ groups the labels, the coarse effect is $G_\\beta=\\sum_{r(a)=\\beta}F_a$. Bilinearity proves\n\\[\n p^{\\rm c}_\\beta=\\sum_{r(a)=\\beta}p_a,\\qquad\n J^{\\rm c}_{\\beta\\alpha}\n =\\sum_{\\substack{r(b)=\\beta","If a map $r:A\\to B$ groups the labels, the coarse effect is $G_\\beta=\\sum_{r(a)=\\beta}F_a$. Bilinearity proves\n\\[\n p^{\\rm c}_\\beta=\\sum_{r(a)=\\beta}p_a,\\qquad\n J^{\\rm c}_{\\beta\\alpha}\n =\\sum_{\\substack{r(b)=\\beta\\\\r(a)=\\alpha}}J^F_{ba}.\n\\]\nNo orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to $J^F$ without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.","r(a)=\\alpha}}J^F_{ba}.\n\\]\nNo orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to $J^F$ without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.","Choose Hilbert spaces $\\KK_a$ and bounded maps $V_a:\\HH\\to\\KK_a$ such that\n\\[\n \\sum_aV_a^*V_a=I.\n\\]\nThe combined analysis map $V:\\HH\\to\\KK=\\bigoplus_a\\KK_a$ is $V\\psi=(V_a\\psi)_a$. Its adjoint is synthesis, $V^*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measu","Choose Hilbert spaces $\\KK_a$ and bounded maps $V_a:\\HH\\to\\KK_a$ such that\n\\[\n \\sum_aV_a^*V_a=I.\n\\]\nThe combined analysis map $V:\\HH\\to\\KK=\\bigoplus_a\\KK_a$ is $V\\psi=(V_a\\psi)_a$. Its adjoint is synthesis, $V^*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this","Choose Hilbert spaces $\\KK_a$ and bounded maps $V_a:\\HH\\to\\KK_a$ such that\n\\[\n \\sum_aV_a^*V_a=I.\n\\]\nThe combined analysis map $V:\\HH\\to\\KK=\\bigoplus_a\\KK_a$ is $V\\psi=(V_a\\psi)_a$. Its adjoint is synthesis, $V^*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this algebraic dilation.","H\\to\\KK_a$ such that\n\\[\n \\sum_aV_a^*V_a=I.\n\\]\nThe combined analysis map $V:\\HH\\to\\KK=\\bigoplus_a\\KK_a$ is $V\\psi=(V_a\\psi)_a$. Its adjoint is synthesis, $V^*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this algebraic dilation.","s synthesis, $V^*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this algebraic dilation.","*(y_a)_a=\\sum_aV_a^*y_a$. Consequently\n\\[\n V^*V=I,\\qquad R=VV^*,\\qquad R^*=R^2=R.\n\\]\nHere $R$ projects onto the closed range of the isometry $V$. The range need not fill $\\KK$. If $\\Pi_a$ is the coordinate projector, the compressed effect is\n\\[\n F_a=V^*\\Pi_aV=V_a^*V_a,\\qquad\n p_a=\\norm{V_a\\psi}^2=\\norm{\\Pi_aV\\psi}^2.\n\\]\nEvery finite effect atlas admits this construction by taking $\\KK_a=\\HH$ and $V_a=F_a^{1/2}$. Positivity supplies the bounded positive square root. No choice of a measurement outcome or post-measurement state is being made by this algebraic dilation.","For $X\\in\\mathcal B(\\HH)$, its channel block is $X_{ba}^{V}=V_bXV_a^*$. Inserting the partition on both sides proves\n\\begin{equation}\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unshar","For $X\\in\\mathcal B(\\HH)$, its channel block is $X_{ba}^{V}=V_bXV_a^*$. Inserting the partition on both sides proves\n\\begin{equation}\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an","For $X\\in\\mathcal B(\\HH)$, its channel block is $X_{ba}^{V}=V_bXV_a^*$. Inserting the partition on both sides proves\n\\begin{equation}\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ di","For $X\\in\\mathcal B(\\HH)$, its channel block is $X_{ba}^{V}=V_bXV_a^*$. Inserting the partition on both sides proves\n\\begin{equation}\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ directly.","bXV_a^*$. Inserting the partition on both sides proves\n\\begin{equation}\n X=\\sum_{a,b}V_b^*X_{ba}^{V}V_a.\n \\label{eq:foundation-channel-reconstruction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ directly.","truction}\n\\end{equation}\nThe lifted operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ directly.","ed operator $\\widetilde X=VXV^*$ is multiplicative on lifted operators, since $V^*V=I$. Its identity is $R$, not $I_{\\KK}$. Conversely, compressing arbitrary ambient operators need not preserve multiplication:\n\\begin{equation}\n V^*ABV-(V^*AV)(V^*BV)=V^*A(I-R)BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ directly.","BV.\n \\label{eq:foundation-compression-defect}\n\\end{equation}\nThis is the exact missing-range term. Setting $A=B=\\Pi_a$ gives\n\\begin{equation}\n \\inner{\\psi}{(F_a-F_a^2)\\psi}\n =\\norm{(I-R)\\Pi_aV\\psi}^2.\n \\label{eq:foundation-unsharpness}\n\\end{equation}\nThus the failure of an effect to be sharp measures how much its coordinate projection leaves the embedded physical range. This identity also proves $F_a-F_a^2\\geq0$ directly.","The dynamics respects the same construction. Put $\\widetilde K=VKV^*$. It is skew-adjoint, and $\\frac{d}{dt}V\\psi=\\widetilde K V\\psi$. Moving adjoints across the inner product shows\n\\[\n 2\\operatorname{Re}\\inner{\\Pi_bV\\psi}{\\widetilde K\\Pi_aV\\psi}\n =2\\operatorname{Re}\\inner{F_b\\psi}{KF_a\\psi}=J^F_{ba}.\n\\]\nThe positive-effect current is therefore exactly the sharp current of this lifted flow evaluated on the embedded state. It is not obtained by treating arbitrary compression as an algebra homomorphism.","\\foundationfigure{08_positive_effects}{An explicit two-effect example with $F_+=\\operatorname{diag}(4/5,1/5)$ and $F_-=I-F_+$. The square-root channel isometry preserves the effect probabilities in an enlarged sharp resolution.}","Let $(P_a)_{a\\in A}$ be a countable orthogonal family of bounded projections with strong sum $I$. For finite $F\\subset A$, write $S_F=\\sum_{a\\in F}P_a$. Strong completeness means $S_Fx\\to x$ for each fixed $x\\in\\HH$, over the directed set of finite subsets. Orthogonality gives\n\\[\n \\norm{S_Fx}^2=\\sum_{a\\in F}\\norm{P_ax}^2,\\qquad\n \\sum_a\\norm{P_ax}^2=\\norm x^2.\n\\]\nThese are Hilbert-space statements. They contain no unbounded operator.","Now let $H:D(H)\\subset\\HH\\to\\HH$ be a densely defined symmetric operator. To write $HP_a\\psi$ for every domain state, require\n\\begin{equation}\n P_aD(H)\\subset D(H).\n \\label{eq:foundation-domain-preserve}\n\\end{equation}\nDefine $\\norm x_H^2=\\norm x^2+\\norm{Hx}^2$. A domain-compatible atlas is graph compatible when\n\\begin{equation}\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n\\end{equation}\nEquivalently, both $S_Fx\\to x$ and $HS_Fx\\to Hx$ hold in $\\HH$. The second convergence does not follow from the first for an unbounded $H$.","ite $HP_a\\psi$ for every domain state, require\n\\begin{equation}\n P_aD(H)\\subset D(H).\n \\label{eq:foundation-domain-preserve}\n\\end{equation}\nDefine $\\norm x_H^2=\\norm x^2+\\norm{Hx}^2$. A domain-compatible atlas is graph compatible when\n\\begin{equation}\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n\\end{equation}\nEquivalently, both $S_Fx\\to x$ and $HS_Fx\\to Hx$ hold in $\\HH$. The second convergence does not follow from the first for an unbounded $H$.","or every domain state, require\n\\begin{equation}\n P_aD(H)\\subset D(H).\n \\label{eq:foundation-domain-preserve}\n\\end{equation}\nDefine $\\norm x_H^2=\\norm x^2+\\norm{Hx}^2$. A domain-compatible atlas is graph compatible when\n\\begin{equation}\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n\\end{equation}\nEquivalently, both $S_Fx\\to x$ and $HS_Fx\\to Hx$ hold in $\\HH$. The second convergence does not follow from the first for an unbounded $H$.","require\n\\begin{equation}\n P_aD(H)\\subset D(H).\n \\label{eq:foundation-domain-preserve}\n\\end{equation}\nDefine $\\norm x_H^2=\\norm x^2+\\norm{Hx}^2$. A domain-compatible atlas is graph compatible when\n\\begin{equation}\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n\\end{equation}\nEquivalently, both $S_Fx\\to x$ and $HS_Fx\\to Hx$ hold in $\\HH$. The second convergence does not follow from the first for an unbounded $H$.","reserve}\n\\end{equation}\nDefine $\\norm x_H^2=\\norm x^2+\\norm{Hx}^2$. A domain-compatible atlas is graph compatible when\n\\begin{equation}\n S_Fx\\longrightarrow x\\quad\\text{in the graph norm for every }x\\in D(H).\n \\label{eq:foundation-graph-compatible}\n\\end{equation}\nEquivalently, both $S_Fx\\to x$ and $HS_Fx\\to Hx$ hold in $\\HH$. The second convergence does not follow from the first for an unbounded $H$.","\\begin{example}[A small state can have a large generator image]\nOn $\\ell^2(\\N_{\\geq1})$, let $He_n=ne_n$ on its maximal domain. The finitely supported vectors $x_N=e_N/N$ satisfy $\\norm{x_N}=1/N\\to0$, while $\\norm{Hx_N}=1$. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because $Hx_N=e_N$ has no norm limit.\n\\end{example}","Let $(\\lambda_n)$ be any real sequence and define\n\\[\n D(H_\\lambda)=\\left\\{c\\in\\ell^2:\n       \\sum_n\\lambda_n^2|c_n|^2<\\infty\\right\\},\\qquad\n (H_\\lambda c)_n=\\lambda_nc_n.\n\\]\nThe operator is dense because it contains every finitely supported sequence. It is symmetric by the reality of the multipliers. Its domain is maximal for this coordinate action.","\\begin{proposition}[Self-adjointness of the diagonal realization]\nThe operator $H_\\lambda$ is self-adjoint. Its coordinate cutoffs converge in graph norm on $D(H_\\lambda)$.\n\\end{proposition}\n\\begin{proof}\nIf $y\\in D(H_\\lambda^*)$ and $H_\\lambda^*y=z$, testing the adjoint identity against $e_n$ gives $z_n=\\lambda_ny_n$. Since $z\\in\\ell^2$, the sequence $y$ belongs to $D(H_\\lambda)$. Symmetry gives the opposite inclusion of domains, and the actions agree there. For $S_Nc=\\sum_{n\\leq N}c_ne_n$,\n\\[\n \\norm{c-S_Nc}_{H_\\lambda}^2\n =\\sum_{n>N}(1+\\lambda_n^2)|c_n|^2\\longrightarrow0.\n\\]\n\\end{proof}","operator $H_\\lambda$ is self-adjoint. Its coordinate cutoffs converge in graph norm on $D(H_\\lambda)$.\n\\end{proposition}\n\\begin{proof}\nIf $y\\in D(H_\\lambda^*)$ and $H_\\lambda^*y=z$, testing the adjoint identity against $e_n$ gives $z_n=\\lambda_ny_n$. Since $z\\in\\ell^2$, the sequence $y$ belongs to $D(H_\\lambda)$. Symmetry gives the opposite inclusion of domains, and the actions agree there. For $S_Nc=\\sum_{n\\leq N}c_ne_n$,\n\\[\n \\norm{c-S_Nc}_{H_\\lambda}^2\n =\\sum_{n>N}(1+\\lambda_n^2)|c_n|^2\\longrightarrow0.\n\\]\n\\end{proof}","adjoint. Its coordinate cutoffs converge in graph norm on $D(H_\\lambda)$.\n\\end{proposition}\n\\begin{proof}\nIf $y\\in D(H_\\lambda^*)$ and $H_\\lambda^*y=z$, testing the adjoint identity against $e_n$ gives $z_n=\\lambda_ny_n$. Since $z\\in\\ell^2$, the sequence $y$ belongs to $D(H_\\lambda)$. Symmetry gives the opposite inclusion of domains, and the actions agree there. For $S_Nc=\\sum_{n\\leq N}c_ne_n$,\n\\[\n \\norm{c-S_Nc}_{H_\\lambda}^2\n =\\sum_{n>N}(1+\\lambda_n^2)|c_n|^2\\longrightarrow0.\n\\]\n\\end{proof}","For positive $\\lambda_n$, the quadratic-form domain instead uses $\\sum_n\\lambda_n|c_n|^2<\\infty$. This is the domain of $H_\\lambda^{1/2}$, not generally the domain of $H_\\lambda$. With $\\lambda_n=n$ and $c_n=n^{-s}$, the three thresholds are\n\\[\n \\begin{array}{c|c}\n \\text{membership}&\\text{condition}","For positive $\\lambda_n$, the quadratic-form domain instead uses $\\sum_n\\lambda_n|c_n|^2<\\infty$. This is the domain of $H_\\lambda^{1/2}$, not generally the domain of $H_\\lambda$. With $\\lambda_n=n$ and $c_n=n^{-s}$, the three thresholds are\n\\[\n \\begin{array}{c|c}\n \\text{membership}&\\text{condition}\\\\ \\hline\n \\ell^2&s>1/2\\\\\n D(H_\\lambda^{1/2})&s>1\\\\\n D(H_\\lambda)&s>3/2.\n \\end{array}\n\\]\nEach equality case diverges harmonically. Thus a state with finite mean positive energy can still lack the Hilbert-space derivative required by the Schr\\\"odinger equation. The harmonic oscillator example has $\\lambda_n=n+1/2$ for $n\\geq0$ and the same asymptotic thresholds after replacing $n$ by $n+1$ in the coefficients.","D(H_\\lambda)&s>3/2.\n \\end{array}\n\\]\nEach equality case diverges harmonically. Thus a state with finite mean positive energy can still lack the Hilbert-space derivative required by the Schr\\\"odinger equation. The harmonic oscillator example has $\\lambda_n=n+1/2$ for $n\\geq0$ and the same asymptotic thresholds after replacing $n$ by $n+1$ in the coefficients.","\\foundationfigure[H]{10_weighted_domains}{For $He_n=ne_n$, $n\\geq1$, coefficients $a_n=n^{-s}$ encounter three different thresholds: state norm, quadratic-form domain, and operator domain. The finite partial-sum curves illustrate the analytic $p$-series criteria.}","Suppose a supplied curve $\\psi(t)\\in D(H)$ is differentiable as a curve in $\\HH$ and satisfies $\\dot\\psi=-iH\\psi$. For a domain-preserving atlas, put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_","Suppose a supplied curve $\\psi(t)\\in D(H)$ is differentiable as a curve in $\\HH$ and satisfies $\\dot\\psi=-iH\\psi$. For a domain-preserving atlas, put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is abs","Suppose a supplied curve $\\psi(t)\\in D(H)$ is differentiable as a curve in $\\HH$ and satisfies $\\dot\\psi=-iH\\psi$. For a domain-preserving atlas, put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{al","Suppose a supplied curve $\\psi(t)\\in D(H)$ is differentiable as a curve in $\\HH$ and satisfies $\\dot\\psi=-iH\\psi$. For a domain-preserving atlas, put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","psi(t)\\in D(H)$ is differentiable as a curve in $\\HH$ and satisfies $\\dot\\psi=-iH\\psi$. For a domain-preserving atlas, put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","put\n\\[\n p_b=\\norm{P_b\\psi}^2,\\qquad\n J_{ba}=2\\operatorname{Im}\\inner{P_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","_b\\psi}{HP_a\\psi}.\n\\]\nSymmetry gives $J_{ba}=-J_{ab}$. Differentiating the bounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","ounded projected norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","norm gives\n\\[\n \\dot p_b=2\\operatorname{Im}\\inner{P_b\\psi}{H\\psi}.\n\\]\nGraph compatibility permits direct insertion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|","rtion of $HS_F\\psi\\to H\\psi$. The formal development also proves a stronger statement for one receiving packet: symmetry moves $H$ onto that packet, so only the strong state sum is needed. Indeed,\n\\[\n \\inner{P_b\\psi}{HP_a\\psi}=\\inner{HP_b\\psi}{P_a\\psi},\n\\]\nand the sum over $a$ converges to $\\inner{HP_b\\psi}{\\psi}=\\inner{P_b\\psi}{H\\psi}$. In fact the row is absolutely summable, since\n\\begin{align*}\n \\sum_a\\left|\\inner{HP_b\\psi}{P_a\\psi}\\right|\n &=\\sum_a\\left|\\inner{P_aHP_b\\psi}{P_a\\psi}\\right|\\\\\n &\\leq\\left(\\sum_a\\norm{P_aHP_b\\psi}^2\\right)^{1/2}\n       \\left(\\sum_a\\norm{P_a\\psi}^2\\right)^{1/2}\n =\\norm{HP_b\\psi}\\norm\\psi.\n\\end{align*}\nConsequently $\\dot p_b=\\sum_aJ_{ba}$ without assuming graph compatibility. The two tail estimates describe the distinct arguments:\n\\begin{align}\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{P_b\\psi}\\norm{HS_F\\psi-H\\psi},\\label{eq:foundation-graph-row}","&\\leq\\left(\\sum_a\\norm{P_aHP_b\\psi}^2\\right)^{1/2}\n       \\left(\\sum_a\\norm{P_a\\psi}^2\\right)^{1/2}\n =\\norm{HP_b\\psi}\\norm\\psi.\n\\end{align*}\nConsequently $\\dot p_b=\\sum_aJ_{ba}$ without assuming graph compatibility. The two tail estimates describe the distinct arguments:\n\\begin{align}\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{P_b\\psi}\\norm{HS_F\\psi-H\\psi},\\label{eq:foundation-graph-row}","&\\leq\\left(\\sum_a\\norm{P_aHP_b\\psi}^2\\right)^{1/2}\n       \\left(\\sum_a\\norm{P_a\\psi}^2\\right)^{1/2}\n =\\norm{HP_b\\psi}\\norm\\psi.\n\\end{align*}\nConsequently $\\dot p_b=\\sum_aJ_{ba}$ without assuming graph compatibility. The two tail estimates describe the distinct arguments:\n\\begin{align}\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{P_b\\psi}\\norm{HS_F\\psi-H\\psi},\\label{eq:foundation-graph-row}\\\\\n \\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{HP_b\\psi}\\norm{S_F\\psi-\\psi}.\\label{eq:foundation-strong-row}\n\\end{align}\nThe first bound uses graph-action control. The second uses symmetry and a fixed receiving packet. Neither states that the entire double array $(J_{ba})$ is absolutely summable.","\\left|\\sum_{a\\in F}J_{ba}-\\dot p_b\\right|\n &\\leq2\\norm{HP_b\\psi}\\norm{S_F\\psi-\\psi}.\\label{eq:foundation-strong-row}\n\\end{align}\nThe first bound uses graph-action control. The second uses symmetry and a fixed receiving packet. Neither states that the entire double array $(J_{ba})$ is absolutely summable.","For a partition $A=\\bigsqcup_{\\alpha}C_\\alpha$, the strong sum $Q_\\alpha=\\sum_{a\\in C_\\alpha}P_a$ is a bounded orthogonal projection. Its weight is always\n\\[\n \\norm{Q_\\alpha\\psi}^2=\\sum_{a\\in C_\\alpha}\\norm{P_a\\psi}^2.\n\\]\nTo define a coarse current with $HQ_\\alpha\\psi$, additionally require $Q_\\alpha D(H)\\subset D(H)$. Fine domain preservation alone does not state this infinite-cell property.","Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately:\n\\begin{align*}\n J^{\\rm c}_{\\beta\\alpha}\n &=2\\operatorname{Im}\\inner{Q_\\beta\\psi}{HQ_\\alpha\\psi}\\\\\n &=\\sum_{b\\in C_\\beta}\\left(\\sum_{a\\in C_\\alpha}J_{ba}\\right)\n =\\sum_{a\\in C_\\alpha}\\left(\\sum_{b\\in C_\\beta}J_{ba}\\right).\n\\end{align*}\nFor example, first summing $a$ moves $H$ onto $P_b\\psi$, then the outer sum pairs $\\sum_bP_b\\psi$ with the fixed vector $HQ_\\alpha\\psi$. The reverse order has its own symmetric argument. Flattening these expressions to a single unordered sum on $C_\\beta\\times C_\\alpha$ requires joint summability. Equality of two justified iterated sums does not supply that hypothesis.","&=\\sum_{b\\in C_\\beta}\\left(\\sum_{a\\in C_\\alpha}J_{ba}\\right)\n =\\sum_{a\\in C_\\alpha}\\left(\\sum_{b\\in C_\\beta}J_{ba}\\right).\n\\end{align*}\nFor example, first summing $a$ moves $H$ onto $P_b\\psi$, then the outer sum pairs $\\sum_bP_b\\psi$ with the fixed vector $HQ_\\alpha\\psi$. The reverse order has its own symmetric argument. Flattening these expressions to a single unordered sum on $C_\\beta\\times C_\\alpha$ requires joint summability. Equality of two justified iterated sums does not supply that hypothesis.","For a simple warning, the antisymmetric scalar array with $a_{2n,2n+1}=1/n$, $a_{2n+1,2n}=-1/n$ for $n\\geq1$, and all other entries zero has finite support in each row but infinite total absolute sum. This array is only a summation example, not an asserted physical realization of the current. It shows why rowwise convergence alone cannot authorize arbitrary rearrangement.","The coarse balance is obtained by differentiating $\\norm{Q_\\beta\\psi(t)}^2$, using the bounded coarse projector. There is no need to interchange an infinite sum of derivatives. Likewise, the total packet norm is identified with $\\norm{\\psi(t)}^2$ by Parseval before its conservation is established. The order of these arguments keeps the domain and summation obligations visible.","For real $(\\lambda_n)$ define\n\\begin{equation}\n (U(t)c)_n=e^{-it\\lambda_n}c_n,\\qquad c\\in\\ell^2.\n \\label{eq:foundation-spectral-flow}\n\\end{equation}\nThe modulus of every multiplier is one. Hence $U(t)$ is a unitary operator, $U(t+s)=U(t)U(s)$, and $U(-t)=U(t)^*$. For each fixed state it is strongly continuous. To prove the latter claim, split $c$ into a finite head and a small $\\ell^2$ tail. The finite head is continuous coordinatewise, while the norm of the difference on the tail is at most twice its initial norm.","\\begin{proposition}[The exact differentiability domain]\nThe orbit $t\\mapsto U(t)c$ is differentiable in $\\ell^2$ at one, equivalently every, time precisely when $c\\in D(H_\\lambda)$. On that domain,\n\\[\n \\frac{d}{dt}U(t)c=-iH_\\lambda U(t)c,\n \\qquad H_\\lambda U(t)c=U(t)H_\\lambda c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $c\\in D(H_\\lambda)$, each coordinate difference quotient tends to $-i\\lambda_ne^{-it\\lambda_n}c_n$. The elementary bound $|e^{-iu}-1|\\leq|u|$ bounds the quotient magnitude by $|\\lambda_nc_n|$. The squared difference from its limit is bounded by $4\\lambda_n^2|c_n|^2$, a summable sequence. Dominated convergence therefore gives convergence in $\\ell^2$.","ability domain]\nThe orbit $t\\mapsto U(t)c$ is differentiable in $\\ell^2$ at one, equivalently every, time precisely when $c\\in D(H_\\lambda)$. On that domain,\n\\[\n \\frac{d}{dt}U(t)c=-iH_\\lambda U(t)c,\n \\qquad H_\\lambda U(t)c=U(t)H_\\lambda c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $c\\in D(H_\\lambda)$, each coordinate difference quotient tends to $-i\\lambda_ne^{-it\\lambda_n}c_n$. The elementary bound $|e^{-iu}-1|\\leq|u|$ bounds the quotient magnitude by $|\\lambda_nc_n|$. The squared difference from its limit is bounded by $4\\lambda_n^2|c_n|^2$, a summable sequence. Dominated convergence therefore gives convergence in $\\ell^2$.","tiable in $\\ell^2$ at one, equivalently every, time precisely when $c\\in D(H_\\lambda)$. On that domain,\n\\[\n \\frac{d}{dt}U(t)c=-iH_\\lambda U(t)c,\n \\qquad H_\\lambda U(t)c=U(t)H_\\lambda c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $c\\in D(H_\\lambda)$, each coordinate difference quotient tends to $-i\\lambda_ne^{-it\\lambda_n}c_n$. The elementary bound $|e^{-iu}-1|\\leq|u|$ bounds the quotient magnitude by $|\\lambda_nc_n|$. The squared difference from its limit is bounded by $4\\lambda_n^2|c_n|^2$, a summable sequence. Dominated convergence therefore gives convergence in $\\ell^2$.","in D(H_\\lambda)$. On that domain,\n\\[\n \\frac{d}{dt}U(t)c=-iH_\\lambda U(t)c,\n \\qquad H_\\lambda U(t)c=U(t)H_\\lambda c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $c\\in D(H_\\lambda)$, each coordinate difference quotient tends to $-i\\lambda_ne^{-it\\lambda_n}c_n$. The elementary bound $|e^{-iu}-1|\\leq|u|$ bounds the quotient magnitude by $|\\lambda_nc_n|$. The squared difference from its limit is bounded by $4\\lambda_n^2|c_n|^2$, a summable sequence. Dominated convergence therefore gives convergence in $\\ell^2$.","c=U(t)H_\\lambda c.\n\\]\n\\end{proposition}\n\\begin{proof}\nIf $c\\in D(H_\\lambda)$, each coordinate difference quotient tends to $-i\\lambda_ne^{-it\\lambda_n}c_n$. The elementary bound $|e^{-iu}-1|\\leq|u|$ bounds the quotient magnitude by $|\\lambda_nc_n|$. The squared difference from its limit is bounded by $4\\lambda_n^2|c_n|^2$, a summable sequence. Dominated convergence therefore gives convergence in $\\ell^2$.","Conversely, if an $\\ell^2$ derivative $d$ exists, bounded coordinate evaluation gives\n\\[\n d_n=-i\\lambda_ne^{-it\\lambda_n}c_n.\n\\]\nSince $d\\in\\ell^2$, one has $\\sum_n\\lambda_n^2|c_n|^2<\\infty$. Commutation and domain preservation follow from the same coordinate formula.\n\\end{proof}","The constructed orbit is the unique global domain-valued solution with a given domain initial state. If two supplied solutions have the same initial value, their difference $w$ satisfies $\\dot w=-iH_\\lambda w$. Symmetry gives\n\\[\n \\frac{d}{dt}\\norm w^2\n =2\\operatorname{Re}\\inner{w}{-iH_\\lambda w}=0.\n\\]\nThe initial zero norm remains zero. This uniqueness proof extends to supplied solutions of a symmetric operator, but the proof of existence above uses the explicit real diagonal construction.","For domain states, the orbit also satisfies the quantitative bound\n\\[\n \\norm{U(t)c-U(s)c}\\leq|t-s|\\norm{H_\\lambda c}.\n\\]\nThe constant belongs to the state. It is not an operator-norm bound for the unbounded generator. Spectral coordinate populations remain constant and their interpacket currents vanish. A different domain-preserving atlas can mix spectral coordinates and carry nonzero currents under the same evolution.","\\foundationfigure[H]{11_graph_tails}{For the normalized state proportional to $n^{-2}$ and $He_n=ne_n$, the state tail and generator tail decay at different rates. Graph control retains both contributions.}","Coordinate cutoffs commute with $U(t)$. Thus for every $c\\in\\ell^2$,\n\\begin{equation}\n \\norm{U(t)c-S_NU(t)c}=\\norm{c-S_Nc}.\n \\label{eq:foundation-time-tail}\n\\end{equation}\nThe right side tends to zero independently of $t\\in\\R$. This gives uniform convergence over all times for each fixed initial state. For $c\\in D(H_\\lambda)$, differentiating the finite cutoff orbit and the full orbit gives a second exact identity:\n\\begin{equation}\n \\norm{\\frac{d}{dt}U(t)c-\n       \\frac{d}{dt}S_NU(t)c}\n =\\norm{H_\\lambda c-S_NH_\\lambda c}.\n \\label{eq:foundation-velocity-tail}\n\\end{equation}\nThis tail also tends to zero uniformly in time. State convergence alone does not imply this velocity convergence.","t)$. Thus for every $c\\in\\ell^2$,\n\\begin{equation}\n \\norm{U(t)c-S_NU(t)c}=\\norm{c-S_Nc}.\n \\label{eq:foundation-time-tail}\n\\end{equation}\nThe right side tends to zero independently of $t\\in\\R$. This gives uniform convergence over all times for each fixed initial state. For $c\\in D(H_\\lambda)$, differentiating the finite cutoff orbit and the full orbit gives a second exact identity:\n\\begin{equation}\n \\norm{\\frac{d}{dt}U(t)c-\n       \\frac{d}{dt}S_NU(t)c}\n =\\norm{H_\\lambda c-S_NH_\\lambda c}.\n \\label{eq:foundation-velocity-tail}\n\\end{equation}\nThis tail also tends to zero uniformly in time. State convergence alone does not imply this velocity convergence.","\\begin{example}[Strong convergence with no differentiable limiting orbit]\nFor $\\lambda_n=n$, take $c_n=C/n$ with $C=(\\sum_{n\\geq1}n^{-2})^{-1/2}$. This is a normalized $\\ell^2$ state outside $D(H_\\lambda)$. Every finite cutoff orbit is differentiable, and \\eqref{eq:foundation-time-tail} gives uniform convergence to the full orbit. Yet the full orbit is nowhere differentiable in $\\ell^2$, by the preceding proposition. In fact the norm of the cutoff velocity is $C\\sqrt N$, which diverges.\n\\end{example}","g orbit]\nFor $\\lambda_n=n$, take $c_n=C/n$ with $C=(\\sum_{n\\geq1}n^{-2})^{-1/2}$. This is a normalized $\\ell^2$ state outside $D(H_\\lambda)$. Every finite cutoff orbit is differentiable, and \\eqref{eq:foundation-time-tail} gives uniform convergence to the full orbit. Yet the full orbit is nowhere differentiable in $\\ell^2$, by the preceding proposition. In fact the norm of the cutoff velocity is $C\\sqrt N$, which diverges.\n\\end{example}","For the faster normalized tail $c_n=C/n^2$, both errors are controlled:\n\\[\n \\norm{c-S_Nc}^2=C^2\\sum_{n>N}n^{-4},\\qquad\n \\norm{H_\\lambda c-S_NH_\\lambda c}^2=C^2\\sum_{n>N}n^{-2}.\n\\]\nThe state tail is $O(N^{-3/2})$ in norm, while the velocity tail is $O(N^{-1/2})$. A small state error can therefore coexist with a much larger derivative error even inside the true domain.","Let $S_N$ be orthogonal projections tending strongly to $I$. Their ranks need not be finite. For a bounded operator $X$, the reconstructed corner sum is $S_NXS_N$. Its natural convergence can be stronger when $X$ belongs to a smaller operator class.","\\begin{theorem}[Three reconstruction statements]\nUnder these hypotheses:\n\\begin{enumerate}\n\\item If $X$ is bounded, $S_NXS_N\\to X$ strongly.\n\\item If $X$ is compact, $\\norm{S_NXS_N-X}\\to0$ in operator norm.\n\\item If $X$ is Hilbert--Schmidt, $\\norm{S_NXS_N-X}_2\\to0$ in Hilbert--Schmidt norm.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nUse the identity\n\\[\n S_NXS_N-X=S_NX(S_N-I)+(S_N-I)X.\n\\]\nSince $\\norm{S_N}\\leq1$, application to a fixed $x$ bounds the error by\n\\[\n \\norm X\\norm{(S_N-I)x}+\\norm{(S_N-I)Xx},\n\\]\nwhich tends to zero.","ements]\nUnder these hypotheses:\n\\begin{enumerate}\n\\item If $X$ is bounded, $S_NXS_N\\to X$ strongly.\n\\item If $X$ is compact, $\\norm{S_NXS_N-X}\\to0$ in operator norm.\n\\item If $X$ is Hilbert--Schmidt, $\\norm{S_NXS_N-X}_2\\to0$ in Hilbert--Schmidt norm.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nUse the identity\n\\[\n S_NXS_N-X=S_NX(S_N-I)+(S_N-I)X.\n\\]\nSince $\\norm{S_N}\\leq1$, application to a fixed $x$ bounds the error by\n\\[\n \\norm X\\norm{(S_N-I)x}+\\norm{(S_N-I)Xx},\n\\]\nwhich tends to zero.","heses:\n\\begin{enumerate}\n\\item If $X$ is bounded, $S_NXS_N\\to X$ strongly.\n\\item If $X$ is compact, $\\norm{S_NXS_N-X}\\to0$ in operator norm.\n\\item If $X$ is Hilbert--Schmidt, $\\norm{S_NXS_N-X}_2\\to0$ in Hilbert--Schmidt norm.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nUse the identity\n\\[\n S_NXS_N-X=S_NX(S_N-I)+(S_N-I)X.\n\\]\nSince $\\norm{S_N}\\leq1$, application to a fixed $x$ bounds the error by\n\\[\n \\norm X\\norm{(S_N-I)x}+\\norm{(S_N-I)Xx},\n\\]\nwhich tends to zero.","For compact $X$, choose finite-rank $Y$ with $\\norm{X-Y}<\\varepsilon$. Strong convergence is uniform on each fixed finite-dimensional subspace, and self-adjointness of $S_N$ controls the right error by the corresponding left error for $Y^*$. Thus $\\norm{S_NYS_N-Y}\\to0$. The difference between the two reconstruction errors is at most $2\\varepsilon$, proving the claim.","For the last assertion, fix an orthonormal basis $(e_j)$ and define\n\\[\n \\norm X_2^2=\\sum_j\\norm{Xe_j}^2<\\infty.\n\\]\nEach column $(S_N-I)Xe_j$ tends to zero, and its squared norm is bounded by $4\\norm{Xe_j}^2$. Dominated convergence gives $\\norm{(S_N-I)X}_2\\to0$. The right error follows by taking adjoints, using $S_N^*=S_N$ and $\\norm{X^*}_2=\\norm X_2$. The ideal inequality $\\norm{AY}_2\\leq\\norm A\\norm Y_2$ and the displayed decomposition finish the proof.\n\\end{proof}","\\begin{example}[The identity never has a small omitted tail in operator norm]\nOn an infinite-dimensional coordinate space, let $S_N$ retain the first $N$ coordinates. For $X=I$,\n\\[\n \\norm{S_NIS_N-I}=\\norm{S_N-I}=1.\n\\]\nA unit vector $e_{N+1}$ witnesses the lower bound. Nevertheless, $S_Nx\\to x$ for every fixed $x\\in\\ell^2$. This is the difference between pointwise control over states and uniform control over the full unit ball.\n\\end{example}","For a diagonal bounded multiplier $f(H_\\lambda)$, zero padding gives the exact formula\n\\[\n \\norm{f(H_\\lambda)-S_Nf(H_\\lambda)S_N}\n =\\sup_{n>N}|f(\\lambda_n)|.\n\\]\nFor the unitary multiplier $f(\\lambda)=e^{-it\\lambda}$, this norm is one at every finite cutoff. Equation \\eqref{eq:foundation-time-tail} remains valid. Uniformity in time for a fixed state and failure of operator-norm convergence are therefore compatible.","For $\\lambda_n=n$, a resolvent at $i\\eta$, $\\eta\\ne0$, has zero-padded error\n\\[\n \\sup_{n>N}\\frac1{\\sqrt{n^2+\\eta^2}}\n =\\frac1{\\sqrt{(N+1)^2+\\eta^2}}\\longrightarrow0.\n\\]\nThe heat multiplier $e^{-tn}$ has error $e^{-t(N+1)}$ for fixed $t>0$. These tails decay because the operators are compact. At $t=0$ the heat operator is the identity, so the same norm-convergence assertion is false.","Two other bounded multipliers approach one along this positive spectrum:\n\\[\n b(n)=\\frac{n}{\\sqrt{1+n^2}},\\qquad\n q(n)=\\frac{n-i}{n+i}.\n\\]\nTheir correct anchored cutoff is\n\\[\n A_N(f)=S_Nf(H_\\lambda)S_N+(I-S_N).\n\\]\nThe errors are then\n\\[\n \\norm{b(H_\\lambda)-A_N(b)}\n =\\sup_{n>N}\\left|\\frac{n}{\\sqrt{1+n^2}}-1\\right|\\longrightarrow0,\n\\]\nand\n\\[\n \\norm{q(H_\\lambda)-A_N(q)}\n =\\sup_{n>N}\\frac2{\\sqrt{n^2+1}}\\longrightarrow0.\n\\]\nThe identity tail is an analytic choice justified by these asymptotic multipliers. It is not a universal prescription for every two-sided or non-diagonal spectrum.","On a graph-compatible atlas, an unbounded generator has another reconstruction statement:\n\\[\n S_NHS_N\\psi-H\\psi\n =S_N(HS_N\\psi-H\\psi)+(S_N-I)H\\psi\\longrightarrow0,\n \\qquad\\psi\\in D(H).\n\\]\nThis is convergence of the displayed output vectors in the ambient Hilbert norm. It neither takes an operator norm of $H$ nor asserts graph-norm convergence of those output vectors, which would require additional domain and second-action information.","\\foundationfigure[H]{09_cutoff_topology}{Zero-padded unitary cutoffs converge on a fixed state but retain operator-norm error one. For the same spectrum $n\\geq1$, the resolvent multiplier has a tail that decays in operator norm.}","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain p","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplie","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct con","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverse","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inver","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspeci","A packet need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","et need not commute with an unbounded Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","d Hamiltonian. Suppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","ppose now that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","that $H$ is self-adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","adjoint, $P$ is a bounded orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","ed orthogonal projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","projection preserving $D(H)$, and $[H,P]$ on $D(H)$ extends to a bounded operator $B$. For nonreal $z$, let $R_z=(H-z)^{-1}$. It maps $\\HH$ into $D(H)$, so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives\n\\[\n [R_z,P]=-R_zBR_z.\n\\]\nIndeed, for $x\\in\\HH$, apply $H-z$ to $PR_zx$ to obtain $Px+BR_zx$, then apply $R_z$. Hence\n\\[\n \\norm{[R_z,P]}\\leq\\norm{R_z}^2\\norm B,\n \\qquad\n \\norm{[R_{i\\eta},P]}\\leq\\frac{\\norm B}{\\eta^2}\n \\quad(\\eta\\ne0).\n\\]\nThe estimate converts a domain-sensitive commutator into a bounded one. In the generic formal certificate, the inverse and its domain properties are supplied explicitly. The diagonal modules also construct concrete inverses. Neither statement licenses a formal inverse for an unspecified operator.","Finite arrays support many useful difference operators. What they cannot support is a nonzero linear derivation of the full pointwise algebra $\\C^N$ with its ordinary product. This obstruction explains why a finite operator calculus must declare which algebra carries its differentiation rule.","\\begin{theorem}[No nonzero derivation of the finite pointwise algebra]\nLet $D:\\C^N\\to\\C^N$ be complex linear and satisfy\n\\[\n D(fg)=D(f)g+fD(g)\n\\]\nfor every pair of vectors, where multiplication is coordinatewise. Then $D=0$.\n\\end{theorem}\n\\begin{proof}\nFor the coordinate idempotent $e_a$, the identity $e_a^2=e_a$ gives $D(e_a)=2e_aD(e_a)$. At a coordinate $b\\ne a$, the right side is zero. At $b=a$, the equality reads $(D(e_a))_a=2(D(e_a))_a$, again forcing zero. Thus $D(e_a)=0$ for every basis vector, and linearity gives $D=0$.\n\\end{proof}","zero derivation of the finite pointwise algebra]\nLet $D:\\C^N\\to\\C^N$ be complex linear and satisfy\n\\[\n D(fg)=D(f)g+fD(g)\n\\]\nfor every pair of vectors, where multiplication is coordinatewise. Then $D=0$.\n\\end{theorem}\n\\begin{proof}\nFor the coordinate idempotent $e_a$, the identity $e_a^2=e_a$ gives $D(e_a)=2e_aD(e_a)$. At a coordinate $b\\ne a$, the right side is zero. At $b=a$, the equality reads $(D(e_a))_a=2(D(e_a))_a$, again forcing zero. Thus $D(e_a)=0$ for every basis vector, and linearity gives $D=0$.\n\\end{proof}","t $D:\\C^N\\to\\C^N$ be complex linear and satisfy\n\\[\n D(fg)=D(f)g+fD(g)\n\\]\nfor every pair of vectors, where multiplication is coordinatewise. Then $D=0$.\n\\end{theorem}\n\\begin{proof}\nFor the coordinate idempotent $e_a$, the identity $e_a^2=e_a$ gives $D(e_a)=2e_aD(e_a)$. At a coordinate $b\\ne a$, the right side is zero. At $b=a$, the equality reads $(D(e_a))_a=2(D(e_a))_a$, again forcing zero. Thus $D(e_a)=0$ for every basis vector, and linearity gives $D=0$.\n\\end{proof}","g)=D(f)g+fD(g)\n\\]\nfor every pair of vectors, where multiplication is coordinatewise. Then $D=0$.\n\\end{theorem}\n\\begin{proof}\nFor the coordinate idempotent $e_a$, the identity $e_a^2=e_a$ gives $D(e_a)=2e_aD(e_a)$. At a coordinate $b\\ne a$, the right side is zero. At $b=a$, the equality reads $(D(e_a))_a=2(D(e_a))_a$, again forcing zero. Thus $D(e_a)=0$ for every basis vector, and linearity gives $D=0$.\n\\end{proof}","The theorem concerns the semisimple pointwise algebra $\\C^N$. It does not assert that every finite-dimensional commutative complex algebra has only the zero derivation. For example, on $\\C[\\varepsilon]/(\\varepsilon^2)$, the map $D(a+b\\varepsilon)=b\\varepsilon$ is a nonzero derivation. The nilpotent element changes the algebraic hypothesis.","\\begin{example}[The precise defect of a forward difference]\nOn a cyclic grid, let $D_hf_j=(f_{j+1}-f_j)/h$, with $h\\ne0$. Expanding the product gives\n\\[\n D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j\n =h(D_hf)_j(D_hg)_j.\n\\]\nThus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go theorem, because that theorem concerns an exact identity for every vector at fixed finite resolution.\n\\end{example}","Move now to $M_N(\\C)$. To avoid confusing a Hermitian observable with the skew-adjoint state generator $K$, denote a Hermitian matrix by $B=B^*$ and define\n\\[\n \\delta_BX=i[B,X]=i(BX-XB).\n\\]\nThen direct multiplication proves\n\\begin{align}\n \\delta_B(XY)&=(\\delta_BX)Y+X\\delta_BY,\\label{eq:foundation-matrix-leibniz}","Move now to $M_N(\\C)$. To avoid confusing a Hermitian observable with the skew-adjoint state generator $K$, denote a Hermitian matrix by $B=B^*$ and define\n\\[\n \\delta_BX=i[B,X]=i(BX-XB).\n\\]\nThen direct multiplication proves\n\\begin{align}\n \\delta_B(XY)&=(\\delta_BX)Y+X\\delta_BY,\\label{eq:foundation-matrix-leibniz}\\\\\n (\\delta_BX)^*&=\\delta_B(X^*),\\label{eq:foundation-matrix-star}\\\\\n \\operatorname{Tr}((\\delta_BX)Y)&=-\\operatorname{Tr}(X\\delta_BY).\n \\label{eq:foundation-matrix-ibp}\n\\end{align}\nFor the first identity, the middle terms $-iXBY$ and $iXBY$ cancel. For the second, taking the adjoint reverses the product order and conjugates $i$, with $B^*=B$ restoring the stated sign. For the third, cyclicity moves the leading $B$ in $\\operatorname{Tr}(BXY)$ to the end. The trace pairing is bilinear here. Combining \\eqref{eq:foundation-matrix-star} and \\eqref{eq:foundation-matrix-ibp} also gives the Hilbert--Schmidt identity\n\\[\n \\inner{\\delta_BX}{Y}_{\\rm HS}=-\\inner{X}{\\delta_BY}_{\\rm HS},\n \\qquad\\inner{X}{Y}_{\\rm HS}=\\operatorname{Tr}(X^*Y).\n\\]","\\operatorname{Tr}((\\delta_BX)Y)&=-\\operatorname{Tr}(X\\delta_BY).\n \\label{eq:foundation-matrix-ibp}\n\\end{align}\nFor the first identity, the middle terms $-iXBY$ and $iXBY$ cancel. For the second, taking the adjoint reverses the product order and conjugates $i$, with $B^*=B$ restoring the stated sign. For the third, cyclicity moves the leading $B$ in $\\operatorname{Tr}(BXY)$ to the end. The trace pairing is bilinear here. Combining \\eqref{eq:foundation-matrix-star} and \\eqref{eq:foundation-matrix-ibp} also gives the Hilbert--Schmidt identity\n\\[\n \\inner{\\delta_BX}{Y}_{\\rm HS}=-\\inner{X}{\\delta_BY}_{\\rm HS},\n \\qquad\\inner{X}{Y}_{\\rm HS}=\\operatorname{Tr}(X^*Y).\n\\]","rms $-iXBY$ and $iXBY$ cancel. For the second, taking the adjoint reverses the product order and conjugates $i$, with $B^*=B$ restoring the stated sign. For the third, cyclicity moves the leading $B$ in $\\operatorname{Tr}(BXY)$ to the end. The trace pairing is bilinear here. Combining \\eqref{eq:foundation-matrix-star} and \\eqref{eq:foundation-matrix-ibp} also gives the Hilbert--Schmidt identity\n\\[\n \\inner{\\delta_BX}{Y}_{\\rm HS}=-\\inner{X}{\\delta_BY}_{\\rm HS},\n \\qquad\\inner{X}{Y}_{\\rm HS}=\\operatorname{Tr}(X^*Y).\n\\]","These identities preserve multiplication order. They do not turn $\\delta_B$ into a derivation from diagonal matrices to diagonal matrices. Usually it leaves that commutative subalgebra.","\\begin{example}[A nonzero derivative leaves the pointwise subalgebra]\nFor\n\\[\n B=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n X=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\n\\]\none obtains\n\\[\n \\delta_BX=\\begin{pmatrix}0&-i","0&0\\end{pmatrix},\n\\]\none obtains\n\\[\n \\delta_BX=\\begin{pmatrix}0&-i\\\\i&0\\end{pmatrix}\\ne0.\n\\]\nThe output is off diagonal. This is exactly how the matrix algebra permits a nonzero derivation without violating the pointwise theorem.\n\\end{example}","For Hermitian generators $B_i,B_j$, put $\\mathcal F_{ij}=i[B_i,B_j]$. This curvature matrix is Hermitian, and the Jacobi identity gives\n\\begin{equation}\n [\\delta_{B_i},\\delta_{B_j}]X\n =-[[B_i,B_j],X]\n =i[\\mathcal F_{ij},X]=\\delta_{\\mathcal F_{ij}}X.\n \\label{eq:foundation-corner-curvature}\n\\end{equation}\nCommuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular $X$ vanishes precisely when that $X$ commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additional structures developed in the gauge and gravity papers.","B_i,B_j]$. This curvature matrix is Hermitian, and the Jacobi identity gives\n\\begin{equation}\n [\\delta_{B_i},\\delta_{B_j}]X\n =-[[B_i,B_j],X]\n =i[\\mathcal F_{ij},X]=\\delta_{\\mathcal F_{ij}}X.\n \\label{eq:foundation-corner-curvature}\n\\end{equation}\nCommuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular $X$ vanishes precisely when that $X$ commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additional structures developed in the gauge and gravity papers.","is Hermitian, and the Jacobi identity gives\n\\begin{equation}\n [\\delta_{B_i},\\delta_{B_j}]X\n =-[[B_i,B_j],X]\n =i[\\mathcal F_{ij},X]=\\delta_{\\mathcal F_{ij}}X.\n \\label{eq:foundation-corner-curvature}\n\\end{equation}\nCommuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular $X$ vanishes precisely when that $X$ commutes with it. This matrix identity provides an algebraic curvature mechanism. A physical geometric interpretation requires the additional structures developed in the gauge and gravity papers.","Let $(P_a)$ be a finite sharp atlas. Writing $X_{ba}=P_bXP_a$ and $B_{ba}=P_bBP_a$, insertion of $I=\\sum_cP_c$ gives\n\\begin{equation}\n P_b\\delta_B(X)P_a\n =i\\sum_c\\left(B_{bc}X_{ca}-X_{bc}B_{ca}\\right).\n \\label{eq:foundation-corner-path}\n\\end{equation}\nEach term has a genuine intermediate packet $c$. One ordering applies $X$ then $B$, while the other applies $B$ then $X$. Replacing a matrix product by an entrywise product would destroy this path composition and the cancellation behind the Leibniz rule.","Summing the differentiated corners reconstructs $\\delta_BX$. For a coarse partition with $Q_\\alpha=\\sum_{a\\in C_\\alpha}P_a$,\n\\[\n Q_\\beta\\delta_B(X)Q_\\alpha\n =\\sum_{\\substack{b\\in C_\\beta","Summing the differentiated corners reconstructs $\\delta_BX$. For a coarse partition with $Q_\\alpha=\\sum_{a\\in C_\\alpha}P_a$,\n\\[\n Q_\\beta\\delta_B(X)Q_\\alpha\n =\\sum_{\\substack{b\\in C_\\beta\\\\a\\in C_\\alpha}}P_b\\delta_B(X)P_a.\n\\]\nThis is exact aggregation of the already differentiated operator. It does not claim that discarding all internal matrix structure produces a new pointwise algebra with a nonzero exact derivation. The retained corners are the structure that makes the finite differential calculus work.","The third route follows analysis. Countable resolutions require an actual summation statement. An unbounded operator requires its domain. A strong limit acts on each fixed vector, while an operator-norm limit controls all unit vectors at once. The final chapters make these distinctions explicit and explain why the finite pointwise algebra $\\C^N$ cannot support a nonzero complex-linear derivation obeying the ordinary Leibniz rule, even though matrix corners support nontrivial commutators.","midrule\\endhead\nInner product & $\\inner{x}{y}$ is conjugate-linear in $x$ and linear in $y$.","Projector & $P_a=P_a^*=P_a^2$, with orthogonality between distinct packet labels.","Transfer block & $L_{ba}=P_bLP_a$ takes a component in packet $a$ to packet $b$.","State equation & $\\dot\\psi=K\\psi$, with $K^*=-K$. The Hamiltonian convention is $K=-iH$ in units $\\hbar=1$.","Packet weight & $p_a=\\norm{P_a\\psi}^2$. It is a probability for a normalized state and a complete orthogonal resolution. It is not an energy expectation by definition.","Current orientation & $J_{b\\leftarrow a}$ is inflow into packet $b$ from packet $a$, so $\\dot p_b=\\sum_aJ_{b\\leftarrow a}$ in the finite closed setting.","Mixed state & $\\rho\\geq0$, $\\operatorname{tr}\\rho=1$ in the finite-dimensional sections.","Unbounded action & An expression such as $H\\psi$ includes the condition $\\psi\\in\\Dom(H)$.","Time and units & $K$ has units of inverse time. We set $\\hbar=1$ when identifying $K=-iH$. All plotted examples state their numerical parameters.","Let $E$ be a real normed vector space and let $f(h)\\in E$ be defined for $0<|h|<\\delta$. Fix a refinement ratio $b>1$. If $f(h)$ represents operators on different Hilbert spaces, comparison maps must first place them in a common space where the following additions are meaningful. The formulas alone do not construct those maps.","\\begin{definition}[The actual refinement tower]\nSet $T_0f(h)=f(h)$ and, for $m\\geq0$, define\n\\[\n T_{m+1}f(h)=\\frac{b^{m+1}T_mf(h/b)-T_mf(h)}{b^{m+1}-1}.\n\\]\n\\end{definition}","\\begin{proposition}[Exact response of a monomial]\nFor $r\\in\\N$ and $v\\in E$,\n\\[\n T_m(h^rv)=c_{m,r}h^rv,\\qquad\n c_{m,r}=\\prod_{k=1}^{m}\\frac{b^{k-r}-1}{b^k-1}.\n\\]\nThe empty product equals one. In particular, $c_{m,0}=1$, and $c_{m,r}=0$ for $1\\leq r\\leq m$.\n\\end{proposition}\n\\begin{proof}\nAt level zero the assertion is the definition. If it holds at level $m$, the next recursion multiplies the coefficient by\n\\[\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n\\]\nThis appends the next factor to the product. For $r=0$, each factor equals one. For $1\\leq r\\leq m$, the factor at $k=r$ vanishes.\n\\end{proof}","ition}[Exact response of a monomial]\nFor $r\\in\\N$ and $v\\in E$,\n\\[\n T_m(h^rv)=c_{m,r}h^rv,\\qquad\n c_{m,r}=\\prod_{k=1}^{m}\\frac{b^{k-r}-1}{b^k-1}.\n\\]\nThe empty product equals one. In particular, $c_{m,0}=1$, and $c_{m,r}=0$ for $1\\leq r\\leq m$.\n\\end{proposition}\n\\begin{proof}\nAt level zero the assertion is the definition. If it holds at level $m$, the next recursion multiplies the coefficient by\n\\[\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n\\]\nThis appends the next factor to the product. For $r=0$, each factor equals one. For $1\\leq r\\leq m$, the factor at $k=r$ vanishes.\n\\end{proof}","\\qquad\n c_{m,r}=\\prod_{k=1}^{m}\\frac{b^{k-r}-1}{b^k-1}.\n\\]\nThe empty product equals one. In particular, $c_{m,0}=1$, and $c_{m,r}=0$ for $1\\leq r\\leq m$.\n\\end{proposition}\n\\begin{proof}\nAt level zero the assertion is the definition. If it holds at level $m$, the next recursion multiplies the coefficient by\n\\[\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n\\]\nThis appends the next factor to the product. For $r=0$, each factor equals one. For $1\\leq r\\leq m$, the factor at $k=r$ vanishes.\n\\end{proof}","c{b^{k-r}-1}{b^k-1}.\n\\]\nThe empty product equals one. In particular, $c_{m,0}=1$, and $c_{m,r}=0$ for $1\\leq r\\leq m$.\n\\end{proposition}\n\\begin{proof}\nAt level zero the assertion is the definition. If it holds at level $m$, the next recursion multiplies the coefficient by\n\\[\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n\\]\nThis appends the next factor to the product. For $r=0$, each factor equals one. For $1\\leq r\\leq m$, the factor at $k=r$ vanishes.\n\\end{proof}","ty product equals one. In particular, $c_{m,0}=1$, and $c_{m,r}=0$ for $1\\leq r\\leq m$.\n\\end{proposition}\n\\begin{proof}\nAt level zero the assertion is the definition. If it holds at level $m$, the next recursion multiplies the coefficient by\n\\[\n \\frac{b^{m+1}b^{-r}-1}{b^{m+1}-1}.\n\\]\nThis appends the next factor to the product. For $r=0$, each factor equals one. For $1\\leq r\\leq m$, the factor at $k=r$ vanishes.\n\\end{proof}","Suppose that a finite expansion is available,\n\\[\n f(h)=F+\\sum_{r=1}^{m}h^rv_r+R(h).\n\\]\nLinearity and the proposition give the exact identity\n\\[\n T_mf(h)-F=T_mR(h).\n\\]\nOnly a finite sum has been used. No convergence of an infinite formal series is needed. The size of the right-hand side remains an analytic question.","\\Needspace{18\\baselineskip}\n\\begin{theorem}[A quantified remainder and perturbation bound]\nLet $p$ be a nonnegative integer and suppose\n\\[\n \\norm{R(h)}\\leq C|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n\\]\nwhere $C,\\varepsilon\\geq0$. Define\n\\[\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n\\]\nThen, throughout the same punctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T","\\Needspace{18\\baselineskip}\n\\begin{theorem}[A quantified remainder and perturbation bound]\nLet $p$ be a nonnegative integer and suppose\n\\[\n \\norm{R(h)}\\leq C|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n\\]\nwhere $C,\\varepsilon\\geq0$. Define\n\\[\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n\\]\nThen, throughout the same punctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two","\\Needspace{18\\baselineskip}\n\\begin{theorem}[A quantified remainder and perturbation bound]\nLet $p$ be a nonnegative integer and suppose\n\\[\n \\norm{R(h)}\\leq C|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n\\]\nwhere $C,\\varepsilon\\geq0$. Define\n\\[\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n\\]\nThen, throughout the same punctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","bound]\nLet $p$ be a nonnegative integer and suppose\n\\[\n \\norm{R(h)}\\leq C|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n\\]\nwhere $C,\\varepsilon\\geq0$. Define\n\\[\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n\\]\nThen, throughout the same punctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","|h|^p,\\qquad\n \\norm{g(h)-f(h)}\\leq\\varepsilon\n \\quad(0<|h|<\\delta),\n\\]\nwhere $C,\\varepsilon\\geq0$. Define\n\\[\n A_{m,p}=\\prod_{k=1}^{m}\\frac{b^{k-p}+1}{b^k-1}.\n\\]\nThen, throughout the same punctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","nctured interval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","nterval,\n\\[\n \\norm{T_mg(h)-F}\\leq A_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","_{m,p}C|h|^p+A_{m,0}\\varepsilon.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","he interval is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","al is stable under $h\\mapsto h/b$. The triangle inequality applied to one tower step gives\n\\[\n \\norm{T_{m+1}R(h)}\\leq\n \\frac{b^{m+1}\\norm{T_mR(h/b)}+\\norm{T_mR(h)}}{b^{m+1}-1}.\n\\]\nStarting with the stated remainder bound and proceeding by induction multiplies its constant at step $m+1$ by $(b^{m+1-p}+1)/(b^{m+1}-1)$. Hence $\\norm{T_mR(h)}\\leq A_{m,p}C|h|^p$. Apply the same argument with $p=0$ to $g-f$. Linearity and the triangle inequality separate the two contributions.\n\\end{proof}","When $p=m+1$, this proves the expected order improvement under an explicit remainder hypothesis. It also retains the effect of noisy samples. Each factor in $A_{m,0}$ exceeds one, so extrapolation can amplify a uniform sample error even while cancelling several deterministic powers of $h$.","\\subsection{A complete three-level example}\nFor $b=2$, direct substitution gives\n\\begin{align*}\n T_1f(h)&=2f(h/2)-f(h),","\\subsection{A complete three-level example}\nFor $b=2$, direct substitution gives\n\\begin{align*}\n T_1f(h)&=2f(h/2)-f(h),\\\\\n T_2f(h)&=\\frac{8f(h/4)-6f(h/2)+f(h)}{3},\\\\\n T_3f(h)&=\\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.\n\\end{align*}\nTake the exact scalar polynomial $f(h)=1+2h-3h^2+4h^3$. The responses are\n\\[\n T_1f(h)=1+\\frac32h^2-3h^3,\\qquad\n T_2f(h)=1+\\frac12h^3,\\qquad T_3f(h)=1.\n\\]\nThese are polynomial identities for every real $h$. If each of the three samples used by $T_2$ has absolute error at most $\\varepsilon$, its output error is at most\n\\[\n \\frac{8+6+1}{3}\\varepsilon=5\\varepsilon.\n\\]\nThis bound is attained by errors with signs matching the three weights. Thus exact removal of the first two powers does not imply a contraction on arbitrary perturbations. For the displayed polynomial, the perturbed second-level approximation has the explicit total bound $|h|^3/2+5\\varepsilon$.","T_3f(h)&=\\frac{64f(h/8)-56f(h/4)+14f(h/2)-f(h)}{21}.\n\\end{align*}\nTake the exact scalar polynomial $f(h)=1+2h-3h^2+4h^3$. The responses are\n\\[\n T_1f(h)=1+\\frac32h^2-3h^3,\\qquad\n T_2f(h)=1+\\frac12h^3,\\qquad T_3f(h)=1.\n\\]\nThese are polynomial identities for every real $h$. If each of the three samples used by $T_2$ has absolute error at most $\\varepsilon$, its output error is at most\n\\[\n \\frac{8+6+1}{3}\\varepsilon=5\\varepsilon.\n\\]\nThis bound is attained by errors with signs matching the three weights. Thus exact removal of the first two powers does not imply a contraction on arbitrary perturbations. For the displayed polynomial, the perturbed second-level approximation has the explicit total bound $|h|^3/2+5\\varepsilon$.","by $T_2$ has absolute error at most $\\varepsilon$, its output error is at most\n\\[\n \\frac{8+6+1}{3}\\varepsilon=5\\varepsilon.\n\\]\nThis bound is attained by errors with signs matching the three weights. Thus exact removal of the first two powers does not imply a contraction on arbitrary perturbations. For the displayed polynomial, the perturbed second-level approximation has the explicit total bound $|h|^3/2+5\\varepsilon$.","Correctly normalized finite fields become $L^2$ step functions. Concrete dyadic boxes identify the approximating subspaces modulo null boundaries, and their projections converge strongly on each fixed field. Comparison of physical generators and evolution remains a separate task.","A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as $\\psi_{x+\\widehat\\mu}-\\psi_x$ therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.","Let $V$ be a finite periodic lattice and let every fiber be a copy of $\\C^r$. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below,\n\\begin{equation}\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n\\end{equation}\nThe gauge $g=(g_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on","Let $V$ be a finite periodic lattice and let every fiber be a copy of $\\C^r$. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below,\n\\begin{equation}\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n\\end{equation}\nThe gauge $g=(g_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_","Let $V$ be a finite periodic lattice and let every fiber be a copy of $\\C^r$. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below,\n\\begin{equation}\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n\\end{equation}\nThe gauge $g=(g_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_{y\\leftarrow x}$ has the transformation $g_yT_{y\\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.","a finite periodic lattice and let every fiber be a copy of $\\C^r$. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below,\n\\begin{equation}\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n\\end{equation}\nThe gauge $g=(g_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_{y\\leftarrow x}$ has the transformation $g_yT_{y\\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.","ing on this fiber. In the convention used for the spatial stencils below,\n\\begin{equation}\\label{g:backward-link}\n U_{x,\\mu}:\\HH_{x+\\widehat\\mu}\\longrightarrow\\HH_x,\n \\qquad U_{x,\\mu}^*U_{x,\\mu}=U_{x,\\mu}U_{x,\\mu}^*=I.\n\\end{equation}\nThe gauge $g=(g_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_{y\\leftarrow x}$ has the transformation $g_yT_{y\\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.","_x)_{x\\in V}$ changes coordinates by $\\psi_x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_{y\\leftarrow x}$ has the transformation $g_yT_{y\\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.","x^g=g_x\\psi_x$. Requiring the transported vector to have the frame of its destination gives\n\\begin{equation}\\label{g:link-transform}\n U_{x,\\mu}^g=g_xU_{x,\\mu}g_{x+\\widehat\\mu}^{-1},\n \\qquad U_{x,\\mu}^g\\psi_{x+\\widehat\\mu}^g\n =g_xU_{x,\\mu}\\psi_{x+\\widehat\\mu}.\n\\end{equation}\nProducts act on column vectors from right to left. More generally, a transporter $T_{y\\leftarrow x}$ has the transformation $g_yT_{y\\leftarrow x}g_x^{-1}$. This notation checks both the input and output of every product.","The forward link is\n\\begin{equation}\\label{g:forward-link}\n u_{x,\\mu}=U_{x,\\mu}^{-1}:\\HH_x\\longrightarrow\\HH_{x+\\widehat\\mu},\n \\qquad u_{x,\\mu}^g=g_{x+\\widehat\\mu}u_{x,\\mu}g_x^{-1}.\n\\end{equation}\nThe older time-dependent Abelian walk writes this forward link as $L_j$. The later geometric Lean sources use $u$. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward $U$. These formulas describe the same edge in opposite directions. In particular, reversing an edge changes its transporter to the inverse, not to the same matrix in a different notation.","\\begin{proposition}[Covariance of linked differences]\\label{g:stencil-covariance}\nFor spacing $h>0$, define\n\\begin{align}\n (D^+_{\\mu,U}\\psi)_x\n &=\\frac{U_{x,\\mu}\\psi_{x+\\widehat\\mu}-\\psi_x}{h},\\label{g:forward-difference}","\\begin{proposition}[Covariance of linked differences]\\label{g:stencil-covariance}\nFor spacing $h>0$, define\n\\begin{align}\n (D^+_{\\mu,U}\\psi)_x\n &=\\frac{U_{x,\\mu}\\psi_{x+\\widehat\\mu}-\\psi_x}{h},\\label{g:forward-difference}\\\\\n (\\Delta_U\\psi)_x\n &=\\frac1{h^2}\\sum_\\mu\\left(\n U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n +U_{x-\\widehat\\mu,\\mu}^{-1}\\psi_{x-\\widehat\\mu}-2\\psi_x\\right).\n \\label{g:laplacian}\n\\end{align}\nThen $D^+_{\\mu,U^g}(g\\psi)=gD^+_{\\mu,U}\\psi$ and\n$\\Delta_{U^g}(g\\psi)=g\\Delta_U\\psi$.\n\\end{proposition}","(\\Delta_U\\psi)_x\n &=\\frac1{h^2}\\sum_\\mu\\left(\n U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n +U_{x-\\widehat\\mu,\\mu}^{-1}\\psi_{x-\\widehat\\mu}-2\\psi_x\\right).\n \\label{g:laplacian}\n\\end{align}\nThen $D^+_{\\mu,U^g}(g\\psi)=gD^+_{\\mu,U}\\psi$ and\n$\\Delta_{U^g}(g\\psi)=g\\Delta_U\\psi$.\n\\end{proposition}","\\begin{proof}\nFor the positive edge, substitute \\eqref{g:link-transform}. The two gauge factors at $x+\\widehat\\mu$ cancel, leaving $g_x$ on the left. For the negative edge,\n\\[\n (U_{x-\\widehat\\mu,\\mu}^g)^{-1}g_{x-\\widehat\\mu}\n =g_xU_{x-\\widehat\\mu,\\mu}^{-1}.\n\\]\nEvery term in either difference therefore has the common factor $g_x$.\n\\end{proof}","The lattice inner product is $\\inner{\\phi}{\\psi}_V=\\sum_x\\phi_x^*\\psi_x$, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity\n\\begin{equation}\\label{g:laplace-form}\n -\\inner{\\psi}{\\Delta_U\\psi}_V\n =\\sum_{x,\\mu}\\left\\lVert(D^+_{\\mu,U}\\psi)_x\\right\\rVert^2.\n\\end{equation}\nIndeed, expanding one squared difference gives\n$h^{-2}(\\norm{\\psi_{x+\\widehat\\mu}}^2+\\norm{\\psi_x}^2\n-2\\operatorname{Re}(\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}))$.\nSumming and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so $\\Delta_U$ is self-adjoint and nonpositive. This finite identity uses the periodic boundary condition; an open domain needs its boundary terms or a specified boundary","The lattice inner product is $\\inner{\\phi}{\\psi}_V=\\sum_x\\phi_x^*\\psi_x$, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity\n\\begin{equation}\\label{g:laplace-form}\n -\\inner{\\psi}{\\Delta_U\\psi}_V\n =\\sum_{x,\\mu}\\left\\lVert(D^+_{\\mu,U}\\psi)_x\\right\\rVert^2.\n\\end{equation}\nIndeed, expanding one squared difference gives\n$h^{-2}(\\norm{\\psi_{x+\\widehat\\mu}}^2+\\norm{\\psi_x}^2\n-2\\operatorname{Re}(\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}))$.\nSumming and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so $\\Delta_U$ is self-adjoint and nonpositive. This finite identity uses the periodic boundary condition; an open domain needs its boundary terms or a specified boundary condition.","e-form}\n -\\inner{\\psi}{\\Delta_U\\psi}_V\n =\\sum_{x,\\mu}\\left\\lVert(D^+_{\\mu,U}\\psi)_x\\right\\rVert^2.\n\\end{equation}\nIndeed, expanding one squared difference gives\n$h^{-2}(\\norm{\\psi_{x+\\widehat\\mu}}^2+\\norm{\\psi_x}^2\n-2\\operatorname{Re}(\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}))$.\nSumming and shifting the first norm term produces the left side. The forward and backward hopping blocks are adjoints, so $\\Delta_U$ is self-adjoint and nonpositive. This finite identity uses the periodic boundary condition; an open domain needs its boundary terms or a specified boundary condition.","\\gaugefigure{01_link_covariance}{A backward link $U$ and its forward inverse $u$ compare the same two fibers. Gauge factors cancel at the input and remain at the destination. The scalar example illustrates the exact transformation law.}","Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let $N\\ge1$ be an integer, let $j\\in\\Z/N\\Z$, let $\\psi_j^n\\in\\C^2$, and let $U_j^n\\in U(1)$ be backward spatial links. Let $T_j^n\\in U(1)$ transport from $(j,n)$ to $(j,n+1)$. At each step choose two on-site unitary coins $C_{1,j}^n,C_{2,j}^n\\in U(2)$ and define\n\\begin{align}\n \\eta_j&=C_{1,j}^n\\psi_j^n,\\label{g:walk-coin}","Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let $N\\ge1$ be an integer, let $j\\in\\Z/N\\Z$, let $\\psi_j^n\\in\\C^2$, and let $U_j^n\\in U(1)$ be backward spatial links. Let $T_j^n\\in U(1)$ transport from $(j,n)$ to $(j,n+1)$. At each step choose two on-site unitary coins $C_{1,j}^n,C_{2,j}^n\\in U(2)$ and define\n\\begin{align}\n \\eta_j&=C_{1,j}^n\\psi_j^n,\\label{g:walk-coin}\\\\\n \\xi_j^+&=(U_{j-1}^n)^{-1}\\eta_{j-1}^+,\n &\\xi_j^-&=U_j^n\\eta_{j+1}^-,\\label{g:walk-shift}\\\\\n \\psi_j^{n+1}&=T_j^n C_{2,j}^n\\xi_j.\\label{g:walk-step}\n\\end{align}\nHere the $+$ channel moves toward increasing $j$, while the $-$ channel moves toward decreasing $j$. The legacy implementation takes the two coins to be equal half mass rotations. The argument needs only their unitarity and the fact that scalar phases commute with them.","\\psi_j^{n+1}&=T_j^n C_{2,j}^n\\xi_j.\\label{g:walk-step}\n\\end{align}\nHere the $+$ channel moves toward increasing $j$, while the $-$ channel moves toward decreasing $j$. The legacy implementation takes the two coins to be equal half mass rotations. The argument needs only their unitarity and the fact that scalar phases commute with them.","For a time-dependent scalar gauge $g_j^n$, use\n\\begin{equation}\\label{g:temporal-link}\n (U_j^n)^g=g_j^nU_j^n(g_{j+1}^n)^{-1},\n \\qquad (T_j^n)^g=g_j^{n+1}T_j^n(g_j^n)^{-1}.\n\\end{equation}\nSpatial transport first brings the gauge factor to site $j$ on the old slice. Temporal transport then brings it to the new slice. Thus the step operator $\\mathcal W_n$ satisfies\n\\begin{equation}\\label{g:walk-covariance}\n \\mathcal W_n[U^g,T^g]G^n=G^{n+1}\\mathcal W_n[U,T],\n \\qquad (G^n\\psi)_j=g_j^n\\psi_j.\n\\end{equation}\nEvery factor in the step is unitary on the finite state space: the two coins and temporal phases act sitewise, and the linked shift is a permutation of the separate channels followed by unit-modulus multiplication.","\\begin{proposition}[Exact one-step continuity]\\label{g:walk-continuity}\nSet $\\rho_j^n=\\norm{\\psi_j^n}^2$ and\n\\begin{equation}\\label{g:walk-current}\n J_{j+1/2}^n=|\\eta_j^+|^2-|\\eta_{j+1}^-|^2.\n\\end{equation}\nThe linked walk obeys\n\\begin{equation}\\label{g:walk-balance}\n \\rho_j^{n+1}-\\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.\n\\end{equation}\nBoth the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc $A$ with specified left and right boundary edges,\n$Q_A^{n+1}-Q_A^n=J_{\\partial_-A}^n-J_{\\partial_+A}^n$.\n\\end{proposition}","]\\label{g:walk-continuity}\nSet $\\rho_j^n=\\norm{\\psi_j^n}^2$ and\n\\begin{equation}\\label{g:walk-current}\n J_{j+1/2}^n=|\\eta_j^+|^2-|\\eta_{j+1}^-|^2.\n\\end{equation}\nThe linked walk obeys\n\\begin{equation}\\label{g:walk-balance}\n \\rho_j^{n+1}-\\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.\n\\end{equation}\nBoth the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc $A$ with specified left and right boundary edges,\n$Q_A^{n+1}-Q_A^n=J_{\\partial_-A}^n-J_{\\partial_+A}^n$.\n\\end{proposition}","i_j^n}^2$ and\n\\begin{equation}\\label{g:walk-current}\n J_{j+1/2}^n=|\\eta_j^+|^2-|\\eta_{j+1}^-|^2.\n\\end{equation}\nThe linked walk obeys\n\\begin{equation}\\label{g:walk-balance}\n \\rho_j^{n+1}-\\rho_j^n=J_{j-1/2}^n-J_{j+1/2}^n.\n\\end{equation}\nBoth the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc $A$ with specified left and right boundary edges,\n$Q_A^{n+1}-Q_A^n=J_{\\partial_-A}^n-J_{\\partial_+A}^n$.\n\\end{proposition}","\\begin{proof}\nThe first coin preserves the local norm, so\n$\\rho_j^n=|\\eta_j^+|^2+|\\eta_j^-|^2$. The second coin and temporal phase preserve the norm after the shift, giving\n$\\rho_j^{n+1}=|\\eta_{j-1}^+|^2+|\\eta_{j+1}^-|^2$.\nSubtract these expressions and group the two terms associated with each boundary edge. This is \\eqref{g:walk-balance}. The transformed $\\eta_j$ is $g_j^n\\eta_j$, so its channel norms are unchanged. Summing the balance over the full cycle or the chosen arc cancels its internal currents.\n\\end{proof}","The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let $A=A^*$ and $A^2=I$ on a finite spinor space. Then\n\\begin{equation}\\label{g:clifford-projections}\n P^+_A=\\tfrac12(I+A),\\qquad P^-_A=\\tfrac12(I-A)\n\\end{equation}\nare self-adjoint complementary orthogonal projections. Idempotence and $P_A^+P_A^-=0$ follow by expanding the products and using $A^2=I$; self-adjointness follows from $A=A^*$. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coordinates to two Boolean channels cannot replace this construction in all three directions.","later formal development makes the directional channels depend on the Clifford matrix of that direction. Let $A=A^*$ and $A^2=I$ on a finite spinor space. Then\n\\begin{equation}\\label{g:clifford-projections}\n P^+_A=\\tfrac12(I+A),\\qquad P^-_A=\\tfrac12(I-A)\n\\end{equation}\nare self-adjoint complementary orthogonal projections. Idempotence and $P_A^+P_A^-=0$ follow by expanding the products and using $A^2=I$; self-adjointness follows from $A=A^*$. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coordinates to two Boolean channels cannot replace this construction in all three directions.","ctional channels depend on the Clifford matrix of that direction. Let $A=A^*$ and $A^2=I$ on a finite spinor space. Then\n\\begin{equation}\\label{g:clifford-projections}\n P^+_A=\\tfrac12(I+A),\\qquad P^-_A=\\tfrac12(I-A)\n\\end{equation}\nare self-adjoint complementary orthogonal projections. Idempotence and $P_A^+P_A^-=0$ follow by expanding the products and using $A^2=I$; self-adjointness follows from $A=A^*$. Different spatial Clifford matrices generally have different spectral projections. A permanent assignment of the four spinor coordinates to two Boolean channels cannot replace this construction in all three directions.","For a bijection $\\tau:V\\to V$ and a forward scalar link $u:V\\to U(1)$, define\n\\begin{equation}\\label{g:clifford-shift}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_A^+\\psi(\\tau^{-1}x)\n +u(x)^{-1}P_A^-\\psi(\\tau x).\n\\end{equation}\nThis agrees with the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\ps","For a bijection $\\tau:V\\to V$ and a forward scalar link $u:V\\to U(1)$, define\n\\begin{equation}\\label{g:clifford-shift}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_A^+\\psi(\\tau^{-1}x)\n +u(x)^{-1}P_A^-\\psi(\\tau x).\n\\end{equation}\nThis agrees with the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{","For a bijection $\\tau:V\\to V$ and a forward scalar link $u:V\\to U(1)$, define\n\\begin{equation}\\label{g:clifford-shift}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_A^+\\psi(\\tau^{-1}x)\n +u(x)^{-1}P_A^-\\psi(\\tau x).\n\\end{equation}\nThis agrees with the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Product","For a bijection $\\tau:V\\to V$ and a forward scalar link $u:V\\to U(1)$, define\n\\begin{equation}\\label{g:clifford-shift}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_A^+\\psi(\\tau^{-1}x)\n +u(x)^{-1}P_A^-\\psi(\\tau x).\n\\end{equation}\nThis agrees with the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","agrees with the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","the orientation dictionary \\eqref{g:forward-link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","link}. Its inverse is\n\\begin{equation}\\label{g:clifford-inverse}\n (S_{A,\\tau,u}^{-1}\\phi)(x)\n =u(x)^{-1}P_A^+\\phi(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","(\\tau x)\n +u(\\tau^{-1}x)P_A^-\\phi(\\tau^{-1}x).\n\\end{equation}\nSubstitution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","tution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing $x$ through $\\tau$ in each channel, gives\n\\begin{equation}\\label{g:clifford-inner}\n \\sum_x\\inner{S\\psi(x)}{S\\phi(x)}\n =\\sum_x\\inner{\\psi(x)}{\\phi(x)}.\n\\end{equation}\nFor $u^g(x)=g(\\tau x)u(x)g(x)^{-1}$, direct endpoint cancellation gives $S_{A,\\tau,u^g}G_g=G_gS_{A,\\tau,u}$. Products of these directional steps and scalar-gauge-equivariant unitary mass steps retain covariance and inner-product preservation.","The genuine unit-circle hypothesis is essential. A nonzero complex scalar is invertible, but multiplication by $2$ multiplies squared norm by $4$. The phase must therefore lie in the unit circle, not merely in $\\C^\\times$. Norm preservation depends on that mathematical hypothesis. The formalization companion identifies its precise type and transport conventions.","The checked four-spinor construction in this layer uses scalar $U(1)$ phases. The $SU(2)$ matrices of the legacy non-Abelian chapter act on an internal two-dimensional fiber. To combine that color action with a Dirac spinor, one would specify a space such as $\\C^4\\otimes\\C^2$, with spin projectors on the first factor and gauge transport on the second. This guarantees their commutation. No assertion here identifies the two-dimensional color fiber with the four-dimensional Clifford spinor or declares the resulting non-Abelian Dirac extension checked.","On a two-dimensional coordinate plane, the backward links define the based plaquette\n\\begin{equation}\\label{g:plaquette}\n W_{x,\\mu\\nu}\n =U_{x,\\mu}U_{x+\\widehat\\mu,\\nu}\n U_{x+\\widehat\\nu,\\mu}^{-1}U_{x,\\nu}^{-1}.\n\\end{equation}\nStarting with a vector at $x$, the rightmost factor moves it to $x+\\widehat\\nu$; the next moves it to the far corner; the last two bring it back to $x$. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving\n\\begin{equation}\\label{g:plaquette-covariance}\n W_{x,\\mu\\nu}^g=g_xW_{x,\\mu\\nu}g_x^{-1}.\n\\end{equation}\nFor $U(1)$ the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on the frame at $x$.","\\nu}\n =U_{x,\\mu}U_{x+\\widehat\\mu,\\nu}\n U_{x+\\widehat\\nu,\\mu}^{-1}U_{x,\\nu}^{-1}.\n\\end{equation}\nStarting with a vector at $x$, the rightmost factor moves it to $x+\\widehat\\nu$; the next moves it to the far corner; the last two bring it back to $x$. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving\n\\begin{equation}\\label{g:plaquette-covariance}\n W_{x,\\mu\\nu}^g=g_xW_{x,\\mu\\nu}g_x^{-1}.\n\\end{equation}\nFor $U(1)$ the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on the frame at $x$.","on}\nStarting with a vector at $x$, the rightmost factor moves it to $x+\\widehat\\nu$; the next moves it to the far corner; the last two bring it back to $x$. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving\n\\begin{equation}\\label{g:plaquette-covariance}\n W_{x,\\mu\\nu}^g=g_xW_{x,\\mu\\nu}g_x^{-1}.\n\\end{equation}\nFor $U(1)$ the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on the frame at $x$.","The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving\n\\begin{equation}\\label{g:plaquette-covariance}\n W_{x,\\mu\\nu}^g=g_xW_{x,\\mu\\nu}g_x^{-1}.\n\\end{equation}\nFor $U(1)$ the plaquette itself is invariant. For a noncommutative group it is its conjugacy class that is invariant; the matrix entries depend on the frame at $x$.","For $SU(2)$ links, the Wilson action of the source is the finite function\n\\begin{equation}\\label{g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum intera","For $SU(2)$ links, the Wilson action of the source is the finite function\n\\begin{equation}\\label{g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.","the Wilson action of the source is the finite function\n\\begin{equation}\\label{g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.","e is the finite function\n\\begin{equation}\\label{g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.","g:wilson-action}\n S_W[U]=\\frac1{g_{\\mathrm{YM}}^2}\n \\sum_{x,\\mu<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.","u<\\nu}\\left(1-\\tfrac12\\operatorname{Re}\\operatorname{Tr}W_{x,\\mu\\nu}\\right),\n \\qquad g_{\\mathrm{YM}}\\in\\R\\setminus\\{0\\}.\n\\end{equation}\nThe subscript distinguishes the coupling from the site gauge $g_x$. Cyclicity of trace proves $S_W[U^g]=S_W[U]$. Since an $SU(2)$ matrix has eigenvalues $e^{i\\theta},e^{-i\\theta}$, a plaquette term is $1-\\cos\\theta\\ge0$. In the Abelian case the analogous term is $1-\\operatorname{Re}W$. Specifying an invariant action is not yet a construction of its Hamiltonian phase space, Gauss constraint, or a continuum interacting field theory.","For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so\n\\begin{equation}\\label{g:abelian-cube}\n \\prod_{f\\subset\\partial c}W_f^{\\operatorname{or}(f,c)}=1.\n\\end{equation}\nThe same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If $W_f=e^{i\\theta_f}$ with each $\\theta_f$ chosen as a principal argument, it implies that the oriented phase sum lies in $2\\pi\\Z$. It does not force that sum to be zero as a real number. An unwrapped small-phase regime or a specified branch is needed for that stronger conclusion.","six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so\n\\begin{equation}\\label{g:abelian-cube}\n \\prod_{f\\subset\\partial c}W_f^{\\operatorname{or}(f,c)}=1.\n\\end{equation}\nThe same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If $W_f=e^{i\\theta_f}$ with each $\\theta_f$ chosen as a principal argument, it implies that the oriented phase sum lies in $2\\pi\\Z$. It does not force that sum to be zero as a real number. An unwrapped small-phase regime or a specified branch is needed for that stronger conclusion.","e occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so\n\\begin{equation}\\label{g:abelian-cube}\n \\prod_{f\\subset\\partial c}W_f^{\\operatorname{or}(f,c)}=1.\n\\end{equation}\nThe same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If $W_f=e^{i\\theta_f}$ with each $\\theta_f$ chosen as a principal argument, it implies that the oriented phase sum lies in $2\\pi\\Z$. It does not force that sum to be zero as a real number. An unwrapped small-phase regime or a specified branch is needed for that stronger conclusion.","ancel, so\n\\begin{equation}\\label{g:abelian-cube}\n \\prod_{f\\subset\\partial c}W_f^{\\operatorname{or}(f,c)}=1.\n\\end{equation}\nThe same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If $W_f=e^{i\\theta_f}$ with each $\\theta_f$ chosen as a principal argument, it implies that the oriented phase sum lies in $2\\pi\\Z$. It does not force that sum to be zero as a real number. An unwrapped small-phase regime or a specified branch is needed for that stronger conclusion.","Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\th","Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$,","Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the","Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$.","Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^","pose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders conc…","ampled by exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison a…","exact line integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-i…","ine integrals, with a consistent loop orientation. Write its holonomy as $W=e^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral…","^{i\\theta}$, where the small real phase is $\\theta=-q\\int_{\\square}F_{\\mu\\nu}\\,dS$. Here $q$ is the charge and $F_{\\mu\\nu}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or …","u}$ is the corresponding oriented curvature component. Take rectangles with side lengths $a_\\mu,a_\\nu=O(a)$ and uniformly controlled derivatives of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","of $F$. If the rectangle is centered at $x_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","_c$ and $F$ is $C^2$, symmetric integration removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","on removes its linear Taylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","ylor terms:\n\\begin{equation}\\label{g:centered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","ered-plaquette}\n \\theta=-qa_\\mu a_\\nu F_{\\mu\\nu}(x_c)+O(a^4).\n\\end{equation}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","}\nConsequently,\n\\begin{equation}\\label{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","abel{g:wilson-expansion}\n 1-\\operatorname{Re}W\n =\\tfrac12q^2a_\\mu^2a_\\nu^2 F_{\\mu\\nu}(x_c)^2+O(a^6).\n\\end{equation}\nTo see the remainder, use $1-\\cos\\theta=\\theta^2/2+O(\\theta^4)$ and $\\theta=O(a^2)$. The cross term from the $O(a^4)$ phase error is $O(a^6)$, while the fourth-power term is $O(a^8)$. Evaluating the curvature at a corner instead generally leaves an $O(a^3)$ phase error and hence an $O(a^5)$ action-density error. These orders concern the chosen pointwise comparison and exact-integral sampling. They do not by themselves justify convergence of an action sum or its solutions.","The finite lattice can also carry dynamical link variables. The following is the particular semidiscrete model in the Abelian source, with time continuous and space a periodic cycle of $N\\ge3$ sites. It is useful because its matter current, electric work, and Gauss propagation can all be checked in the same equations.","Fix $h>0$, $m\\in\\R$, and Hermitian spin matrices $\\alpha=\\alpha^*$ and $\\beta=\\beta^*$ on $\\C^s$. Let $\\psi_j\\in\\C^s$ and $A_j,E_j\\in\\R$, and set $U_j=e^{-ihA_j}$. The source absorbs the charge into $A$. Its real Hamiltonian is\n\\begin{align}\n H_h&=H_{\\mathrm m}+H_E,\\label{g:hamiltonian}","Fix $h>0$, $m\\in\\R$, and Hermitian spin matrices $\\alpha=\\alpha^*$ and $\\beta=\\beta^*$ on $\\C^s$. Let $\\psi_j\\in\\C^s$ and $A_j,E_j\\in\\R$, and set $U_j=e^{-ihA_j}$. The source absorbs the charge into $A$. Its real Hamiltonian is\n\\begin{align}\n H_h&=H_{\\mathrm m}+H_E,\\label{g:hamiltonian}\\\\\n H_{\\mathrm m}\n &=hm\\sum_j\\psi_j^*\\beta\\psi_j\n   +\\sum_j\\operatorname{Im}(\\psi_j^*\\alpha U_j\\psi_{j+1}),\n &H_E&=\\frac h2\\sum_jE_j^2.\\label{g:hamiltonian-parts}\n\\end{align}\nHermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_…","\\beta\\psi_j\n   +\\sum_j\\operatorname{Im}(\\psi_j^*\\alpha U_j\\psi_{j+1}),\n &H_E&=\\frac h2\\sum_jE_j^2.\\label{g:hamiltonian-parts}\n\\end{align}\nHermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}","al Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}","but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}","y conservation in this finite model. The action fixes all normalization and sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}","sign conventions:\n\\begin{equation}\\label{g:action}\n \\mathcal S_h=\\int\\left[\n \\frac{ih}{2}\\sum_j(\\psi_j^*\\dot\\psi_j-\\dot\\psi_j^*\\psi_j)\n +h\\sum_jE_j\\dot A_j-H_h\\right]dt.\n\\end{equation}\nVariations with fixed endpoint data yield $ih\\dot\\psi_j=\\partial H_h/\\partial\\overline\\psi_j$, $h\\dot A_j=\\partial H_h/\\partial E_j$, and $h\\dot E_j=-\\partial H_h/\\partial A_j$. Written out, these are\n\\begin{align}\n \\dot\\psi_j&=-im\\beta\\psi_j\n -\\frac1{2h}\\alpha U_j\\psi_{j+1}\n +\\frac1{2h}U_{j-1}^*\\alpha\\psi_{j-1},\\label{g:matter-ode}\\\\\n \\dot A_j&=E_j,\n &\\dot E_j&=J_j,\n &J_j&=\\operatorname{Re}(\\psi_j^*\\alpha U_j\\psi_{j+1}).\n \\label{g:field-ode}\n\\end{align}\nFor example, if $z_j=\\psi_j^*\\alpha U_j\\psi_{j+1}$, then\n$\\partial_{A_j}\\operatorname{Im}z_j\n=\\operatorname{Im}(-ihz_j)=-h\\operatorname{Re}z_j$.\nThis fixes the positive sign of $\\dot E_j=J_j$ for the conventions in the action.","\\dot A_j&=E_j,\n &\\dot E_j&=J_j,\n &J_j&=\\operatorname{Re}(\\psi_j^*\\alpha U_j\\psi_{j+1}).\n \\label{g:field-ode}\n\\end{align}\nFor example, if $z_j=\\psi_j^*\\alpha U_j\\psi_{j+1}$, then\n$\\partial_{A_j}\\operatorname{Im}z_j\n=\\operatorname{Im}(-ihz_j)=-h\\operatorname{Re}z_j$.\nThis fixes the positive sign of $\\dot E_j=J_j$ for the conventions in the action.","\\begin{theorem}[Charge balance and Gauss propagation]\\label{g:gauss}\nFor every differentiable solution of \\eqref{g:matter-ode}--\\eqref{g:field-ode}, define\n\\begin{equation}\\label{g:gauss-def}\n \\rho_j=\\psi_j^*\\psi_j,\\qquad\n \\bar\\rho=\\frac1N\\sum_j\\rho_j,\\qquad\n \\mathcal G_j=\\frac{E_j-E_{j-1}}h+\\rho_j-\\bar\\rho.\n\\end{equation}\nThen\n\\begin{equation}\\label{g:gauss-balance}\n \\dot\\rho_j+\\frac{J_j-J_{j-1}}h=0,\n \\qquad\\dot{\\bar\\rho}=0,\n \\qquad\\dot{\\mathcal G}_j=0.\n\\end{equation}\nIn particular, initial data satisfying $\\mathcal G_j=0$ retain that constraint on their interval of existence.\n\\end{theorem}",":matter-ode}--\\eqref{g:field-ode}, define\n\\begin{equation}\\label{g:gauss-def}\n \\rho_j=\\psi_j^*\\psi_j,\\qquad\n \\bar\\rho=\\frac1N\\sum_j\\rho_j,\\qquad\n \\mathcal G_j=\\frac{E_j-E_{j-1}}h+\\rho_j-\\bar\\rho.\n\\end{equation}\nThen\n\\begin{equation}\\label{g:gauss-balance}\n \\dot\\rho_j+\\frac{J_j-J_{j-1}}h=0,\n \\qquad\\dot{\\bar\\rho}=0,\n \\qquad\\dot{\\mathcal G}_j=0.\n\\end{equation}\nIn particular, initial data satisfying $\\mathcal G_j=0$ retain that constraint on their interval of existence.\n\\end{theorem}","\\begin{proof}\nDifferentiate $\\rho_j$ and use $\\dot\\rho_j=2\\operatorname{Re}(\\psi_j^*\\dot\\psi_j)$. The mass term contributes zero, since $\\psi_j^*\\beta\\psi_j$ is real. The forward hopping term contributes $-J_j/h$. The backward scalar is the conjugate of $\\psi_{j-1}^*\\alpha U_{j-1}\\psi_j$, using $\\alpha=\\alpha^*$ and the scalar nature of $U$. Its contribution is $J_{j-1}/h$. This proves continuity. Periodic summation gives $\\dot{\\bar\\rho}=0$. Finally,\n\\[\n \\dot{\\mathcal G}_j\n =\\frac{\\dot E_j-\\dot E_{j-1}}h+\\dot\\rho_j-\\dot{\\bar\\rho}\n =\\frac{J_j-J_{j-1}}h-\\frac{J_j-J_{j-1}}h=0.\n\\]\n\\end{proof}","$\\rho_j$ and use $\\dot\\rho_j=2\\operatorname{Re}(\\psi_j^*\\dot\\psi_j)$. The mass term contributes zero, since $\\psi_j^*\\beta\\psi_j$ is real. The forward hopping term contributes $-J_j/h$. The backward scalar is the conjugate of $\\psi_{j-1}^*\\alpha U_{j-1}\\psi_j$, using $\\alpha=\\alpha^*$ and the scalar nature of $U$. Its contribution is $J_{j-1}/h$. This proves continuity. Periodic summation gives $\\dot{\\bar\\rho}=0$. Finally,\n\\[\n \\dot{\\mathcal G}_j\n =\\frac{\\dot E_j-\\dot E_{j-1}}h+\\dot\\rho_j-\\dot{\\bar\\rho}\n =\\frac{J_j-J_{j-1}}h-\\frac{J_j-J_{j-1}}h=0.\n\\]\n\\end{proof}","2\\operatorname{Re}(\\psi_j^*\\dot\\psi_j)$. The mass term contributes zero, since $\\psi_j^*\\beta\\psi_j$ is real. The forward hopping term contributes $-J_j/h$. The backward scalar is the conjugate of $\\psi_{j-1}^*\\alpha U_{j-1}\\psi_j$, using $\\alpha=\\alpha^*$ and the scalar nature of $U$. Its contribution is $J_{j-1}/h$. This proves continuity. Periodic summation gives $\\dot{\\bar\\rho}=0$. Finally,\n\\[\n \\dot{\\mathcal G}_j\n =\\frac{\\dot E_j-\\dot E_{j-1}}h+\\dot\\rho_j-\\dot{\\bar\\rho}\n =\\frac{J_j-J_{j-1}}h-\\frac{J_j-J_{j-1}}h=0.\n\\]\n\\end{proof}","The subtraction of $\\bar\\rho$ is structural on a periodic cycle: the discrete divergence sums to zero, so a constraint containing a nonzero total charge without compensating background would be inconsistent. Equation \\eqref{g:gauss-def} supplies that neutralizing mean. Conservation of $\\mathcal G_j$ is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over $a\\le j\\le b$ gives the physical-length-weighted charge balance\n\\[\n \\frac{d}{dt}\\left(h\\sum_{j=a}^{b}\\rho_j\\right)=J_{a-1}-J_b.\n\\]","ence sums to zero, so a constraint containing a nonzero total charge without compensating background would be inconsistent. Equation \\eqref{g:gauss-def} supplies that neutralizing mean. Conservation of $\\mathcal G_j$ is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over $a\\le j\\le b$ gives the physical-length-weighted charge balance\n\\[\n \\frac{d}{dt}\\left(h\\sum_{j=a}^{b}\\rho_j\\right)=J_{a-1}-J_b.\n\\]","rge without compensating background would be inconsistent. Equation \\eqref{g:gauss-def} supplies that neutralizing mean. Conservation of $\\mathcal G_j$ is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over $a\\le j\\le b$ gives the physical-length-weighted charge balance\n\\[\n \\frac{d}{dt}\\left(h\\sum_{j=a}^{b}\\rho_j\\right)=J_{a-1}-J_b.\n\\]","\\begin{proposition}[Electric work and total energy]\\label{g:work}\nAlong the same solutions,\n\\begin{equation}\\label{g:work-balance}\n \\dot H_E=h\\sum_jE_jJ_j,\\qquad\n \\dot H_{\\mathrm m}=-h\\sum_jE_jJ_j,\\qquad\n \\dot H_h=0.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe first formula follows from $\\dot E_j=J_j$. For the matter contribution, let $q_j=\\partial H_{\\mathrm m}/\\partial\\overline\\psi_j=ih\\dot\\psi_j$. The state-dependent part of the real derivative is\n$2\\operatorname{Re}\\sum_jq_j^*\\dot\\psi_j=0$, because $q_j^*\\dot\\psi_j=-ih\\norm{\\dot\\psi_j}^2$ is purely imaginary. The remaining explicit link derivative is\n\\[\n \\sum_j\\frac{\\partial H_{\\mathrm m}}{\\partial A_j}\\dot A_j\n =-h\\sum_jJ_jE_j.\n\\]\nAdding the two sectors proves total conservation.\n\\end{proof}","For time-independent real $\\chi_j$, the transformation\n\\begin{equation}\\label{g:residual-gauge}\n \\psi_j\\mapsto e^{i\\chi_j}\\psi_j,\n \\qquad A_j\\mapsto A_j+\\frac{\\chi_{j+1}-\\chi_j}{h},\n \\qquad E_j\\mapsto E_j\n\\end{equation}\nleaves the action and equations invariant. In fact $U_j$ then has precisely the backward-link law \\eqref{g:link-transform}, every hopping scalar $z_j$ is unchanged, and the time derivatives of the gauge parameters vanish.","This is the residual symmetry of the stated temporal-gauge model. If $\\chi_j$ depends on time, the matter time derivative acquires $i\\dot\\chi_j\\psi_j$, while $\\dot A_j$ acquires $(\\dot\\chi_{j+1}-\\dot\\chi_j)/h$. With $E_j$ fixed and no scalar potential in \\eqref{g:action}, these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link $T$; it must not be transferred to this action after that degree of freedom has been removed.","dual symmetry of the stated temporal-gauge model. If $\\chi_j$ depends on time, the matter time derivative acquires $i\\dot\\chi_j\\psi_j$, while $\\dot A_j$ acquires $(\\dot\\chi_{j+1}-\\dot\\chi_j)/h$. With $E_j$ fixed and no scalar potential in \\eqref{g:action}, these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete linked walk uses its temporal link $T$; it must not be transferred to this action after that degree of freedom has been removed.","The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let $A$ be a real $C^2$ function near $x$, let $\\psi$ be $C^3$ there, and set\n\\begin{equation}\\label{g:integrated-link}\n U_h(x)=\\exp\\left(-i\\int_x^{x+h}A(s)\\,ds\\right),\\qquad\n (D_h^A\\psi)(x)\n =\\frac{U_h(x)\\psi(x+h)-U_h(x-h)^{-1}\\psi(x-h)}{2h}.\n\\end{equation}\nThe smooth $A$ in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking $A_j$ to be its edge average $h^{-1}\\int_{x_j}^{x_j+h}A(s)\\,ds$. This is an additional sampling prescription.","comparison does. Let $A$ be a real $C^2$ function near $x$, let $\\psi$ be $C^3$ there, and set\n\\begin{equation}\\label{g:integrated-link}\n U_h(x)=\\exp\\left(-i\\int_x^{x+h}A(s)\\,ds\\right),\\qquad\n (D_h^A\\psi)(x)\n =\\frac{U_h(x)\\psi(x+h)-U_h(x-h)^{-1}\\psi(x-h)}{2h}.\n\\end{equation}\nThe smooth $A$ in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking $A_j$ to be its edge average $h^{-1}\\int_{x_j}^{x_j+h}A(s)\\,ds$. This is an additional sampling prescription.","real $C^2$ function near $x$, let $\\psi$ be $C^3$ there, and set\n\\begin{equation}\\label{g:integrated-link}\n U_h(x)=\\exp\\left(-i\\int_x^{x+h}A(s)\\,ds\\right),\\qquad\n (D_h^A\\psi)(x)\n =\\frac{U_h(x)\\psi(x+h)-U_h(x-h)^{-1}\\psi(x-h)}{2h}.\n\\end{equation}\nThe smooth $A$ in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking $A_j$ to be its edge average $h^{-1}\\int_{x_j}^{x_j+h}A(s)\\,ds$. This is an additional sampling prescription.","\\begin{proposition}[Local second-order consistency]\\label{g:second-order}\nWith uniform bounds on $A$ and its derivatives through order two and on $\\psi$ and its derivatives through order three on a fixed neighborhood,\n\\begin{equation}\\label{g:stencil-limit}\n (D_h^A\\psi)(x)=(\\partial_x-iA(x))\\psi(x)+O(h^2)\n \\qquad(h\\to0).\n\\end{equation}\nThe remainder is uniform over points whose neighborhoods have the same derivative bounds.\n\\end{proposition}","\\begin{proof}\nFor fixed $x$, put\n$f_x(s)=\\exp(-i\\int_x^{x+s}A(r)\\,dr)\\psi(x+s)$.\nThen the two numerator terms in \\eqref{g:integrated-link} are $f_x(h)$ and $f_x(-h)$. The derivative at zero is\n$f_x'(0)=\\psi'(x)-iA(x)\\psi(x)$.\nTaylor expansion of $f_x$ at zero to second order with bounded third derivative gives\n$f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)$.\nDivision by $2h$ proves the assertion. The derivative bounds for $A$ and $\\psi$ give the required uniform bound on $f_x'''$.\n\\end{proof}","For fixed $x$, put\n$f_x(s)=\\exp(-i\\int_x^{x+s}A(r)\\,dr)\\psi(x+s)$.\nThen the two numerator terms in \\eqref{g:integrated-link} are $f_x(h)$ and $f_x(-h)$. The derivative at zero is\n$f_x'(0)=\\psi'(x)-iA(x)\\psi(x)$.\nTaylor expansion of $f_x$ at zero to second order with bounded third derivative gives\n$f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)$.\nDivision by $2h$ proves the assertion. The derivative bounds for $A$ and $\\psi$ give the required uniform bound on $f_x'''$.\n\\end{proof}","ed $x$, put\n$f_x(s)=\\exp(-i\\int_x^{x+s}A(r)\\,dr)\\psi(x+s)$.\nThen the two numerator terms in \\eqref{g:integrated-link} are $f_x(h)$ and $f_x(-h)$. The derivative at zero is\n$f_x'(0)=\\psi'(x)-iA(x)\\psi(x)$.\nTaylor expansion of $f_x$ at zero to second order with bounded third derivative gives\n$f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)$.\nDivision by $2h$ proves the assertion. The derivative bounds for $A$ and $\\psi$ give the required uniform bound on $f_x'''$.\n\\end{proof}",")\\,dr)\\psi(x+s)$.\nThen the two numerator terms in \\eqref{g:integrated-link} are $f_x(h)$ and $f_x(-h)$. The derivative at zero is\n$f_x'(0)=\\psi'(x)-iA(x)\\psi(x)$.\nTaylor expansion of $f_x$ at zero to second order with bounded third derivative gives\n$f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3)$.\nDivision by $2h$ proves the assertion. The derivative bounds for $A$ and $\\psi$ give the required uniform bound on $f_x'''$.\n\\end{proof}","A midpoint link $\\widetilde U_h(x)=e^{-ihA(x+h/2)}$ has the same conclusion under these bounds. The midpoint quadrature error in the integral is $O(h^3)$, and $|e^{ia}-e^{ib}|\\le|a-b|$ for real $a,b$ transfers that error to the link. The two numerator perturbations are each $O(h^3)$ for bounded $\\psi$, hence the stencil perturbation is $O(h^2)$.","\\begin{example}[Left-endpoint sampling can lose an order]\\label{g:endpoint-counterexample}\nTake $A(x)=x$ near the origin and $\\psi(x)=1$. With the left-endpoint prescription $\\widehat U_h(x)=e^{-ihA(x)}$, the stencil at zero is\n\\[\n \\frac{\\widehat U_h(0)-\\widehat U_h(-h)^{-1}}{2h}\n =\\frac{1-e^{-ih^2}}{2h}=\\frac{i}{2}h+O(h^3).\n\\]\nThe continuum value $(\\partial_x-iA(0))\\psi(0)$ is zero. Thus this perfectly smooth input has a first-order error. The formula $U_j=e^{-ihA_j}$ by itself does not identify a second-order sampling scheme.\n\\end{example}","label{g:endpoint-counterexample}\nTake $A(x)=x$ near the origin and $\\psi(x)=1$. With the left-endpoint prescription $\\widehat U_h(x)=e^{-ihA(x)}$, the stencil at zero is\n\\[\n \\frac{\\widehat U_h(0)-\\widehat U_h(-h)^{-1}}{2h}\n =\\frac{1-e^{-ih^2}}{2h}=\\frac{i}{2}h+O(h^3).\n\\]\nThe continuum value $(\\partial_x-iA(0))\\psi(0)$ is zero. Thus this perfectly smooth input has a first-order error. The formula $U_j=e^{-ihA_j}$ by itself does not identify a second-order sampling scheme.\n\\end{example}","Now take $\\psi_x\\in\\C^2$, $U_{x,\\mu}\\in SU(2)$, and local gauges $g_x\\in SU(2)$, as in the legacy non-Abelian chapter. The linked scalar\n\\begin{equation}\\label{g:linked-scalar}\n z_{x,\\mu}=\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n\\end{equation}\nis gauge invariant: the adjoint factor $g_x^*$ cancels $g_x$, and the factors at $x+\\widehat\\mu$ also cancel. It follows that the source's scalar link observable\n\\begin{equation}\\label{g:su2-current}\n j_{x,\\mu}=2\\operatorname{Im}z_{x,\\mu}\n\\end{equation}\nis invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of motion for color charge.","\\C^2$, $U_{x,\\mu}\\in SU(2)$, and local gauges $g_x\\in SU(2)$, as in the legacy non-Abelian chapter. The linked scalar\n\\begin{equation}\\label{g:linked-scalar}\n z_{x,\\mu}=\\psi_x^*U_{x,\\mu}\\psi_{x+\\widehat\\mu}\n\\end{equation}\nis gauge invariant: the adjoint factor $g_x^*$ cancels $g_x$, and the factors at $x+\\widehat\\mu$ also cancel. It follows that the source's scalar link observable\n\\begin{equation}\\label{g:su2-current}\n j_{x,\\mu}=2\\operatorname{Im}z_{x,\\mu}\n\\end{equation}\nis invariant. To obtain a continuity equation with its particular sign and normalization one must specify the hopping generator. Invariance of the observable alone is not an equation of motion for color charge.","Let $\\sigma^1,\\sigma^2,\\sigma^3$ be the Pauli matrices and define the real components $q_x^a=\\psi_x^*\\sigma^a\\psi_x$. Expand the conjugated Pauli basis as\n\\begin{equation}\\label{g:adjoint-color}\n g_x^*\\sigma^a g_x=\\sum_b R_x{}^a{}_b\\sigma^b,\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n\\end{equation}\nThe coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\\tfrac12\\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\\psi_x^*\\psi_x$ is separately invariant. A color component relative to a selected Pauli axi","Let $\\sigma^1,\\sigma^2,\\sigma^3$ be the Pauli matrices and define the real components $q_x^a=\\psi_x^*\\sigma^a\\psi_x$. Expand the conjugated Pauli basis as\n\\begin{equation}\\label{g:adjoint-color}\n g_x^*\\sigma^a g_x=\\sum_b R_x{}^a{}_b\\sigma^b,\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n\\end{equation}\nThe coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\\tfrac12\\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\\psi_x^*\\psi_x$ is separately invariant. A color component relative to a selected Pauli axis is not generally invariant under all local frames.","a^1,\\sigma^2,\\sigma^3$ be the Pauli matrices and define the real components $q_x^a=\\psi_x^*\\sigma^a\\psi_x$. Expand the conjugated Pauli basis as\n\\begin{equation}\\label{g:adjoint-color}\n g_x^*\\sigma^a g_x=\\sum_b R_x{}^a{}_b\\sigma^b,\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n\\end{equation}\nThe coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\\tfrac12\\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\\psi_x^*\\psi_x$ is separately invariant. A color component relative to a selected Pauli axis is not generally invariant under all local frames.","pand the conjugated Pauli basis as\n\\begin{equation}\\label{g:adjoint-color}\n g_x^*\\sigma^a g_x=\\sum_b R_x{}^a{}_b\\sigma^b,\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n\\end{equation}\nThe coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\\tfrac12\\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\\psi_x^*\\psi_x$ is separately invariant. A color component relative to a selected Pauli axis is not generally invariant under all local frames.",",\n \\qquad q_x^{g,a}=\\sum_bR_x{}^a{}_bq_x^b.\n\\end{equation}\nThe coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product $\\tfrac12\\operatorname{Tr}(AB)$ is preserved by conjugation, and the Pauli basis is orthonormal for it; hence $R_xR_x^{\\mathsf T}=I$. Thus the color vector rotates while its Euclidean squared length stays fixed. The scalar charge $\\psi_x^*\\psi_x$ is separately invariant. A color component relative to a selected Pauli axis is not generally invariant under all local frames.","The same distinction applies to a plaquette: $W$ changes by conjugation, while its trace does not. Keeping the full matrix retains transport information that a single trace can lose. Even for $SU(2)$, identical traces at separate plaquettes do not determine the relative frames needed to combine them into a larger loop.","\\gaugefigure{03_nonabelian_order}{Noncommuting $SU(2)$ factors retain their order. A based holonomy changes by conjugation, while the normalized Wilson trace is unchanged. The color fiber here is separate from the four-component Clifford spinor.}","The Abelian rule ``multiply the fine plaquettes'' hides two choices in a noncommutative group: where each plaquette is based, and in which order the transported matrices are multiplied. A $2\\times2$ patch makes both choices visible without an asymptotic expansion.","Write $X_{ij}$ for the backward horizontal link from $(i+1,j)$ to $(i,j)$, where $i=0,1$ and $j=0,1,2$. Write $Y_{ij}$ for the backward vertical link from $(i,j+1)$ to $(i,j)$, where $i=0,1,2$ and $j=0,1$. Define the four fine plaquettes by\n\\begin{equation}\\label{g:fine-plaquette}\n W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},\n \\qquad i,j\\in\\{0,1\\}.\n\\end{equation}\nEach $W_{ij}$ acts at its own lower-left basepoint $(i,j)$. The two row holonomies, based at $(0,j)$, are\n\\begin{equation}\\label{g:row-holonomies}\n R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},\n \\qquad j=0,1.\n\\end{equation}\nThe conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.","backward horizontal link from $(i+1,j)$ to $(i,j)$, where $i=0,1$ and $j=0,1,2$. Write $Y_{ij}$ for the backward vertical link from $(i,j+1)$ to $(i,j)$, where $i=0,1,2$ and $j=0,1$. Define the four fine plaquettes by\n\\begin{equation}\\label{g:fine-plaquette}\n W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},\n \\qquad i,j\\in\\{0,1\\}.\n\\end{equation}\nEach $W_{ij}$ acts at its own lower-left basepoint $(i,j)$. The two row holonomies, based at $(0,j)$, are\n\\begin{equation}\\label{g:row-holonomies}\n R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},\n \\qquad j=0,1.\n\\end{equation}\nThe conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.","o $(i,j)$, where $i=0,1$ and $j=0,1,2$. Write $Y_{ij}$ for the backward vertical link from $(i,j+1)$ to $(i,j)$, where $i=0,1,2$ and $j=0,1$. Define the four fine plaquettes by\n\\begin{equation}\\label{g:fine-plaquette}\n W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},\n \\qquad i,j\\in\\{0,1\\}.\n\\end{equation}\nEach $W_{ij}$ acts at its own lower-left basepoint $(i,j)$. The two row holonomies, based at $(0,j)$, are\n\\begin{equation}\\label{g:row-holonomies}\n R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},\n \\qquad j=0,1.\n\\end{equation}\nThe conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.","$j=0,1,2$. Write $Y_{ij}$ for the backward vertical link from $(i,j+1)$ to $(i,j)$, where $i=0,1,2$ and $j=0,1$. Define the four fine plaquettes by\n\\begin{equation}\\label{g:fine-plaquette}\n W_{ij}=X_{ij}Y_{i+1,j}X_{i,j+1}^{-1}Y_{ij}^{-1},\n \\qquad i,j\\in\\{0,1\\}.\n\\end{equation}\nEach $W_{ij}$ acts at its own lower-left basepoint $(i,j)$. The two row holonomies, based at $(0,j)$, are\n\\begin{equation}\\label{g:row-holonomies}\n R_j=(X_{0j}W_{1j}X_{0j}^{-1})W_{0j},\n \\qquad j=0,1.\n\\end{equation}\nThe conjugation transports the right plaquette into the left base fiber before multiplying it with the left plaquette.","\\begin{theorem}[Ordered patch identity]\\label{g:patch-theorem}\nFor invertible links in any matrix group, the coarse boundary holonomy satisfies\n\\begin{align}\n W^{(2)}\n &:=X_{00}X_{10}Y_{20}Y_{21}\n X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}\n \\label{g:coarse-boundary}\\\\\n &=R_0Y_{00}R_1Y_{00}^{-1}.\\label{g:patch-identity}\n\\end{align}\nThis identity requires neither small links nor commuting links.\n\\end{theorem}","\\Needspace{10\\baselineskip}\n\\begin{proof}\nFirst expand one row while retaining the product order:\n\\begin{align*}\n R_j\n &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}Y_{1j}^{-1}\n X_{0j}^{-1}X_{0j}Y_{1j}X_{0,j+1}^{-1}Y_{0j}^{-1}\\\\\n &=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.\n\\end{align*}\nThe adjacent pairs $X_{0j}^{-1}X_{0j}$ and $Y_{1j}^{-1}Y_{1j}$ cancel. Substituting these row expressions gives\n\\begin{align*}\n R_0Y_{00}R_1Y_{00}^{-1}\n ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}\n \\underbrace{Y_{00}^{-1}Y_{00}}_{I}","&=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.\n\\end{align*}\nThe adjacent pairs $X_{0j}^{-1}X_{0j}$ and $Y_{1j}^{-1}Y_{1j}$ cancel. Substituting these row expressions gives\n\\begin{align*}\n R_0Y_{00}R_1Y_{00}^{-1}\n ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}\n \\underbrace{Y_{00}^{-1}Y_{00}}_{I}","&=X_{0j}X_{1j}Y_{2j}X_{1,j+1}^{-1}X_{0,j+1}^{-1}Y_{0j}^{-1}.\n\\end{align*}\nThe adjacent pairs $X_{0j}^{-1}X_{0j}$ and $Y_{1j}^{-1}Y_{1j}$ cancel. Substituting these row expressions gives\n\\begin{align*}\n R_0Y_{00}R_1Y_{00}^{-1}\n ={}&X_{00}X_{10}Y_{20}X_{11}^{-1}X_{01}^{-1}\n \\underbrace{Y_{00}^{-1}Y_{00}}_{I}\\\\\n &\\quad\\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.\n\\end{align*}\nNow $X_{01}^{-1}X_{01}$ cancels, followed by $X_{11}^{-1}X_{11}$. The remaining factors are exactly \\eqref{g:coarse-boundary}.\n\\end{proof}","&\\quad\\cdot X_{01}X_{11}Y_{21}X_{12}^{-1}X_{02}^{-1}Y_{01}^{-1}Y_{00}^{-1}.\n\\end{align*}\nNow $X_{01}^{-1}X_{01}$ cancels, followed by $X_{11}^{-1}X_{11}$. The remaining factors are exactly \\eqref{g:coarse-boundary}.\n\\end{proof}","All terms in \\eqref{g:patch-identity} transform by conjugation at $(0,0)$ once their displayed transports are included. This checks the basepoint typing as well as the algebra. In the Abelian case conjugations disappear and the fine plaquettes can be reordered, recovering the familiar surface product. In the non-Abelian case a naive product of the four untransported matrices can be formed after identifying their coordinate spaces, but it has no reason to obey the required local gauge transformation or to equal the boundary holonomy.","Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed $A,B\\in\\mathfrak{su}(2)$ define the commutator loop\n\\begin{equation}\\label{g:commutator-loop}\n W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.\n\\end{equation}\nHere $A$ and $B$ are constant skew-Hermitian traceless matrices, and $h$ is a real scalar near zero. Because $W_0=I$, a single analytic matrix logarithm near the identity can be used for all sufficiently small $|h|$. Define\n\\begin{equation}\\label{g:log-curvature}\n F_h=h^{-2}\\log W_h,\\qquad h\\ne0.\n\\end{equation}\nThis construction describes a local constant-matrix model of noncommuting transport. A varying connection has additional derivative terms and must be analyzed with those terms present.","curvature are different operations. For fixed $A,B\\in\\mathfrak{su}(2)$ define the commutator loop\n\\begin{equation}\\label{g:commutator-loop}\n W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.\n\\end{equation}\nHere $A$ and $B$ are constant skew-Hermitian traceless matrices, and $h$ is a real scalar near zero. Because $W_0=I$, a single analytic matrix logarithm near the identity can be used for all sufficiently small $|h|$. Define\n\\begin{equation}\\label{g:log-curvature}\n F_h=h^{-2}\\log W_h,\\qquad h\\ne0.\n\\end{equation}\nThis construction describes a local constant-matrix model of noncommuting transport. A varying connection has additional derivative terms and must be analyzed with those terms present.","ations. For fixed $A,B\\in\\mathfrak{su}(2)$ define the commutator loop\n\\begin{equation}\\label{g:commutator-loop}\n W_h=e^{hA}e^{hB}e^{-hA}e^{-hB}.\n\\end{equation}\nHere $A$ and $B$ are constant skew-Hermitian traceless matrices, and $h$ is a real scalar near zero. Because $W_0=I$, a single analytic matrix logarithm near the identity can be used for all sufficiently small $|h|$. Define\n\\begin{equation}\\label{g:log-curvature}\n F_h=h^{-2}\\log W_h,\\qquad h\\ne0.\n\\end{equation}\nThis construction describes a local constant-matrix model of noncommuting transport. A varying connection has additional derivative terms and must be analyzed with those terms present.","\\begin{proposition}[Local logarithmic refinement]\\label{g:curvature-order}\nOn the common analytic logarithm branch, in any fixed matrix norm,\n\\begin{align}\n F_h&=[A,B]+hC_3(A,B)+O(h^2),\\label{g:log-series}\\\\\n C_3(A,B)&=\\tfrac12[A+B,[A,B]],\\label{g:cubic-coefficient}\\\\\n F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).\n \\label{g:richardson-formula}\n\\end{align}\nFor constant $G\\in SU(2)$, simultaneous conjugation of $A,B$ conjugates both $F_h$ and $F_h^{[1]}$ by $G$.\n\\end{proposition}","F_h^{[1]}&:=2F_{h/2}-F_h=[A,B]+O(h^2).\n \\label{g:richardson-formula}\n\\end{align}\nFor constant $G\\in SU(2)$, simultaneous conjugation of $A,B$ conjugates both $F_h$ and $F_h^{[1]}$ by $G$.\n\\end{proposition}","\\begin{proof}\nMultiplying the four exponential power series through degree three gives\n\\[\n W_h=I+h^2[A,B]+\\frac{h^3}{2}[A+B,[A,B]]+O(h^4).\n\\]\nThe analytic logarithm has $\\log(I+Z)=Z+O(\\norm Z^2)$ near zero. Since $Z=W_h-I=O(h^2)$, its correction begins at order $h^4$. Dividing by $h^2$ proves \\eqref{g:log-series}. At half spacing the linear term is $hC_3/2$, so $2F_{h/2}-F_h$ removes it exactly and leaves an $O(h^2)$ remainder. Finally, the power series for both the exponential and the local logarithm commute with conjugation. Applying them term by term proves the covariance statements.\n\\end{proof}","Multiplying the four exponential power series through degree three gives\n\\[\n W_h=I+h^2[A,B]+\\frac{h^3}{2}[A+B,[A,B]]+O(h^4).\n\\]\nThe analytic logarithm has $\\log(I+Z)=Z+O(\\norm Z^2)$ near zero. Since $Z=W_h-I=O(h^2)$, its correction begins at order $h^4$. Dividing by $h^2$ proves \\eqref{g:log-series}. At half spacing the linear term is $hC_3/2$, so $2F_{h/2}-F_h$ removes it exactly and leaves an $O(h^2)$ remainder. Finally, the power series for both the exponential and the local logarithm commute with conjugation. Applying them term by term proves the covariance statements.\n\\end{proof}","The common logarithm branch is essential: unrelated choices can introduce terms of size $h^{-2}$ after rescaling and invalidate the cancellation. The Richardson value is a linear combination of Lie-algebra estimates; it is not asserted to be the logarithm obtained by the exact patch rule.","midrule\\endhead\nInner product & $\\inner{v}{w}$ is conjugate-linear in $v$ and linear in $w$. Matrix adjoints are written $T^*$.","Endpoint transport & $T_{y\\leftarrow x}$ maps the fiber at $x$ to the fiber at $y$. Under a local unitary frame $g$, it becomes $g_yT_{y\\leftarrow x}g_x^{-1}$.","Matrix order & $T_{z\\leftarrow y}T_{y\\leftarrow x}$ transports first from $x$ to $y$, then from $y$ to $z$. The rightmost factor acts first.","Linked shifts & A coefficient $U_j$ in $U_j\\psi_{j+1}$ transports from $j+1$ back to $j$. Thus $U_j'=g_jU_jg_{j+1}^{-1}$. An outward link uses the opposite endpoint convention.","Gauge groups & $U(1)$ links are unit complex numbers. The finite non-Abelian construction uses unitary matrices, with $SU(2)$ as an explicit example. Scalar multiplication identities are not silently generalized to noncommuting matrices.","Mesh and time & Mesh widths are positive when dividing by them. The finite matter system uses $\\hbar=1$ and the displayed charge convention. Refinement level and evolution time are different parameters.","We use the forward convention of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner p","We use the forward convention of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established e","We use the forward convention of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","use the forward convention of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","forward convention of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","on of the formal refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","refinement modules. A link $u(x)$ transports from $x$ to $\\tau x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","x$, so\n\\[\n u^g(x)=g(\\tau x)u(x)g(x)^{-1}.\n\\]\nThus $u(x)=U(x)^{-1}$ when $U(x)$ denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to $U(1)$; the unit modulus assumption is used in every Hilbert norm statement. Let $A=A^*$, $A^2=1$, act on $\\C^4$, and put $P_\\pm=(1\\pm A)/2$. The directional transport is\n\\begin{equation}\n (S_{A,\\tau,u}\\psi)(x)\n =u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n  +u(x)^{-1}P_-\\psi(\\tau x).\n \\label{g:source-shift}\n\\end{equation}\nThe channels are orthogonal. A bijective site translation and unit links therefore make this an isometry for the finite counting-space inner product established earlier.","\\subsection{Midpoint geometry and independent gauges}\nWrite $e_x$ for the retained endpoint and $m_x$ for the midpoint of the edge $x\\to\\tau x$. The fine translation is the bijection\n\\[\n \\sigma e_x=m_x,\\qquad \\sigma m_x=e_{\\tau x}.\n\\]\nConsequently $\\sigma^2e_x=e_{\\tau x}$ and $\\sigma^2m_x=m_{\\tau x}$. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to $\\sigma$ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.","y and independent gauges}\nWrite $e_x$ for the retained endpoint and $m_x$ for the midpoint of the edge $x\\to\\tau x$. The fine translation is the bijection\n\\[\n \\sigma e_x=m_x,\\qquad \\sigma m_x=e_{\\tau x}.\n\\]\nConsequently $\\sigma^2e_x=e_{\\tau x}$ and $\\sigma^2m_x=m_{\\tau x}$. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing to $\\sigma$ is compatible with this combinatorics; it is not by itself a chart or continuum limit theorem.","Choose unit links $a(x)$ and $b(x)$ on the first and second half-edge with\n\\begin{equation}\n b(x)a(x)=u(x).\n \\label{g:ordered-product}\n\\end{equation}\nThe associated fine link is $v(e_x)=a(x)$, $v(m_x)=b(x)$. The interpolation and reconstruction maps are\n\\begin{align}\n (I_a\\psi)(e_x)&=\\psi(x),&\n (I_a\\psi)(m_x)&=a(x)\\psi(x),\\label{g:edge-interpolation}","Choose unit links $a(x)$ and $b(x)$ on the first and second half-edge with\n\\begin{equation}\n b(x)a(x)=u(x).\n \\label{g:ordered-product}\n\\end{equation}\nThe associated fine link is $v(e_x)=a(x)$, $v(m_x)=b(x)$. The interpolation and reconstruction maps are\n\\begin{align}\n (I_a\\psi)(e_x)&=\\psi(x),&\n (I_a\\psi)(m_x)&=a(x)\\psi(x),\\label{g:edge-interpolation}\\\\\n (R_a\\Psi)(x)&=\\tfrac12\\bigl(\\Psi(e_x)+a(x)^{-1}\\Psi(m_x)\\bigr).\n \\label{g:edge-restriction}\n\\end{align}\nThe midpoint is a transported value, not an independently sampled field value. This identifies exactly the subspace on which the refinement theorem acts.","Coarse endpoint gauges $g(x)$ and midpoint gauges $q(x)$ may be chosen independently. The two sublinks transform by\n\\[\n a^g(x)=q(x)a(x)g(x)^{-1},\\qquad\n b^g(x)=g(\\tau x)b(x)q(x)^{-1}.\n\\]\nThe midpoint factors cancel in $b^ga^g$, leaving the required coarse transformation. At the midpoint,\n\\[\n a^g(x)g(x)\\psi(x)=q(x)a(x)\\psi(x),\n\\]\nwhile endpoint covariance is immediate. Hence interpolation commutes exactly with the corresponding gauge actions. This argument allows independent new gauge data at every inserted midpoint.","\\begin{theorem}[Exact two-half-step refinement]\nFor every coarse state $\\psi$ and subdivision satisfying \\eqref{g:ordered-product},\n\\begin{equation}\n S_{A,\\sigma,v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n\\end{equation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\ps","\\begin{theorem}[Exact two-half-step refinement]\nFor every coarse state $\\psi$ and subdivision satisfying \\eqref{g:ordered-product},\n\\begin{equation}\n S_{A,\\sigma,v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n\\end{equation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:e","\\begin{theorem}[Exact two-half-step refinement]\nFor every coarse state $\\psi$ and subdivision satisfying \\eqref{g:ordered-product},\n\\begin{equation}\n S_{A,\\sigma,v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n\\end{equation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","ct},\n\\begin{equation}\n S_{A,\\sigma,v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n\\end{equation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","v}^{\\,2}I_a\\psi=I_aS_{A,\\tau,u}\\psi.\n \\label{g:half-intertwining}\n\\end{equation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","quation}\nNo flatness hypothesis is needed.\n\\end{theorem}\n\\begin{proof}\nWhen the fine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","ine shift is applied twice, the mixed projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","d projector terms vanish because $P_+P_-=P_-P_+=0$. The remaining positive channel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","nel travels two half-edges forward, and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","and the negative channel travels the same pair backward. At $e_x$, the positive coefficient is $b(\\tau^{-1}x)a(\\tau^{-1}x)=u(\\tau^{-1}x)$; the negative coefficient is $a(x)^{-1}b(x)^{-1}=u(x)^{-1}$. The result is exactly \\eqref{g:source-shift}. At $m_x$, the two surviving terms are\n\\[\n a(x)u(\\tau^{-1}x)P_+\\psi(\\tau^{-1}x)\n   +b(x)^{-1}P_-\\psi(\\tau x).\n\\]\nSince $b(x)^{-1}=a(x)u(x)^{-1}$, this is $a(x)$ times the coarse shifted state, as required by \\eqref{g:edge-interpolation}.\n\\end{proof}","\\subsection{The measure is part of the comparison}\nFor finite sites, give each fine point half the coarse weight:\n\\[\n \\inner{\\Psi}{\\Phi}_{1/2}\n =\\tfrac12\\sum_x\\bigl(\n    \\inner{\\Psi(e_x)}{\\Phi(e_x)}\n   +\\inner{\\Psi(m_x)}{\\Phi(m_x)}\\bigr).\n\\]\nUnitarity of $a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.","ine point half the coarse weight:\n\\[\n \\inner{\\Psi}{\\Phi}_{1/2}\n =\\tfrac12\\sum_x\\bigl(\n    \\inner{\\Psi(e_x)}{\\Phi(e_x)}\n   +\\inner{\\Psi(m_x)}{\\Phi(m_x)}\\bigr).\n\\]\nUnitarity of $a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.","ac12\\sum_x\\bigl(\n    \\inner{\\Psi(e_x)}{\\Phi(e_x)}\n   +\\inner{\\Psi(m_x)}{\\Phi(m_x)}\\bigr).\n\\]\nUnitarity of $a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.",")}{\\Phi(m_x)}\\bigr).\n\\]\nUnitarity of $a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.","\\]\nUnitarity of $a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.","$a(x)$ then gives\n\\begin{equation}\n \\inner{I_a\\psi}{I_a\\phi}_{1/2}=\\inner{\\psi}{\\phi},\n \\qquad\n \\inner{I_a\\psi}{\\Psi}_{1/2}=\\inner{\\psi}{R_a\\Psi}.\n \\label{g:edge-adjoint}\n\\end{equation}\nThe second identity follows by moving $a(x)$ from the first argument of the inner product to the second as $a(x)^{-1}$. It explains the averaging formula, including its factor $1/2$. In particular $R_aI_a=1$ and $I_aR_a$ is the orthogonal projection onto the transported midpoint subspace. Endpoint evaluation also gives a left inverse, but does not have this adjoint interpretation.","A local mass step can be refined factor by factor. If a local linear map $N$ satisfies $N^2=M$, then $N$ commutes with the scalar interpolation. Consequently\n\\[\n N^2S_{A,\\sigma,v}^{\\,2}N^2I_a\n   =I_aMS_{A,\\tau,u}M.\n\\]\nUnitary spectral phases provide explicit choices of $N$. The refined expression replaces each coarse factor by its two half factors. It does not assert that two repetitions of a complete fine $N S_{A,\\sigma,v}N$ walk equal the coarse walk: the intervening order would be different.","s step can be refined factor by factor. If a local linear map $N$ satisfies $N^2=M$, then $N$ commutes with the scalar interpolation. Consequently\n\\[\n N^2S_{A,\\sigma,v}^{\\,2}N^2I_a\n   =I_aMS_{A,\\tau,u}M.\n\\]\nUnitary spectral phases provide explicit choices of $N$. The refined expression replaces each coarse factor by its two half factors. It does not assert that two repetitions of a complete fine $N S_{A,\\sigma,v}N$ walk equal the coarse walk: the intervening order would be different.","\\gaugefigure[H]{07_edge_subdivision}{Integrated abelian phases compose exactly under edge subdivision. For the displayed coefficient $a(s)=\\sin s$, the two subinterval integrals sum to the whole-edge integral. Independently sampled coarse and fine phases can disagree; that approximation error is separate from exact transport composition.}","\\section{Cubical transport and the curvature obstruction}\n\\label{g:cube}\nThree one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one $2\\times2\\times2$ vertex cube. Let the vertices be $(x,y,z)\\in\\{0,1\\}^3$, with positive links $\\ell_x(y,z)$, $\\ell_y(x,z)$ and $\\ell_z(x,y)$. There are twelve links and six faces. Under a vertex gauge, every link acquires its target gauge on the left and its source inverse on the right.","For example, the two positive paths across an $xy$ face at height $z$ are\n\\[\n p_{xy}(z)=\\ell_y(1,z)\\ell_x(0,z),\\qquad\n p_{yx}(z)=\\ell_x(1,z)\\ell_y(0,z).\n\\]\nDefine $\\Omega_{xy}(z)=p_{xy}(z)p_{yx}(z)^{-1}$, and define the $xz$ and $yz$ face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each $\\Omega$ is gauge invariant, and $\\Omega=1$ precisely when the two path phases agree.","\\subsection{Six paths and one Bianchi relation}\nThe three coordinate directions can be traversed in six orders, denoted $p_{XYZ}$, $p_{XZY}$, $p_{YXZ}$, $p_{YZX}$, $p_{ZXY}$ and $p_{ZYX}$. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give\n\\Needspace{6\\baselineskip}\n\\begin{align*}\n p_{XYZ}&=\\Omega_{xy}(0)p_{YXZ},&\n p_{XYZ}&=\\Omega_{yz}(1)p_{XZY},","\\subsection{Six paths and one Bianchi relation}\nThe three coordinate directions can be traversed in six orders, denoted $p_{XYZ}$, $p_{XZY}$, $p_{YXZ}$, $p_{YZX}$, $p_{ZXY}$ and $p_{ZYX}$. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give\n\\Needspace{6\\baselineskip}\n\\begin{align*}\n p_{XYZ}&=\\Omega_{xy}(0)p_{YXZ},&\n p_{XYZ}&=\\Omega_{yz}(1)p_{XZY},\\\\\n p_{YXZ}&=\\Omega_{xz}(1)p_{YZX},&\n p_{XZY}&=\\Omega_{xz}(0)p_{ZXY},\\\\\n p_{YZX}&=\\Omega_{yz}(0)p_{ZYX},&\n p_{ZXY}&=\\Omega_{xy}(1)p_{ZYX}.\n\\end{align*}\nEach identity follows by replacing the two-edge portion being swapped by its alternative path and its face ratio. The scalar nature of the links permits the common third edge to commute past the ratio.","\\begin{proposition}[Abelian cubical Bianchi identity]\nThe six face phases satisfy\n\\begin{equation}\n \\Omega_{yz}(1)\\Omega_{xz}(0)\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","ty]\nThe six face phases satisfy\n\\begin{equation}\n \\Omega_{yz}(1)\\Omega_{xz}(0)\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","ce phases satisfy\n\\begin{equation}\n \\Omega_{yz}(1)\\Omega_{xz}(0)\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","isfy\n\\begin{equation}\n \\Omega_{yz}(1)\\Omega_{xz}(0)\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}",")\\Omega_{xy}(1)\n =\\Omega_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","_{xy}(0)\\Omega_{xz}(1)\\Omega_{yz}(0).\n \\label{g:bianchi}\n\\end{equation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","uation}\nIf all six faces are flat, the six corner path phases agree. Flatness of the five faces other than $xy(1)$ already forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","ady forces the sixth to be flat.\n\\end{proposition}\n\\begin{proof}\nUse the three identities down the left route from $XYZ$ through $YXZ,YZX$ to $ZYX$, and the three down the right route through $XZY,ZXY$. They express $p_{XYZ}$ as the respective products in \\eqref{g:bianchi} times $p_{ZYX}$. Cancel the unit terminal phase. If the five displayed independent face phases equal one, the identity gives $\\Omega_{xy}(1)=1$. When all faces are flat, every adjacent path swap leaves the phase unchanged, and the displayed relations connect all six paths.\n\\end{proof}","\\subsection{Interpolation is possible even when it depends on a path}\nChoose one reference path from the origin to each vertex and let $r(v)$ be its transport. Then\n\\[\n (I_\\ell\\psi)(v)=r(v)\\psi,\n \\qquad\n \\inner{\\Psi}{\\Phi}_{\\rm cube}\n =\\tfrac18\\sum_{v\\in\\{0,1\\}^3}\\inner{\\Psi(v)}{\\Phi(v)}.\n\\]\nEvery $r(v)$ is unitary, so this map is a Hilbert isometry from one spinor to the eight-child cell. Its gauge covariance follows from telescoping along the selected paths. Neither property requires flatness. What curvature obstructs is independence from the reference-path choice.","Suppose two such paths have phases $p$ and $q$ with $p=\\Omega q$. Their discrepancy has the exact form\n\\begin{equation}\n D(p,q;\\psi)=p\\psi-q\\psi=q(\\Omega-1)\\psi,\n \\qquad\n \\norm{D(p,q;\\psi)}^2=|\\Omega-1|^2\\norm{\\psi}^2.\n \\label{g:curvature-defect}\n\\end{equation}\nThis is an equality, not a small-curvature approximation. It follows by factoring $q$ and using $|q|=1$. For a nonzero test spinor, the defect vanishes exactly when $\\Omega=1$. The formal nonflat cube has $p_{XYZ}=-1$ and $p_{YXZ}=1$, producing a nonzero defect on the first basis spinor. Thus the distinction between covariant interpolation and path-independent interpolation is witnessed by explicit data.","phases $p$ and $q$ with $p=\\Omega q$. Their discrepancy has the exact form\n\\begin{equation}\n D(p,q;\\psi)=p\\psi-q\\psi=q(\\Omega-1)\\psi,\n \\qquad\n \\norm{D(p,q;\\psi)}^2=|\\Omega-1|^2\\norm{\\psi}^2.\n \\label{g:curvature-defect}\n\\end{equation}\nThis is an equality, not a small-curvature approximation. It follows by factoring $q$ and using $|q|=1$. For a nonzero test spinor, the defect vanishes exactly when $\\Omega=1$. The formal nonflat cube has $p_{XYZ}=-1$ and $p_{YXZ}=1$, producing a nonzero defect on the first basis spinor. Thus the distinction between covariant interpolation and path-independent interpolation is witnessed by explicit data.","eir discrepancy has the exact form\n\\begin{equation}\n D(p,q;\\psi)=p\\psi-q\\psi=q(\\Omega-1)\\psi,\n \\qquad\n \\norm{D(p,q;\\psi)}^2=|\\Omega-1|^2\\norm{\\psi}^2.\n \\label{g:curvature-defect}\n\\end{equation}\nThis is an equality, not a small-curvature approximation. It follows by factoring $q$ and using $|q|=1$. For a nonzero test spinor, the defect vanishes exactly when $\\Omega=1$. The formal nonflat cube has $p_{XYZ}=-1$ and $p_{YXZ}=1$, producing a nonzero defect on the first basis spinor. Thus the distinction between covariant interpolation and path-independent interpolation is witnessed by explicit data.","as the exact form\n\\begin{equation}\n D(p,q;\\psi)=p\\psi-q\\psi=q(\\Omega-1)\\psi,\n \\qquad\n \\norm{D(p,q;\\psi)}^2=|\\Omega-1|^2\\norm{\\psi}^2.\n \\label{g:curvature-defect}\n\\end{equation}\nThis is an equality, not a small-curvature approximation. It follows by factoring $q$ and using $|q|=1$. For a nonzero test spinor, the defect vanishes exactly when $\\Omega=1$. The formal nonflat cube has $p_{XYZ}=-1$ and $p_{YXZ}=1$, producing a nonzero defect on the first basis spinor. Thus the distinction between covariant interpolation and path-independent interpolation is witnessed by explicit data.","For later estimates, retain the five-face quantity\n\\begin{align}\n \\kappa(\\ell)={}&|\\Omega_{xy}(0)-1|+|\\Omega_{xz}(1)-1|\n  +|\\Omega_{yz}(0)-1|\\notag\\\\\n &+|\\Omega_{yz}(1)-1|+|\\Omega_{xz}(0)-1|.\n \\label{g:fiveface}\n\\end{align}\nThe omitted face is $xy(1)$. This selection is the one used by the formal five-swap composition; the multiplicities in later constants come from this choice.","&+|\\Omega_{yz}(1)-1|+|\\Omega_{xz}(0)-1|.\n \\label{g:fiveface}\n\\end{align}\nThe omitted face is $xy(1)$. This selection is the one used by the formal five-swap composition; the multiplicities in later constants come from this choice.","\\section{Exact straight refinement and curvature are independent}\n\\label{g:independence}\nThe one-edge identity can be assembled into a three-axis refinement on the fine site set $\\mathrm{Site}\\times\\{0,1\\}^3$. The half translation $\\sigma_i$ toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation $\\tau_i$ to the site. Thus $\\sigma_i^2$ is the lift of $\\tau_i$. A phase frame $r(s,c)\\in U(1)$ defines\n\\[\n (I_r\\psi)(s,c)=r(s,c)\\psi(s).\n\\]\nWith one-eighth fine weights this is again an isometry.","Exact straight refinement and curvature are independent}\n\\label{g:independence}\nThe one-edge identity can be assembled into a three-axis refinement on the fine site set $\\mathrm{Site}\\times\\{0,1\\}^3$. The half translation $\\sigma_i$ toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation $\\tau_i$ to the site. Thus $\\sigma_i^2$ is the lift of $\\tau_i$. A phase frame $r(s,c)\\in U(1)$ defines\n\\[\n (I_r\\psi)(s,c)=r(s,c)\\psi(s).\n\\]\nWith one-eighth fine weights this is again an isometry.","t and curvature are independent}\n\\label{g:independence}\nThe one-edge identity can be assembled into a three-axis refinement on the fine site set $\\mathrm{Site}\\times\\{0,1\\}^3$. The half translation $\\sigma_i$ toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation $\\tau_i$ to the site. Thus $\\sigma_i^2$ is the lift of $\\tau_i$. A phase frame $r(s,c)\\in U(1)$ defines\n\\[\n (I_r\\psi)(s,c)=r(s,c)\\psi(s).\n\\]\nWith one-eighth fine weights this is again an isometry.","ndent}\n\\label{g:independence}\nThe one-edge identity can be assembled into a three-axis refinement on the fine site set $\\mathrm{Site}\\times\\{0,1\\}^3$. The half translation $\\sigma_i$ toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation $\\tau_i$ to the site. Thus $\\sigma_i^2$ is the lift of $\\tau_i$. A phase frame $r(s,c)\\in U(1)$ defines\n\\[\n (I_r\\psi)(s,c)=r(s,c)\\psi(s).\n\\]\nWith one-eighth fine weights this is again an isometry.","\\subsection{The straight compatibility contract}\nFor each direction, canonical fine links can be defined by\n\\[\n v_i(p)=r(\\sigma_i p)\\,\\gamma_i(p)\\,r(p)^{-1},\n\\]\nwhere $\\gamma_i(p)=1$ on a noncarrying half step and $\\gamma_i(p)=u_i(s)$ on a carrying half step. Cancellation of the intermediate frame gives\n\\begin{equation}\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n\\end{equation}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here","\\subsection{The straight compatibility contract}\nFor each direction, canonical fine links can be defined by\n\\[\n v_i(p)=r(\\sigma_i p)\\,\\gamma_i(p)\\,r(p)^{-1},\n\\]\nwhere $\\gamma_i(p)=1$ on a noncarrying half step and $\\gamma_i(p)=u_i(s)$ on a carrying half step. Cancellation of the intermediate frame gives\n\\begin{equation}\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n\\end{equation}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here replaces each coarse sp","\\subsection{The straight compatibility contract}\nFor each direction, canonical fine links can be defined by\n\\[\n v_i(p)=r(\\sigma_i p)\\,\\gamma_i(p)\\,r(p)^{-1},\n\\]\nwhere $\\gamma_i(p)=1$ on a noncarrying half step and $\\gamma_i(p)=u_i(s)$ on a carrying half step. Cancellation of the intermediate frame gives\n\\begin{equation}\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n\\end{equation}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here replaces each coarse spatial factor by two half factors and each coarse loc","\\subsection{The straight compatibility contract}\nFor each direction, canonical fine links can be defined by\n\\[\n v_i(p)=r(\\sigma_i p)\\,\\gamma_i(p)\\,r(p)^{-1},\n\\]\nwhere $\\gamma_i(p)=1$ on a noncarrying half step and $\\gamma_i(p)=u_i(s)$ on a carrying half step. Cancellation of the intermediate frame gives\n\\begin{equation}\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n\\end{equation}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here replaces each coarse spatial factor by two half factors and each coarse local factor by its supplied subdivision. The result is exact on the framed subspace.","lation of the intermediate frame gives\n\\begin{equation}\n v_i(\\sigma_i p)v_i(p)\n =r(\\sigma_i^2p)u_i(s)r(p)^{-1}.\n \\label{g:straight-contract}\n\\end{equation}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here replaces each coarse spatial factor by two half factors and each coarse local factor by its supplied subdivision. The result is exact on the framed subspace.","}\nFor arbitrary fine links, equation \\eqref{g:straight-contract} is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of \\eqref{g:half-intertwining}:\n\\[\n S_{i,\\mathrm{fine}}^2 I_r=I_rS_{i,\\mathrm{coarse}}.\n\\]\nIf the mass substeps also intertwine, the identities compose through the entire execution order $M-X-Y-Z-Y-X-M$. The refined walk here replaces each coarse spatial factor by two half factors and each coarse local factor by its supplied subdivision. The result is exact on the framed subspace.","The construction's internal cube links are pure frame ratios and therefore locally flat. Nevertheless, when the coarse $x$ and $y$ translations commute, the large two-by-two fine Wilson loop equals the coarse plaquette. Coarse curvature survives the refinement. Hence exact straight refinement can preserve a nontrivial coarse plaquette; it does not require that every loop become flat.","\\subsection{A flat cube need not reconstruct the prescribed edge}\nConversely, take an arbitrary ordered half-link pair $(a,b)$ and call its actual product $v=ba$. It is always a subdivision of $v$, even when the separately prescribed coarse link is $u\\ne v$. The exact directional defect is\n\\begin{align}\n D_{\\rm fine}\\psi\n  &:=S_{A,\\sigma,(a,b)}^2I_a\\psi-I_aS_{A,\\tau,u}\\psi\\notag","\\subsection{A flat cube need not reconstruct the prescribed edge}\nConversely, take an arbitrary ordered half-link pair $(a,b)$ and call its actual product $v=ba$. It is always a subdivision of $v$, even when the separately prescribed coarse link is $u\\ne v$. The exact directional defect is\n\\begin{align}\n D_{\\rm fine}\\psi\n  &:=S_{A,\\sigma,(a,b)}^2I_a\\psi-I_aS_{A,\\tau,u}\\psi\\notag\\\\\n  &=I_a(S_{A,\\tau,v}-S_{A,\\tau,u})\\psi.\n \\label{g:exact-mismatch}\n\\end{align}\nThe equality follows by applying the exact theorem to the product $v$ and using linearity of $I_a$. Endpoint restriction makes $I_a$ injective, so the fine defect vanishes on a state precisely when its coarse mismatch vanishes on that state. This statewise statement is not a claim that one test state detects every link error.","&=I_a(S_{A,\\tau,v}-S_{A,\\tau,u})\\psi.\n \\label{g:exact-mismatch}\n\\end{align}\nThe equality follows by applying the exact theorem to the product $v$ and using linearity of $I_a$. Endpoint restriction makes $I_a$ injective, so the fine defect vanishes on a state precisely when its coarse mismatch vanishes on that state. This statewise statement is not a claim that one test state detects every link error.","A concrete counterexample removes any possible inference from flatness alone. Let both half-links be $i$, and prescribe $u=1$. Their product is $v=-1$. A cube whose twelve positive links are all $i$ has every face plaquette equal to one. On a one-site periodic lattice, however, $\\tau=1$ and\n\\[\n S_{A,1,1}=P_++P_-=1,\\qquad\n S_{A,1,-1}=-P_+-P_-=-1.\n\\]\nThus the coarse mismatch is $-2\\psi$, and the normalized fine defect has norm $2\\norm{\\psi}$. The same link data are perfectly flat in the path-comparison sense and maximally wrong relative to the prescribed straight link. This is the role of the quarter-turn witness in the formal source.","ves any possible inference from flatness alone. Let both half-links be $i$, and prescribe $u=1$. Their product is $v=-1$. A cube whose twelve positive links are all $i$ has every face plaquette equal to one. On a one-site periodic lattice, however, $\\tau=1$ and\n\\[\n S_{A,1,1}=P_++P_-=1,\\qquad\n S_{A,1,-1}=-P_+-P_-=-1.\n\\]\nThus the coarse mismatch is $-2\\psi$, and the normalized fine defect has norm $2\\norm{\\psi}$. The same link data are perfectly flat in the path-comparison sense and maximally wrong relative to the prescribed straight link. This is the role of the quarter-turn witness in the formal source.","\\section{Hilbert estimates with explicit constants}\n\\label{g:hilbert-estimates}\nFinite-dimensional continuity is not enough to identify an operator norm with physical $\\ell^2$ energy. The earlier continuous-linear packaging retains the ambient norm on a function space. The Hilbert scalar layer instead realizes a finite four-spinor field as\n\\[\n \\HH_S=\\ell^2(S\\times\\{0,1,2,3\\};\\C),\\qquad\n \\norm{\\psi}^2=\\sum_{x\\in S}\\norm{\\psi(x)}_{\\C^4}^2.\n\\]\nThe formal realization is \\code{EuclideanSpace}; its norm is identified explicitly with the real part of the previously defined state inner product. All operator norms from this point use this Hilbert structure. Assume $S$ is finite and nonempty when taking a maximum over sites.","ace. The Hilbert scalar layer instead realizes a finite four-spinor field as\n\\[\n \\HH_S=\\ell^2(S\\times\\{0,1,2,3\\};\\C),\\qquad\n \\norm{\\psi}^2=\\sum_{x\\in S}\\norm{\\psi(x)}_{\\C^4}^2.\n\\]\nThe formal realization is \\code{EuclideanSpace}; its norm is identified explicitly with the real part of the previously defined state inner product. All operator norms from this point use this Hilbert structure. Assume $S$ is finite and nonempty when taking a maximum over sites.","\\begin{theorem}[Constant-one directional mismatch bound]\nLet $A$ be a Hermitian Clifford involution, $\\tau$ a site bijection, and $u,v:S\\to U(1)$. Then\n\\begin{equation}\n \\norm{S_{A,\\tau,v}-S_{A,\\tau,u}}\n \\leq\\delta(u,v):=\\max_{x\\in S}|v(x)-u(x)|\n =\\max_{x\\in S}|v(x)u(x)^{-1}-1|.\n \\label{g:constant-one}\n\\end{equation}\nThe constant is independent of the number of sites. The actual half-grid defect in \\eqref{g:exact-mismatch}, with half-cell Hilbert measure, obeys the same bound.\n\\end{theorem}\n\\begin{proof}\nAt each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity\n\\begin{align*}\n \\norm{(S_v-S_u)\\psi}^2\n =\\sum_x\\bigl[&|v(\\tau^{-1}x)-u(\\tau^{-1}x)|^2","\\begin{theorem}[Constant-one directional mismatch bound]\nLet $A$ be a Hermitian Clifford involution, $\\tau$ a site bijection, and $u,v:S\\to U(1)$. Then\n\\begin{equation}\n \\norm{S_{A,\\tau,v}-S_{A,\\tau,u}}\n \\leq\\delta(u,v):=\\max_{x\\in S}|v(x)-u(x)|\n =\\max_{x\\in S}|v(x)u(x)^{-1}-1|.\n \\label{g:constant-one}\n\\end{equation}\nThe constant is independent of the number of sites. The actual half-grid defect in \\eqref{g:exact-mismatch}, with half-cell Hilbert measure, obeys the same bound.\n\\end{theorem}\n\\begin{proof}\nAt each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity\n\\begin{align*}\n \\norm{(S_v-S_u)\\psi}^2\n =\\sum_x\\bigl[&|v(\\tau^{-1}x)-u(\\tau^{-1}x)|^2\n                   \\norm{P_+\\psi(\\tau^{-1}x)}^2","1|.\n \\label{g:constant-one}\n\\end{equation}\nThe constant is independent of the number of sites. The actual half-grid defect in \\eqref{g:exact-mismatch}, with half-cell Hilbert measure, obeys the same bound.\n\\end{theorem}\n\\begin{proof}\nAt each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity\n\\begin{align*}\n \\norm{(S_v-S_u)\\psi}^2\n =\\sum_x\\bigl[&|v(\\tau^{-1}x)-u(\\tau^{-1}x)|^2\n                   \\norm{P_+\\psi(\\tau^{-1}x)}^2\\\\\n &+|v(x)^{-1}-u(x)^{-1}|^2\n                   \\norm{P_-\\psi(\\tau x)}^2\\bigr].\n\\end{align*}\nThere is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies $|v^{-1}-u^{-1}|=|v-u|$. Bound every coefficient by $\\delta^2$, reindex the two sums through $\\tau$, and use $\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2$. The result is at most $\\delta^2\\norm{\\psi}^2$. Taking square roots and then the operator supremum proves the first inequality. Multiplication by $u(x)^{-1}$ proves the equality of sc…","&+|v(x)^{-1}-u(x)^{-1}|^2\n                   \\norm{P_-\\psi(\\tau x)}^2\\bigr].\n\\end{align*}\nThere is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies $|v^{-1}-u^{-1}|=|v-u|$. Bound every coefficient by $\\delta^2$, reindex the two sums through $\\tau$, and use $\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2$. The result is at most $\\delta^2\\norm{\\psi}^2$. Taking square roots and then the operator supremum proves the first inequality. Multiplication by $u(x)^{-1}$ proves the equality of scalar maxima. Finally \\eqref{g:exact-mismatch} and the isometry of $I_a$ identify the fine defect norm on every input with the coarse mismatch norm.\n\\end{proof}","{-1}|^2\n                   \\norm{P_-\\psi(\\tau x)}^2\\bigr].\n\\end{align*}\nThere is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies $|v^{-1}-u^{-1}|=|v-u|$. Bound every coefficient by $\\delta^2$, reindex the two sums through $\\tau$, and use $\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2$. The result is at most $\\delta^2\\norm{\\psi}^2$. Taking square roots and then the operator supremum proves the first inequality. Multiplication by $u(x)^{-1}$ proves the equality of scalar maxima. Finally \\eqref{g:exact-mismatch} and the isometry of $I_a$ identify the fine defect norm on every input with the coarse mismatch norm.\n\\end{proof}","come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies $|v^{-1}-u^{-1}|=|v-u|$. Bound every coefficient by $\\delta^2$, reindex the two sums through $\\tau$, and use $\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2$. The result is at most $\\delta^2\\norm{\\psi}^2$. Taking square roots and then the operator supremum proves the first inequality. Multiplication by $u(x)^{-1}$ proves the equality of scalar maxima. Finally \\eqref{g:exact-mismatch} and the isometry of $I_a$ identify the fine defect norm on every input with the coarse mismatch norm.\n\\end{proof}","u^{-1}|=|v-u|$. Bound every coefficient by $\\delta^2$, reindex the two sums through $\\tau$, and use $\\norm{P_+\\eta}^2+\\norm{P_-\\eta}^2=\\norm{\\eta}^2$. The result is at most $\\delta^2\\norm{\\psi}^2$. Taking square roots and then the operator supremum proves the first inequality. Multiplication by $u(x)^{-1}$ proves the equality of scalar maxima. Finally \\eqref{g:exact-mismatch} and the isometry of $I_a$ identify the fine defect norm on every input with the coarse mismatch norm.\n\\end{proof}","In ordinary Euclidean coordinates on the fine site set, the half-cell measure is implemented by multiplying the interpolated coordinates by $1/\\sqrt2$. Omitting this factor changes the norm and inserts an artificial $\\sqrt2$ into the comparison. The quarter-turn example attains the value two, showing that the constant one cannot be reduced uniformly over all allowed data.","\\subsection{The telescoping mechanism for a complete walk}\nLet $I:X\\to Y$ be a bounded linear comparison map and let $F_j:Y\\to Y$, $C_j:X\\to X$ be fine and coarse factors. Write $D_j=F_jI-IC_j$. Two factors satisfy the exact identity\n\\[\n F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.\n\\]\nRepeated insertion gives, in execution order $1,\\ldots,N$,\n\\begin{equation}\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j","\\subsection{The telescoping mechanism for a complete walk}\nLet $I:X\\to Y$ be a bounded linear comparison map and let $F_j:Y\\to Y$, $C_j:X\\to X$ be fine and coarse factors. Write $D_j=F_jI-IC_j$. Two factors satisfy the exact identity\n\\[\n F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.\n\\]\nRepeated insertion gives, in execution order $1,\\ldots,N$,\n\\begin{equation}\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j$. Exact local intertwining makes every term zero.","let $F_j:Y\\to Y$, $C_j:X\\to X$ be fine and coarse factors. Write $D_j=F_jI-IC_j$. Two factors satisfy the exact identity\n\\[\n F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.\n\\]\nRepeated insertion gives, in execution order $1,\\ldots,N$,\n\\begin{equation}\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j$. Exact local intertwining makes every term zero.",". Two factors satisfy the exact identity\n\\[\n F_2F_1I-IC_2C_1=F_2D_1+D_2C_1.\n\\]\nRepeated insertion gives, in execution order $1,\\ldots,N$,\n\\begin{equation}\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j$. Exact local intertwining makes every term zero.","tion gives, in execution order $1,\\ldots,N$,\n\\begin{equation}\n F_N\\cdots F_1I-IC_N\\cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j$. Exact local intertwining makes every term zero.","cdots C_1\n =\\sum_{j=1}^N F_N\\cdots F_{j+1}D_jC_{j-1}\\cdots C_1.\n \\label{g:telescoping}\n\\end{equation}\nEmpty products are identities. If every $F_j,C_j$ is contractive, the triangle inequality and submultiplicativity yield\n\\[\n \\norm{F_N\\cdots F_1I-IC_N\\cdots C_1}\n \\leq\\sum_{j=1}^N\\norm{D_j}.\n\\]\nNo separate bound on $I$ is needed for this form, since its contribution is already present inside each $D_j$. Exact local intertwining makes every term zero.","For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the","For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a","For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis compariso","For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis comparison still has to provide its actual embeddings and local factor","For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis comparison still has to provide its actual embeddings and local factor defects.","both walks act on the same finite Hilbert space, with $I=1$. One uses the reconstructed links $v_x,v_y,v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis comparison still has to provide its actual embeddings and local factor defects.",",v_z$, the other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis comparison still has to provide its actual embeddings and local factor defects.","he other the prescribed links $u_x,u_y,u_z$, and both use the same two contractive mass factors $M_0,M_1$. With execution order $M_0-X-Y-Z-Y-X-M_1$, define $\\delta_i=\\delta(u_i,v_i)$. The mass defects vanish. Equation \\eqref{g:telescoping} and \\eqref{g:constant-one} prove\n\\begin{equation}\n \\norm{W_v-W_u}\\leq 2\\delta_x+2\\delta_y+\\delta_z.\n \\label{g:sevenbudget}\n\\end{equation}\nThe repeated $X$ and $Y$ factors explain the coefficients two. The upper bound need not be attained by the complete walk: its individual error terms can cancel. It is an operator estimate between effective common-space walks. The separate half-grid theorem supplies the genuine normalized fine directional estimate; a general spatially varying three-axis comparison still has to provide its actual embeddings and local factor defects.","\\section{A two-stage edge and path estimate}\n\\label{g:two-stage}\nThe cube phase discrepancies can be made into bounded operators on the same finite Hilbert state space. For $z\\in U(1)$, let $P_z\\psi=z\\psi$ uniformly at every site. If $p=\\Omega q$, then\n\\[\n P_p-P_q=q(\\Omega-1)\\id,\\qquad\n \\norm{P_p-P_q}=|\\Omega-1|.\n\\]\nThe norm equality uses a nonzero Hilbert space. It is the operator form of \\eqref{g:curvature-defect}.","The formal construction selects five ordered phase pairs, in this order:\n\\[\n (p_{XYZ},p_{YXZ}),\\quad(p_{YXZ},p_{YZX}),\\quad\n (p_{YZX},p_{ZYX}),\\quad(p_{XYZ},p_{XZY}),\\quad\n (p_{XZY},p_{ZXY}).\n\\]\nLet $Q_f$ be the execution-order product of the first phase operators in these pairs, and $Q_c$ the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. Applying \\eqref{g:telescoping} yields\n\\[\n \\norm{Q_f-Q_c}\\leq\\kappa(\\ell).\n\\]\nThese are explicitly constructed products of uniform phase operators. Calling them a path-reordering stage describes their algebraic role; it does not assert that a global spatially varying inter-cell assembly has already been constructed.","red phase pairs, in this order:\n\\[\n (p_{XYZ},p_{YXZ}),\\quad(p_{YXZ},p_{YZX}),\\quad\n (p_{YZX},p_{ZYX}),\\quad(p_{XYZ},p_{XZY}),\\quad\n (p_{XZY},p_{ZXY}).\n\\]\nLet $Q_f$ be the execution-order product of the first phase operators in these pairs, and $Q_c$ the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. Applying \\eqref{g:telescoping} yields\n\\[\n \\norm{Q_f-Q_c}\\leq\\kappa(\\ell).\n\\]\nThese are explicitly constructed products of uniform phase operators. Calling them a path-reordering stage describes their algebraic role; it does not assert that a global spatially varying inter-cell assembly has already been constructed.","\\begin{theorem}[Combined finite Hilbert estimate]\nWith the common-space walks and contractive mass factors from \\eqref{g:sevenbudget}, define\n\\[\n D_{\\rm total}=Q_fW_v-Q_cW_u.\n\\]\nThen\n\\begin{equation}\n \\norm{D_{\\rm total}}\n \\leq 2\\delta_x+2\\delta_y+\\delta_z+\\kappa(\\ell).\n \\label{g:combined-budget}\n\\end{equation}\nIf $v_i=u_i$ for all three directions and the cube is flat, the defect is exactly zero.\n\\end{theorem}\n\\begin{proof}\nInsert and subtract $Q_fW_u$:\n\\[\n D_{\\rm total}=Q_f(W_v-W_u)+(Q_f-Q_c)W_u.\n\\]\nAll phase factors are unitary, and $W_u$ is a product of contractions. Taking norms bounds the first term by the straight-edge budget and the second by the path budget. In the exact sector the walks agree and every selected phase pair agrees, so both terms vanish before taking norms.\n\\end{proof}","With the common-space walks and contractive mass factors from \\eqref{g:sevenbudget}, define\n\\[\n D_{\\rm total}=Q_fW_v-Q_cW_u.\n\\]\nThen\n\\begin{equation}\n \\norm{D_{\\rm total}}\n \\leq 2\\delta_x+2\\delta_y+\\delta_z+\\kappa(\\ell).\n \\label{g:combined-budget}\n\\end{equation}\nIf $v_i=u_i$ for all three directions and the cube is flat, the defect is exactly zero.\n\\end{theorem}\n\\begin{proof}\nInsert and subtract $Q_fW_u$:\n\\[\n D_{\\rm total}=Q_f(W_v-W_u)+(Q_f-Q_c)W_u.\n\\]\nAll phase factors are unitary, and $W_u$ is a product of contractions. Taking norms bounds the first term by the straight-edge budget and the second by the path budget. In the exact sector the walks agree and every selected phase pair agrees, so both terms vanish before taking norms.\n\\end{proof}","mmon-space walks and contractive mass factors from \\eqref{g:sevenbudget}, define\n\\[\n D_{\\rm total}=Q_fW_v-Q_cW_u.\n\\]\nThen\n\\begin{equation}\n \\norm{D_{\\rm total}}\n \\leq 2\\delta_x+2\\delta_y+\\delta_z+\\kappa(\\ell).\n \\label{g:combined-budget}\n\\end{equation}\nIf $v_i=u_i$ for all three directions and the cube is flat, the defect is exactly zero.\n\\end{theorem}\n\\begin{proof}\nInsert and subtract $Q_fW_u$:\n\\[\n D_{\\rm total}=Q_f(W_v-W_u)+(Q_f-Q_c)W_u.\n\\]\nAll phase factors are unitary, and $W_u$ is a product of contractions. Taking norms bounds the first term by the straight-edge budget and the second by the path budget. In the exact sector the walks agree and every selected phase pair agrees, so both terms vanish before taking norms.\n\\end{proof}","[\n D_{\\rm total}=Q_fW_v-Q_cW_u.\n\\]\nThen\n\\begin{equation}\n \\norm{D_{\\rm total}}\n \\leq 2\\delta_x+2\\delta_y+\\delta_z+\\kappa(\\ell).\n \\label{g:combined-budget}\n\\end{equation}\nIf $v_i=u_i$ for all three directions and the cube is flat, the defect is exactly zero.\n\\end{theorem}\n\\begin{proof}\nInsert and subtract $Q_fW_u$:\n\\[\n D_{\\rm total}=Q_f(W_v-W_u)+(Q_f-Q_c)W_u.\n\\]\nAll phase factors are unitary, and $W_u$ is a product of contractions. Taking norms bounds the first term by the straight-edge budget and the second by the path budget. In the exact sector the walks agree and every selected phase pair agrees, so both terms vanish before taking norms.\n\\end{proof}","\\section{Connection-generated links and a quadratic mesh bound}\n\\label{g:regular}\nThe scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components $A_x,A_y,A_z$ on a Euclidean chart. Define exact coordinate-edge angles and forward links by\n\\begin{equation}\n \\theta_i(p,h)=\\int_0^h A_i(p+t e_i)\\,dt,\n \\qquad u_i(p,h)=e^{i\\theta_i(p,h)}.\n \\label{g:integrated-links}\n\\end{equation}\nThe coupling is absorbed into $A_i$. These links are automatically unitary, and the interval integral is oriented when $h<0$.","s become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components $A_x,A_y,A_z$ on a Euclidean chart. Define exact coordinate-edge angles and forward links by\n\\begin{equation}\n \\theta_i(p,h)=\\int_0^h A_i(p+t e_i)\\,dt,\n \\qquad u_i(p,h)=e^{i\\theta_i(p,h)}.\n \\label{g:integrated-links}\n\\end{equation}\nThe coupling is absorbed into $A_i$. These links are automatically unitary, and the interval integral is oriented when $h<0$.","es. Consider real abelian connection components $A_x,A_y,A_z$ on a Euclidean chart. Define exact coordinate-edge angles and forward links by\n\\begin{equation}\n \\theta_i(p,h)=\\int_0^h A_i(p+t e_i)\\,dt,\n \\qquad u_i(p,h)=e^{i\\theta_i(p,h)}.\n \\label{g:integrated-links}\n\\end{equation}\nThe coupling is absorbed into $A_i$. These links are automatically unitary, and the interval integral is oriented when $h<0$.","\\subsection{Exact composition precedes every estimate}\nContinuity of $A_i$ along its own coordinate lines permits interval-integral additivity and a change of variable:\n\\begin{align*}\n \\theta_i(p,h)\n &=\\int_0^{h/2}A_i(p+t e_i)\\,dt\n   +\\int_{h/2}^{h}A_i(p+t e_i)\\,dt","\\subsection{Exact composition precedes every estimate}\nContinuity of $A_i$ along its own coordinate lines permits interval-integral additivity and a change of variable:\n\\begin{align*}\n \\theta_i(p,h)\n &=\\int_0^{h/2}A_i(p+t e_i)\\,dt\n   +\\int_{h/2}^{h}A_i(p+t e_i)\\,dt\\\\\n &=\\theta_i(p,h/2)+\\theta_i(p+(h/2)e_i,h/2).\n\\end{align*}\nExponentiation gives the ordered identity\n\\begin{equation}\n u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h).\n \\label{g:integral-composition}\n\\end{equation}\nThus the reconstructed and prescribed links agree exactly and all three straight-edge errors vanish. This is an additivity theorem for exact line integrals. Independently approximating each integral by a quadrature rule would introduce another error term, which is not silently included in the present bound.","&=\\theta_i(p,h/2)+\\theta_i(p+(h/2)e_i,h/2).\n\\end{align*}\nExponentiation gives the ordered identity\n\\begin{equation}\n u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h).\n \\label{g:integral-composition}\n\\end{equation}\nThus the reconstructed and prescribed links agree exactly and all three straight-edge errors vanish. This is an additivity theorem for exact line integrals. Independently approximating each integral by a quadrature rule would introduce another error term, which is not silently included in the present bound.","For an affine component, $A_i(p+t e_i)=A_i(p)+t\\,\\partial_iA_i$, so\n\\[\n \\theta_i(p,h)=hA_i(p)+\\tfrac12h^2\\partial_iA_i.\n\\]\nThe face angle is exactly $h^2F_{ij}$, where $F_{ij}=\\partial_iA_j-\\partial_jA_i$. Hence $\\Omega_{ij}=e^{ih^2F_{ij}}$. Using $|e^{it}-1|\\leq|t|$ and the selected five faces yields\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\n \\leq h^2\\bigl(|F_{xy}|+2|F_{xz}|+2|F_{yz}|\\bigr).\n \\label{g:affine-bound}\n\\end{equation}\nThis exact affine calculation is stronger than a truncated Taylor argument. It also fixes the orientation sign of the curvature convention.","\\subsection{The actual coordinate regularity assumptions}\nThe more general formal connection structure records the three components, the six cross derivatives $\\partial_iA_j$ for $i\\ne j$, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants\n\\[\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n\\]\nFor example, $B_{y\\mid x}$ bounds the derivative of $A_y$ in the $x$ direction. These explicit hypotheses are sufficient for the coordinate arguments below. They avoid appealing to an unspecified stronger multivariable smoothness class.","formal connection structure records the three components, the six cross derivatives $\\partial_iA_j$ for $i\\ne j$, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants\n\\[\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n\\]\nFor example, $B_{y\\mid x}$ bounds the derivative of $A_y$ in the $x$ direction. These explicit hypotheses are sufficient for the coordinate arguments below. They avoid appealing to an unspecified stronger multivariable smoothness class.","oss derivatives $\\partial_iA_j$ for $i\\ne j$, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants\n\\[\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n\\]\nFor example, $B_{y\\mid x}$ bounds the derivative of $A_y$ in the $x$ direction. These explicit hypotheses are sufficient for the coordinate arguments below. They avoid appealing to an unspecified stronger multivariable smoothness class.","ne j$, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants\n\\[\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n\\]\nFor example, $B_{y\\mid x}$ bounds the derivative of $A_y$ in the $x$ direction. These explicit hypotheses are sufficient for the coordinate arguments below. They avoid appealing to an unspecified stronger multivariable smoothness class.","eir one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants\n\\[\n |\\partial_iA_j(p)|\\leq B_{j\\mid i}\\qquad(i\\ne j).\n\\]\nFor example, $B_{y\\mid x}$ bounds the derivative of $A_y$ in the $x$ direction. These explicit hypotheses are sufficient for the coordinate arguments below. They avoid appealing to an unspecified stronger multivariable smoothness class.","The fundamental theorem of calculus first gives\n\\[\n |A_j(p+h e_i)-A_j(p)|\\leq B_{j\\mid i}|h|.\n\\]\nIntegrating along the transverse $j$ edge then gives\n\\begin{equation}\n |\\theta_j(p+h e_i,h)-\\theta_j(p,h)|\n \\leq B_{j\\mid i}|h|^2.\n \\label{g:angle-variation}\n\\end{equation}\nIndeed the integrand difference is bounded by $B_{j\\mid i}|h|$ at every point of an interval of length $|h|$. Oriented integration changes the sign but not this absolute bound.","For the $xy$ face, the two-path angle difference is\n\\begin{align*}\n \\Theta_{xy}(p,h)\n ={}&\\theta_y(p+h e_x,h)-\\theta_y(p,h)","For the $xy$ face, the two-path angle difference is\n\\begin{align*}\n \\Theta_{xy}(p,h)\n ={}&\\theta_y(p+h e_x,h)-\\theta_y(p,h)\\\\\n    &-\\bigl(\\theta_x(p+h e_y,h)-\\theta_x(p,h)\\bigr).\n\\end{align*}\nTherefore\n\\[\n |\\Theta_{xy}(p,h)|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2,\n \\qquad\n |\\Omega_{xy}(p,h)-1|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2.\n\\]\nFor the last implication, $e^{it}-1=\\int_0^t i e^{is}\\,ds$ has absolute value at most $|t|$. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancellation inside $F_{ij}$.","&-\\bigl(\\theta_x(p+h e_y,h)-\\theta_x(p,h)\\bigr).\n\\end{align*}\nTherefore\n\\[\n |\\Theta_{xy}(p,h)|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2,\n \\qquad\n |\\Omega_{xy}(p,h)-1|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2.\n\\]\nFor the last implication, $e^{it}-1=\\int_0^t i e^{is}\\,ds$ has absolute value at most $|t|$. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancellation inside $F_{ij}$.","y})h^2,\n \\qquad\n |\\Omega_{xy}(p,h)-1|\\leq\n   (B_{y\\mid x}+B_{x\\mid y})h^2.\n\\]\nFor the last implication, $e^{it}-1=\\int_0^t i e^{is}\\,ds$ has absolute value at most $|t|$. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a sharper general surface-integral formula that exploits cancellation inside $F_{ij}$.","\\begin{theorem}[Regular background estimate]\nSuppose the coordinate regularity and derivative bounds above hold. Set\n\\begin{align}\n \\mathcal A_{\\partial}={}&B_{y\\mid x}+B_{x\\mid y}\n +2(B_{z\\mid x}+B_{x\\mid z})\\notag\\\\\n &+2(B_{z\\mid y}+B_{y\\mid z}).\n \\label{g:derivative-amplitude}\n\\end{align}\nGenerate the reconstructed and prescribed finite-site links from the exact integrals \\eqref{g:integrated-links}, and generate the twelve links of the comparison cube from the same connection at a chart point $p$. For the two-stage finite Hilbert operator of Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms,","&+2(B_{z\\mid y}+B_{y\\mid z}).\n \\label{g:derivative-amplitude}\n\\end{align}\nGenerate the reconstructed and prescribed finite-site links from the exact integrals \\eqref{g:integrated-links}, and generate the twelve links of the comparison cube from the same connection at a chart point $p$. For the two-stage finite Hilbert operator of Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","\\label{g:derivative-amplitude}\n\\end{align}\nGenerate the reconstructed and prescribed finite-site links from the exact integrals \\eqref{g:integrated-links}, and generate the twelve links of the comparison cube from the same connection at a chart point $p$. For the two-stage finite Hilbert operator of Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","the twelve links of the comparison cube from the same connection at a chart point $p$. For the two-stage finite Hilbert operator of Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","rt operator of Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","Section~\\ref{g:two-stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","stage}, with the same contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","me contractive mass factors on both sides,\n\\begin{equation}\n \\norm{D_{\\rm total}(h)}\\leq h^2\\mathcal A_{\\partial}.\n \\label{g:regular-bound}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nEquation \\eqref{g:integral-composition} makes $\\delta_x=\\delta_y=\\delta_z=0$. Apply the appropriate face estimate to each term in \\eqref{g:fiveface}. There is one $xy$ term, two $xz$ terms and two $yz$ terms, giving $\\kappa\\leq h^2\\mathcal A_{\\partial}$. Substitute in \\eqref{g:combined-budget}.\n\\end{proof}","\\subsection{A spatially varying witness}\nTake $A_x=A_z=0$ and $A_y(x,y,z)=\\sin x$. Its curvature $F_{xy}=\\cos x$ is nonconstant. The only nonzero derivative envelope is $B_{y\\mid x}=1$, so $\\mathcal A_{\\partial}=1$. Here\n\\[\n \\theta_y(p,h)=h\\sin x,\\qquad\n \\Omega_{xy}(p,h)=\\exp\\!\\bigl(ih[\\sin(x+h)-\\sin x]\\bigr).\n\\]\nThe other plane holonomies are one, and the two-stage defect is bounded by $h^2$. The bound is uniform in the basepoint. The trigonometric formula illustrates how actual phases can be substantially smaller than a global envelope; it does not change the envelope's role in a uniform theorem.","\\gaugefigure[H]{10_regular_connection}{The non-affine background $A_y=\\sin x$ has varying curvature and exact line-integral links. The quadratic envelope follows from $|\\partial_xA_y|\\leq1$; the computed phase amplitudes illustrate this bound over the plotted finite range.}","\\section{Summable refinement and what a limit still requires}\n\\label{g:summable}\nFor the dyadic mesh $h_n=a_0 2^{-n}$, the regular estimate produces the nonnegative sequence\n\\[\n \\varepsilon_n=C4^{-n},\\qquad C=a_0^2\\mathcal A_{\\partial}.\n\\]\nThe actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely $\\varepsilon_n\\to0$. A sequence with adjacent errors tending to zero need not converge; summability controls the accumulated tail.","\\Needspace{11\\baselineskip}\n\\begin{proposition}[A complete-space limit with an explicit tail]\nLet $f_n$ be a sequence in a complete metric space and suppose\n\\[\n d(f_n,f_{n+1})\\leq C4^{-n}\\qquad(n\\geq0),\\quad C\\geq0.\n\\]\nThen $f_n$ converges to some $f_\\infty$, and\n\\begin{equation}\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor $m>n$, the triangle inequality gives\n\\[\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n\\]\nThe upper bound tends to zero with $n$, so the sequence is Cauchy. Completeness provides a limit. Passing $m\\to\\infty$ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.\n\\end{proof}","skip}\n\\begin{proposition}[A complete-space limit with an explicit tail]\nLet $f_n$ be a sequence in a complete metric space and suppose\n\\[\n d(f_n,f_{n+1})\\leq C4^{-n}\\qquad(n\\geq0),\\quad C\\geq0.\n\\]\nThen $f_n$ converges to some $f_\\infty$, and\n\\begin{equation}\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor $m>n$, the triangle inequality gives\n\\[\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n\\]\nThe upper bound tends to zero with $n$, so the sequence is Cauchy. Completeness provides a limit. Passing $m\\to\\infty$ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.\n\\end{proof}","pace limit with an explicit tail]\nLet $f_n$ be a sequence in a complete metric space and suppose\n\\[\n d(f_n,f_{n+1})\\leq C4^{-n}\\qquad(n\\geq0),\\quad C\\geq0.\n\\]\nThen $f_n$ converges to some $f_\\infty$, and\n\\begin{equation}\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor $m>n$, the triangle inequality gives\n\\[\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n\\]\nThe upper bound tends to zero with $n$, so the sequence is Cauchy. Completeness provides a limit. Passing $m\\to\\infty$ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.\n\\end{proof}","d(f_n,f_{n+1})\\leq C4^{-n}\\qquad(n\\geq0),\\quad C\\geq0.\n\\]\nThen $f_n$ converges to some $f_\\infty$, and\n\\begin{equation}\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor $m>n$, the triangle inequality gives\n\\[\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n\\]\nThe upper bound tends to zero with $n$, so the sequence is Cauchy. Completeness provides a limit. Passing $m\\to\\infty$ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.\n\\end{proof}","nverges to some $f_\\infty$, and\n\\begin{equation}\n d(f_n,f_\\infty)\\leq\\sum_{k=n}^{\\infty}C4^{-k}\n =\\tfrac43C4^{-n}.\n \\label{g:dyadic-tail}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor $m>n$, the triangle inequality gives\n\\[\n d(f_n,f_m)\\leq\\sum_{k=n}^{m-1}C4^{-k}\n \\leq\\tfrac43C4^{-n}.\n\\]\nThe upper bound tends to zero with $n$, so the sequence is Cauchy. Completeness provides a limit. Passing $m\\to\\infty$ in the distance estimate proves the tail bound, and the displayed evaluation is the geometric series.\n\\end{proof}","formal regular-connection limit theorem has precisely this conditional shape: it takes a sequence in a named complete metric space and an adjacent-distance hypothesis bounded by the regular dyadic budget. The source also proves summability of the actual connection-generated finite-operator norms. These are two compatible ingredients, but the operator bound alone does not manufacture the sequence $f_n$ or identify its terms across different Hilbert spaces.","\\gaugefigure[H]{11_dyadic_budget}{For the regular connection envelope, halving the mesh quarters the one-level budget. The remaining accumulated error is bounded by the geometric tail $\\frac43C4^{-n}$ once the adjacent-distance hypothesis has been established in a common complete space.}","\\section{From weighted cells to a continuum Hilbert space}\n\\label{g:geometry-interface}\nThe reconstruction blocks that follow the connection estimates supply an important part of the common-space infrastructure. They establish what finite cell coefficients mean as $L^2$ fields, then identify a concrete family of approximating subspaces. They do not by themselves transfer the gauge-coupled walk or its generator through those maps.","\\subsection{Cell values and normalized coefficients}\nFor a measurable finite partition into cells of equal volume $v>0$, distinguish the physical constant value $q_a\\in\\C^4$ on cell $C_a$ from the normalized coefficient $\\psi_a=\\sqrt v\\,q_a$. The step interpolation is\n\\[\n (J\\psi)(x)=v^{-1/2}\\psi_a\\quad(x\\in C_a),\n\\]\nwith equality understood almost everywhere. Direct integration gives\n\\[\n \\norm{J\\psi}_{L^2}^2\n =\\sum_a\\int_{C_a}v^{-1}\\norm{\\psi_a}^2\n =\\sum_a\\norm{\\psi_a}^2.\n\\]\nEquivalently, physical cell values carry the weighted norm $v\\sum_a\\norm{q_a}^2$. The adjoint averaging map yields the orthogonal projection onto the finite step subspace. Its Pythagorean decomposition proves best approximation in $L^2$.","sical constant value $q_a\\in\\C^4$ on cell $C_a$ from the normalized coefficient $\\psi_a=\\sqrt v\\,q_a$. The step interpolation is\n\\[\n (J\\psi)(x)=v^{-1/2}\\psi_a\\quad(x\\in C_a),\n\\]\nwith equality understood almost everywhere. Direct integration gives\n\\[\n \\norm{J\\psi}_{L^2}^2\n =\\sum_a\\int_{C_a}v^{-1}\\norm{\\psi_a}^2\n =\\sum_a\\norm{\\psi_a}^2.\n\\]\nEquivalently, physical cell values carry the weighted norm $v\\sum_a\\norm{q_a}^2$. The adjoint averaging map yields the orthogonal projection onto the finite step subspace. Its Pythagorean decomposition proves best approximation in $L^2$.","),\n\\]\nwith equality understood almost everywhere. Direct integration gives\n\\[\n \\norm{J\\psi}_{L^2}^2\n =\\sum_a\\int_{C_a}v^{-1}\\norm{\\psi_a}^2\n =\\sum_a\\norm{\\psi_a}^2.\n\\]\nEquivalently, physical cell values carry the weighted norm $v\\sum_a\\norm{q_a}^2$. The adjoint averaging map yields the orthogonal projection onto the finite step subspace. Its Pythagorean decomposition proves best approximation in $L^2$.","At octree depth $n$, the number of cells is $8^n$, their side scale is $2^{-n}$ and their volume is $8^{-n}$. The first construction uses infinite octant addresses with the uniform product probability measure. Finite-prefix step subspaces are nested and have dense union. The weighted normalization is thus tied to the actual measure of a prefix cell, rather than inferred from the number of coordinates alone. The later source convention \\code{euclideanDyadicLevel d} uses prefix depth $d+1$; the geometric depth $n$ used here counts digits directly.","r volume is $8^{-n}$. The first construction uses infinite octant addresses with the uniform product probability measure. Finite-prefix step subspaces are nested and have dense union. The weighted normalization is thus tied to the actual measure of a prefix cell, rather than inferred from the number of coordinates alone. The later source convention \\code{euclideanDyadicLevel d} uses prefix depth $d+1$; the geometric depth $n$ used here counts digits directly.","irst construction uses infinite octant addresses with the uniform product probability measure. Finite-prefix step subspaces are nested and have dense union. The weighted normalization is thus tied to the actual measure of a prefix cell, rather than inferred from the number of coordinates alone. The later source convention \\code{euclideanDyadicLevel d} uses prefix depth $d+1$; the geometric depth $n$ used here counts digits directly.","ng each octant digit into three bits and applying positional binary decoding gives a continuous surjection from address space onto the unit cube. The decoder is not pointwise injective: dyadic coordinates admit the familiar terminating and nonterminating binary descriptions. The measure result then proves that fair binary product measure pushes forward to cube volume, producing a norm-preserving $L^2$ pullback.","y and strong convergence of their orthogonal projections can be proved at that stage, but it would be premature to call them ordinary dyadic-box subspaces. The next block closes this identification: a prefix cell equals the preimage of its corresponding concrete Cartesian dyadic box almost everywhere. The formal boxes are closed and share boundary faces; those faces have zero volume. Equality in $L^2$ therefore survives the binary boundary ambiguity.","The transported and concrete closed levels are then proved equal. Their union is dense, and the orthogonal projections $P_n$ satisfy\n\\[\n \\norm{P_nf-f}_{L^2}\\longrightarrow0\n \\qquad\\text{for every }f\\in L^2([0,1]^3;\\C^4).\n\\]\nThis is a state-space approximation theorem. Strong convergence of projections does not control an unbounded Dirac operator, establish graph convergence, or identify the limit of gauge-coupled propagators. Those statements need consistency on suitable domains and stability estimates for the transported operators.","\\subsection{An exact matrix momentum sector}\nLet the fine canonical coordinates be Hermitian matrix pairs $(Z^A,P_A)$, with real components and the trace bracket\n\\begin{equation}\n \\{F,G\\}=\\sum_A\\operatorname{Tr}\\left(\n \\nabla_{Z^A}F\\nabla_{P_A}G-\n \\nabla_{P_A}F\\nabla_{Z^A}G\\right).\n \\label{c:constraints:trace-bracket}\n\\end{equation}\nThe trace gradients are defined by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","exact matrix momentum sector}\nLet the fine canonical coordinates be Hermitian matrix pairs $(Z^A,P_A)$, with real components and the trace bracket\n\\begin{equation}\n \\{F,G\\}=\\sum_A\\operatorname{Tr}\\left(\n \\nabla_{Z^A}F\\nabla_{P_A}G-\n \\nabla_{P_A}F\\nabla_{Z^A}G\\right).\n \\label{c:constraints:trace-bracket}\n\\end{equation}\nThe trace gradients are defined by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","Let the fine canonical coordinates be Hermitian matrix pairs $(Z^A,P_A)$, with real components and the trace bracket\n\\begin{equation}\n \\{F,G\\}=\\sum_A\\operatorname{Tr}\\left(\n \\nabla_{Z^A}F\\nabla_{P_A}G-\n \\nabla_{P_A}F\\nabla_{Z^A}G\\right).\n \\label{c:constraints:trace-bracket}\n\\end{equation}\nThe trace gradients are defined by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","atorname{Tr}\\left(\n \\nabla_{Z^A}F\\nabla_{P_A}G-\n \\nabla_{P_A}F\\nabla_{Z^A}G\\right).\n \\label{c:constraints:trace-bracket}\n\\end{equation}\nThe trace gradients are defined by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","equation}\nThe trace gradients are defined by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","ned by real variations. For a Hermitian parameter $\\Lambda$, put\n\\begin{equation}\n D[\\Lambda]=i\\sum_A\\operatorname{Tr}\\bigl(P_A[\\Lambda,Z^A]\\bigr).\n\\end{equation}\nThis real functional generates simultaneous conjugation:\n\\begin{equation}\n \\{Z^A,D[\\Lambda]\\}=i[\\Lambda,Z^A],\\qquad\n \\{P_A,D[\\Lambda]\\}=i[\\Lambda,P_A].\n\\end{equation}\nIndeed, cyclic variation gives\n$\\nabla_PD[\\Lambda]=i[\\Lambda,Z]$ and\n$\\nabla_ZD[\\Lambda]=i[P,\\Lambda]$ for one pair.","\\begin{proposition}[Finite inner momentum closure]\nWith these conventions,\n\\begin{equation}\n \\{D[\\Lambda],D[\\Gamma]\\}=D[i[\\Lambda,\\Gamma]].\n \\label{c:constraints:inner-closure}\n\\end{equation}\nIf a matrix density $\\mathcal E$ obeys\n$\\{\\mathcal E,D[\\Lambda]\\}=i[\\Lambda,\\mathcal E]$,\nthen $H[N]=\\operatorname{Tr}(N\\mathcal E)$ satisfies\n\\begin{equation}\n \\{D[\\Lambda],H[N]\\}=H[i[\\Lambda,N]].\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor one pair, substitution of the two cyclic gradients yields\n\\begin{align}\n \\{D[\\Lambda],D[\\Gamma]\\}\n &=\\operatorname{Tr}\\left(\n   -[P,\\Lambda][\\Gamma,Z]+[\\Lambda,Z][P,\\Gamma]\\right)","ma]\\}=D[i[\\Lambda,\\Gamma]].\n \\label{c:constraints:inner-closure}\n\\end{equation}\nIf a matrix density $\\mathcal E$ obeys\n$\\{\\mathcal E,D[\\Lambda]\\}=i[\\Lambda,\\mathcal E]$,\nthen $H[N]=\\operatorname{Tr}(N\\mathcal E)$ satisfies\n\\begin{equation}\n \\{D[\\Lambda],H[N]\\}=H[i[\\Lambda,N]].\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFor one pair, substitution of the two cyclic gradients yields\n\\begin{align}\n \\{D[\\Lambda],D[\\Gamma]\\}\n &=\\operatorname{Tr}\\left(\n   -[P,\\Lambda][\\Gamma,Z]+[\\Lambda,Z][P,\\Gamma]\\right)\\\\\n &=\\operatorname{Tr}\\left(P\\bigl(\n   -[\\Lambda,[\\Gamma,Z]]+[\\Gamma,[\\Lambda,Z]]\\bigr)\\right)\\\\\n &=-\\operatorname{Tr}\\bigl(P[[\\Lambda,\\Gamma],Z]\\bigr)\n   =D[i[\\Lambda,\\Gamma]].\n\\end{align}\nSummation handles all pairs. For the second statement, antisymmetry and cyclicity give\n\\begin{equation}\n \\{D[\\Lambda],H[N]\\}\n =-i\\operatorname{Tr}(N[\\Lambda,\\mathcal E])\n =\\operatorname{Tr}(i[\\Lambda,N]\\mathcal E).\n\\end{equation}\n\\end{proof}","&=-\\operatorname{Tr}\\bigl(P[[\\Lambda,\\Gamma],Z]\\bigr)\n   =D[i[\\Lambda,\\Gamma]].\n\\end{align}\nSummation handles all pairs. For the second statement, antisymmetry and cyclicity give\n\\begin{equation}\n \\{D[\\Lambda],H[N]\\}\n =-i\\operatorname{Tr}(N[\\Lambda,\\mathcal E])\n =\\operatorname{Tr}(i[\\Lambda,N]\\mathcal E).\n\\end{equation}\n\\end{proof}","The closed finite object is the full inner algebra. To identify a continuum spatial diffeomorphism, one must additionally construct parameters $\\Lambda_h[\\xi]$ whose action approximates the Lie derivative on resolved observables. An arbitrary subset of matrices chosen to resemble first derivatives need not be closed under commutators. Equation~\\eqref{c:constraints:inner-closure} does not establish that extra geometric identification.","\\subsection{What direct projection loses}\nLet $(\\mathcal C_F,s_F)$ be a graded fine complex with $s_F^2=0$. Let $P:\\mathcal C_F\\to\\mathcal C_C$ and $I:\\mathcal C_C\\to\\mathcal C_F$ be degree-zero maps satisfying $PI=\\mathrm{id}_C$, and set\n\\begin{equation}\n Q=\\mathrm{id}_F-IP,\\qquad \\bar s_C=Ps_FI.\n\\end{equation}\nHere $I$ is the inclusion map, whereas $\\mathrm{id}$ denotes an identity operator. Insertion of $IP=\\mathrm{id}_F-Q$ proves the exact identity\n\\begin{equation}\n \\bar s_C^2\n =Ps_F(IP)s_FI\n =-Ps_FQs_FI.\n \\label{c:constraints:leakage}\n\\end{equation}\nThe lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no reason for it to vanish merely because the fine di","\\subsection{What direct projection loses}\nLet $(\\mathcal C_F,s_F)$ be a graded fine complex with $s_F^2=0$. Let $P:\\mathcal C_F\\to\\mathcal C_C$ and $I:\\mathcal C_C\\to\\mathcal C_F$ be degree-zero maps satisfying $PI=\\mathrm{id}_C$, and set\n\\begin{equation}\n Q=\\mathrm{id}_F-IP,\\qquad \\bar s_C=Ps_FI.\n\\end{equation}\nHere $I$ is the inclusion map, whereas $\\mathrm{id}$ denotes an identity operator. Insertion of $IP=\\mathrm{id}_F-Q$ proves the exact identity\n\\begin{equation}\n \\bar s_C^2\n =Ps_F(IP)s_FI\n =-Ps_FQs_FI.\n \\label{c:constraints:leakage}\n\\end{equation}\nThe lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no reason for it to vanish merely because the fine differential is nilpotent.","{What direct projection loses}\nLet $(\\mathcal C_F,s_F)$ be a graded fine complex with $s_F^2=0$. Let $P:\\mathcal C_F\\to\\mathcal C_C$ and $I:\\mathcal C_C\\to\\mathcal C_F$ be degree-zero maps satisfying $PI=\\mathrm{id}_C$, and set\n\\begin{equation}\n Q=\\mathrm{id}_F-IP,\\qquad \\bar s_C=Ps_FI.\n\\end{equation}\nHere $I$ is the inclusion map, whereas $\\mathrm{id}$ denotes an identity operator. Insertion of $IP=\\mathrm{id}_F-Q$ proves the exact identity\n\\begin{equation}\n \\bar s_C^2\n =Ps_F(IP)s_FI\n =-Ps_FQs_FI.\n \\label{c:constraints:leakage}\n\\end{equation}\nThe lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no reason for it to vanish merely because the fine differential is nilpotent.","raded fine complex with $s_F^2=0$. Let $P:\\mathcal C_F\\to\\mathcal C_C$ and $I:\\mathcal C_C\\to\\mathcal C_F$ be degree-zero maps satisfying $PI=\\mathrm{id}_C$, and set\n\\begin{equation}\n Q=\\mathrm{id}_F-IP,\\qquad \\bar s_C=Ps_FI.\n\\end{equation}\nHere $I$ is the inclusion map, whereas $\\mathrm{id}$ denotes an identity operator. Insertion of $IP=\\mathrm{id}_F-Q$ proves the exact identity\n\\begin{equation}\n \\bar s_C^2\n =Ps_F(IP)s_FI\n =-Ps_FQs_FI.\n \\label{c:constraints:leakage}\n\\end{equation}\nThe lost path starts in retained data, enters a discarded channel, and returns to retained data. Its sign and order are fixed by the formula. There is no reason for it to vanish merely because the fine differential is nilpotent.","For example, the finite transfer test has the naive coarse block\n\\begin{equation}\n \\bar s_C=\\begin{pmatrix}\n -1/25&-1/25&-26/625\\\\\n 1&0&1/25\\\\\n 0&1/25&0\n \\end{pmatrix}.\n\\end{equation}\nIts square is nonzero, as its $(2,2)$ entry is $-1/25$. The transferred block in that same test is\n\\begin{equation}\n s_C=\\begin{pmatrix}0&0&0","0&1/25&0\n \\end{pmatrix}.\n\\end{equation}\nIts square is nonzero, as its $(2,2)$ entry is $-1/25$. The transferred block in that same test is\n\\begin{equation}\n s_C=\\begin{pmatrix}0&0&0","0&1/25&0\n \\end{pmatrix}.\n\\end{equation}\nIts square is nonzero, as its $(2,2)$ entry is $-1/25$. The transferred block in that same test is\n\\begin{equation}\n s_C=\\begin{pmatrix}0&0&0\\\\1&0&1/25\\\\0&0&0\\end{pmatrix},\n \\qquad s_C^2=0.\n\\end{equation}\nThis small rational example illustrates the distinction between a projection and a transfer. The matrices by themselves are an algebraic diagnostic. They do not define gravitational constraints, a symplectic BFV theory, or a continuum model.","\\subsection{A precise conditional transfer theorem}\nSplit $s_F=s_0+\\delta$, where $s_0^2=0$ and $\\delta$ has degree one. Suppose $(\\mathcal C_C,s_{0,C})$ is a coarse complex and the maps are chain maps:\n\\begin{equation}\n s_0I=Is_{0,C},\\qquad Ps_0=s_{0,C}P.\n\\end{equation}\nAssume a degree-minus-one homotopy $H$ satisfies\n\\begin{equation}\n \\begin{gathered}\n PI=\\mathrm{id}_C,\\qquad\n \\mathrm{id}_F-IP=s_0H+Hs_0,","\\subsection{A precise conditional transfer theorem}\nSplit $s_F=s_0+\\delta$, where $s_0^2=0$ and $\\delta$ has degree one. Suppose $(\\mathcal C_C,s_{0,C})$ is a coarse complex and the maps are chain maps:\n\\begin{equation}\n s_0I=Is_{0,C},\\qquad Ps_0=s_{0,C}P.\n\\end{equation}\nAssume a degree-minus-one homotopy $H$ satisfies\n\\begin{equation}\n \\begin{gathered}\n PI=\\mathrm{id}_C,\\qquad\n \\mathrm{id}_F-IP=s_0H+Hs_0,\\\\\n PH=0,\\qquad HI=0,\\qquad H^2=0.\n \\end{gathered}\n \\label{c:constraints:contraction}\n\\end{equation}\nFinally, require $(s_0+\\delta)^2=0$ and an inverse of $\\mathrm{id}_F+H\\delta$. The inverse may exist in a declared operator topology, or in a complete filtration in which the Neumann series makes formal sense. A formal inverse is a formal conclusion, not an analytic convergence theorem.","PH=0,\\qquad HI=0,\\qquad H^2=0.\n \\end{gathered}\n \\label{c:constraints:contraction}\n\\end{equation}\nFinally, require $(s_0+\\delta)^2=0$ and an inverse of $\\mathrm{id}_F+H\\delta$. The inverse may exist in a declared operator topology, or in a complete filtration in which the Neumann series makes formal sense. A formal inverse is a formal conclusion, not an analytic convergence theorem.","\\begin{theorem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibil","\\begin{theorem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine different","\\begin{theorem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yiel","\\begin{theorem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta","\\begin{theorem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","rem}[Transfer of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","er of a differential]\nUnder these assumptions, define\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","fine\n\\begin{equation}\n I_\\delta=(\\mathrm{id}_F+H\\delta)^{-1}I,\n \\qquad\n s_C=s_{0,C}+P\\delta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","lta I_\\delta.\n \\label{c:constraints:transfer-formula}\n\\end{equation}\nThen $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","Then $PI_\\delta=\\mathrm{id}_C$,\n$(s_0+\\delta)I_\\delta=I_\\delta s_C$, and $s_C^2=0$.\n\\end{theorem}\n\\begin{proof}\nThe inverse equation is $I_\\delta=I-H\\delta I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","I_\\delta$.\nApplication of $P$ proves $PI_\\delta=PI=\\mathrm{id}_C$.\nLet\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","Let\n\\begin{equation}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","}\n K=(s_0+\\delta)I_\\delta-I_\\delta s_C.\n\\end{equation}\nUsing the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","Using the chain-map identity and\n$\\mathrm{id}_F-IP=s_0H+Hs_0$ in this expression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","ression gives\n\\begin{equation}\n K=H\\left(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","eft(s_0\\delta I_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","_\\delta+\n \\delta I_\\delta s_{0,C}+\\delta I_\\delta P\\delta I_\\delta\\right).\n\\end{equation}\nNilpotency of the fine differential gives\n$s_0\\delta+\\delta s_0+\\delta^2=0$. The parenthesis therefore equals $-\\delta K$. Hence\n$(\\mathrm{id}_F+H\\delta)K=0$, and invertibility gives $K=0$.\nApplying the fine differential to the intertwining identity yields\n$0=I_\\delta s_C^2$. The left inverse $P$ then gives $s_C^2=0$.\n\\end{proof}","Expansion of the inverse exhibits the returned detail:\n\\begin{equation}\n \\begin{aligned}\n s_C={}&s_{0,C}+P\\delta I-P\\delta H\\delta I\\\\\n       &+P\\delta H\\delta H\\delta I-\\cdots.\n \\end{aligned}\n\\end{equation}\nThe first correction describes one propagation through $H$ between two nonlinear vertices. More interactions require further terms. Truncating this series generally changes the nilpotency statement.","&+P\\delta H\\delta H\\delta I-\\cdots.\n \\end{aligned}\n\\end{equation}\nThe first correction describes one propagation through $H$ between two nonlinear vertices. More interactions require further terms. Truncating this series generally changes the nilpotency statement.","There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts $c^A$ of ghost number one and their odd conjugate momenta $\\varpi_A$ of ghost number minus one. A charge of ghost number one has the form\n\\begin{equation}\n \\Omega_F=c^AC_A-\\tfrac12c^Ac^BU_{AB}{}^C\\varpi_C\n             +\\Omega_F^{(\\ge3)}.\n \\label{c:constraints:bfv-charge}\n\\end{equation}\nThe superscript counts ghost factors, not total ghost number. Higher terms can have further ghost momenta and still have ghost number one. They carry the additional identities needed for field-dependent, open, or reducible algebras. They cannot be suppressed just because the displayed constraint brackets close.","If $\\{\\Omega_F,\\Omega_F\\}=0$, then\n$s_F=\\{\\Omega_F,\\cdot\\}$ is a differential. To transfer a nonlinear Hamiltonian gauge structure, the contraction must also lift compatibly to the completed nonlinear coalgebra and respect its cyclic or symplectic structure. On a local exact canonical chart, a resulting Hamiltonian vector field can have a charge $\\Omega_C$. Its square then has Hamiltonian $\\tfrac12\\{\\Omega_C,\\Omega_C\\}$, which vanishes when the canonical center has no ghost-number-two constants. These are additional assumptions. Connectedness alone does not turn every symplectic vector field into a global Hamiltonian vector field.","\\subsection{A quantitative return bound}\nFor bounded maps on declared normed spaces, assume\n\\begin{equation}\n \\|H\\delta\\|\\le\\rho<1,\\qquad\n \\|Q\\delta I\\|\\le C h^r.\n\\end{equation}\nThe resolvent identity and $H=HQ$ give\n\\begin{align}\n s_C-(s_{0,C}+P\\delta I)\n &=-P\\delta(\\mathrm{id}_F+H\\delta)^{-1}H\\delta I,","\\subsection{A quantitative return bound}\nFor bounded maps on declared normed spaces, assume\n\\begin{equation}\n \\|H\\delta\\|\\le\\rho<1,\\qquad\n \\|Q\\delta I\\|\\le C h^r.\n\\end{equation}\nThe resolvent identity and $H=HQ$ give\n\\begin{align}\n s_C-(s_{0,C}+P\\delta I)\n &=-P\\delta(\\mathrm{id}_F+H\\delta)^{-1}H\\delta I,\\\\\n \\|s_C-(s_{0,C}+P\\delta I)\\|\n &\\le\\frac{\\|P\\delta\\|\\,\\|H\\|}{1-\\rho}\\,C h^r.\n \\label{c:constraints:return-bound}\n\\end{align}\nTo make this an approximation theorem along a refinement sequence, the prefactor must be bounded uniformly in $h$. An unbounded detail Green operator can defeat the conclusion even when $Q\\delta I$ is small. The estimate controls the stated differential maps. It does not by itself prove convergence of constrained trajectories, physical observables, or a quantum measure.","\\|s_C-(s_{0,C}+P\\delta I)\\|\n &\\le\\frac{\\|P\\delta\\|\\,\\|H\\|}{1-\\rho}\\,C h^r.\n \\label{c:constraints:return-bound}\n\\end{align}\nTo make this an approximation theorem along a refinement sequence, the prefactor must be bounded uniformly in $h$. An unbounded detail Green operator can defeat the conclusion even when $Q\\delta I$ is small. The estimate controls the stated differential maps. It does not by itself prove convergence of constrained trajectories, physical observables, or a quantum measure.","\\subsection{A filtered algebra with room for products}\nOn $\\mathbb T^3=(\\mathbb R/2\\pi\\mathbb Z)^3$, let\n\\begin{equation}\n \\mathcal T_K=\\left\\{\\sum_{\\|k\\|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","for products}\nOn $\\mathbb T^3=(\\mathbb R/2\\pi\\mathbb Z)^3$, let\n\\begin{equation}\n \\mathcal T_K=\\left\\{\\sum_{\\|k\\|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","quation}\n \\mathcal T_K=\\left\\{\\sum_{\\|k\\|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","T_K=\\left\\{\\sum_{\\|k\\|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","|_\\infty\\le K}\n              \\widehat f_k e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","e^{ik\\cdot x}\\right\\}.\n\\end{equation}\nThe algebraic relations are\n\\begin{equation}\n \\mathcal T_K\\mathcal T_L\\subset\\mathcal T_{K+L},\\qquad\n \\partial_a\\mathcal T_K\\subset\\mathcal T_K.\n\\end{equation}\nThus $\\mathcal T_K$ is a resolved space, while\n$\\bigcup_K\\mathcal T_K$ is the product-closed differential algebra.\nLet $S_N$ sample on an $N^3$ periodic grid, $\\delta_a$ be its Fourier derivative, and\n$\\tau_N=N^{-3}\\sum_x$ its normalized trace.","If $f_j\\in\\mathcal T_{K_j}$ and $\\sum_jK_j<N/2$, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives\n\\begin{align}\n \\delta_a\\prod_jS_Nf_j\n &=S_N\\partial_a\\left(\\prod_jf_j\\right),","If $f_j\\in\\mathcal T_{K_j}$ and $\\sum_jK_j<N/2$, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives\n\\begin{align}\n \\delta_a\\prod_jS_Nf_j\n &=S_N\\partial_a\\left(\\prod_jf_j\\right),\\\\\n \\tau_N\\left(\\prod_jS_Nf_j\\right)\n &=(2\\pi)^{-3}\\int_{\\mathbb T^3}\\prod_jf_j\\,\\mathrm d^3x.\n \\label{c:constraints:alias-free}\n\\end{align}\nThe Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled pointwise products. This is how the resolved construction avoids the finite pointwise derivation obstruction: it retains enough detail instead of demanding a derivation on the whole finite function algebra.","\\subsection{Constraints and their normalizations}\nTake connection coefficients $A_a^i$, densitized triads $E^a_i$, internal parameters $\\lambda^i$, shifts $\\xi^a$, and densitized lapse $n$ in $\\mathcal T_K$. Set the canonical normalization to one and define\n\\begin{align}\n F_{ab}^i&=\\delta_aA_b^i-\\delta_bA_a^i\n                    +\\epsilon^i{}_{jk}A_a^jA_b^k,","\\subsection{Constraints and their normalizations}\nTake connection coefficients $A_a^i$, densitized triads $E^a_i$, internal parameters $\\lambda^i$, shifts $\\xi^a$, and densitized lapse $n$ in $\\mathcal T_K$. Set the canonical normalization to one and define\n\\begin{align}\n F_{ab}^i&=\\delta_aA_b^i-\\delta_bA_a^i\n                    +\\epsilon^i{}_{jk}A_a^jA_b^k,\\\\\n \\mathcal G_i&=\\delta_aE^a_i+\\epsilon_{ij}{}^kA_a^jE^a_k,\\\\\n \\mathcal V_a&=F_{ab}^iE^b_i,\\qquad\n \\mathcal D_a=\\mathcal V_a-A_a^i\\mathcal G_i,\\\\\n \\mathcal H&=\\tfrac12\\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k\n                                  +\\Lambda\\det E.\n\\end{align}\nHere $\\det E=\\tfrac16\\epsilon_{abc}\\epsilon^{ijk}E^a_iE^b_jE^c_k$.\nThe smeared generators are\n\\begin{align}\nG[\\lambda]&=\\tau_N(\\lambda^i\\mathcal G_i),&\nD[\\xi]&=\\tau_N(\\xi^a\\mathcal D_a),","\\mathcal H&=\\tfrac12\\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k\n                                  +\\Lambda\\det E.\n\\end{align}\nHere $\\det E=\\tfrac16\\epsilon_{abc}\\epsilon^{ijk}E^a_iE^b_jE^c_k$.\nThe smeared generators are\n\\begin{align}\nG[\\lambda]&=\\tau_N(\\lambda^i\\mathcal G_i),&\nD[\\xi]&=\\tau_N(\\xi^a\\mathcal D_a),","\\mathcal H&=\\tfrac12\\epsilon^{ij}{}_kE^a_iE^b_jF_{ab}^k\n                                  +\\Lambda\\det E.\n\\end{align}\nHere $\\det E=\\tfrac16\\epsilon_{abc}\\epsilon^{ijk}E^a_iE^b_jE^c_k$.\nThe smeared generators are\n\\begin{align}\nG[\\lambda]&=\\tau_N(\\lambda^i\\mathcal G_i),&\nD[\\xi]&=\\tau_N(\\xi^a\\mathcal D_a),\\\\\nV[\\xi]&=\\tau_N(\\xi^a\\mathcal V_a),&\nH[n]&=\\tau_N(n\\mathcal H).\n\\end{align}","On the full grid canonical phase space the bracket is\n\\begin{equation}\n \\{F,G\\}_N=N^3\\sum_{x,a,i}\n \\left(\\frac{\\partial F}{\\partial A_a^i(x)}\n       \\frac{\\partial G}{\\partial E^a_i(x)}\n      -\\frac{\\partial F}{\\partial E^a_i(x)}\n       \\frac{\\partial G}{\\partial A_a^i(x)}\\right).\n\\end{equation}\nThe factor $N^3$ matches the normalized trace in the action. The resolved identities below are evaluated at band-limited data in this phase space. Replacing this bracket by a bracket that first projects every gradient back to $\\mathcal T_K$ is a different operation and can reintroduce the leakage of equation~\\eqref{c:constraints:leakage}.","partial A_a^i(x)}\n       \\frac{\\partial G}{\\partial E^a_i(x)}\n      -\\frac{\\partial F}{\\partial E^a_i(x)}\n       \\frac{\\partial G}{\\partial A_a^i(x)}\\right).\n\\end{equation}\nThe factor $N^3$ matches the normalized trace in the action. The resolved identities below are evaluated at band-limited data in this phase space. Replacing this bracket by a bracket that first projects every gradient back to $\\mathcal T_K$ is a different operation and can reintroduce the leakage of equation~\\eqref{c:constraints:leakage}.","Put\n\\begin{align}\n (\\lambda\\times\\mu)^i&=\\epsilon^i{}_{jk}\\lambda^j\\mu^k,\\\\\n [\\xi,\\eta]^a&=\\xi^b\\delta_b\\eta^a-\\eta^b\\delta_b\\xi^a,\\\\\n \\mathcal L_\\xi^{(-1)}n&=\\xi^a\\delta_an-n\\delta_a\\xi^a,\\\\\n S^a(n,m)&=E^a_iE^{bi}(m\\delta_bn-n\\delta_bm).\n\\end{align}\nThe lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","\\begin{proposition}[The resolved algebraic transfer step]\nAssume the continuum polynomial constraint identities in this convention and $16K<N$. Their full-grid sampled evaluations obey\n\\begin{align}\n \\{G[\\lambda],G[\\mu]\\}_N&=G[\\lambda\\times\\mu],","\\begin{proposition}[The resolved algebraic transfer step]\nAssume the continuum polynomial constraint identities in this convention and $16K<N$. Their full-grid sampled evaluations obey\n\\begin{align}\n \\{G[\\lambda],G[\\mu]\\}_N&=G[\\lambda\\times\\mu],\\\\\n \\{D[\\xi],G[\\lambda]\\}_N&=G[\\xi^a\\delta_a\\lambda],\\\\\n \\{D[\\xi],D[\\eta]\\}_N&=D[[\\xi,\\eta]],\\\\\n \\{G[\\lambda],H[n]\\}_N&=0,\\\\\n \\{D[\\xi],H[n]\\}_N&=H[\\mathcal L_\\xi^{(-1)}n],\\\\\n \\{H[n],H[m]\\}_N&=V[S(n,m)]\n                  =D[S(n,m)]+G[S^a(n,m)A_a].\n \\label{c:constraints:resolved-algebra}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nIncluding the smearing, the Gauss, improved momentum, and Hamiltonian functionals contain at most three, four, and five band-$K$ factors. Their canonical gradients contain at most two, three, and four. A bracket therefore contains at most eight factors. The hypothesis gives $8K<N/2$. Equation~\\eqref{c:constraints:alias-free} identifies every product, derivative, integration by parts, and zero-mode extraction in the continuum derivation with its sampled value. The continuum identities transfer term by term. The cosmolog…","\\{H[n],H[m]\\}_N&=V[S(n,m)]\n                  =D[S(n,m)]+G[S^a(n,m)A_a].\n \\label{c:constraints:resolved-algebra}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nIncluding the smearing, the Gauss, improved momentum, and Hamiltonian functionals contain at most three, four, and five band-$K$ factors. Their canonical gradients contain at most two, three, and four. A bracket therefore contains at most eight factors. The hypothesis gives $8K<N/2$. Equation~\\eqref{c:constraints:alias-free} identifies every product, derivative, integration by parts, and zero-mode extraction in the continuum derivation with its sampled value. The continuum identities transfer term by term. The cosmological term is gauge and diffeomorphism invariant. Its cross terms in the Hamiltonian bracket cancel by antisymmetry. The identity $V[S]=D[S]+G[S^aA_a]$ follows directly from the definition of $\\mathcal D_a$.\n\\end{proof}","=D[S(n,m)]+G[S^a(n,m)A_a].\n \\label{c:constraints:resolved-algebra}\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nIncluding the smearing, the Gauss, improved momentum, and Hamiltonian functionals contain at most three, four, and five band-$K$ factors. Their canonical gradients contain at most two, three, and four. A bracket therefore contains at most eight factors. The hypothesis gives $8K<N/2$. Equation~\\eqref{c:constraints:alias-free} identifies every product, derivative, integration by parts, and zero-mode extraction in the continuum derivation with its sampled value. The continuum identities transfer term by term. The cosmological term is gauge and diffeomorphism invariant. Its cross terms in the Hamiltonian bracket cancel by antisymmetry. The identity $V[S]=D[S]+G[S^aA_a]$ follows directly from the definition of $\\mathcal D_a$.\n\\end{proof}","The hypothesis gives $8K<N/2$. Equation~\\eqref{c:constraints:alias-free} identifies every product, derivative, integration by parts, and zero-mode extraction in the continuum derivation with its sampled value. The continuum identities transfer term by term. The cosmological term is gauge and diffeomorphism invariant. Its cross terms in the Hamiltonian bracket cancel by antisymmetry. The identity $V[S]=D[S]+G[S^aA_a]$ follows directly from the definition of $\\mathcal D_a$.\n\\end{proof}","lias-free} identifies every product, derivative, integration by parts, and zero-mode extraction in the continuum derivation with its sampled value. The continuum identities transfer term by term. The cosmological term is gauge and diffeomorphism invariant. Its cross terms in the Hamiltonian bracket cancel by antisymmetry. The identity $V[S]=D[S]+G[S^aA_a]$ follows directly from the definition of $\\mathcal D_a$.\n\\end{proof}","\\subsection{What the closure tower does and does not preserve}\nAt a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space $\\mathcal T_K$ is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies\n\\begin{equation}\n I_{L\\to M}I_{K\\to L}=I_{K\\to M}.\n\\end{equation}\nWhen the larger band and its grid resolve every generated product, the embedded constraints and their bracket values agree. This is compatibility of finite algebraic evaluations. A tower of such evaluations does not yet supply a uniform interval of existence, a bound on band growth, or a continuum evolution theorem.","The source certificate tests the algebra at $N=18$, $K=1$, where $16K<N$. Its bracket residuals are of order $10^{-17}$. A band-three negative control violates the hypothesis and produces a Hamiltonian-bracket residual near $3.3\\times10^{-8}$. These are useful diagnostics of the implementation and of the support boundary. A finite test is not a proof for all admissible fields.","The Lorentzian limitation of this self-dual chart is also explicit. A single Lorentzian self-dual connection is complex, with a physical sector requiring real $E$ and\n$A+\\overline A=2\\Gamma(E)$. The source chapter does not construct a finite step preserving that sector. The real parent below addresses reality by changing variables and retaining both chiral sectors, rather than by declaring this missing step solved inside the single complex chart.","\\subsection{The separate Gowdy BFV target}\nThe Gowdy target chapter changes from $\\tau$ to areal time $t=e^{-\\tau}>0$. Its wave-map Hamiltonian is\n\\begin{equation}\n \\mathcal H_t=\\frac{\\Pi_P^2+e^{-2P}\\Pi_Q^2}{2t}\n                   +\\frac t2(P_\\theta^2+e^{2P}Q_\\theta^2).\n\\end{equation}\nThe physical Hamiltonian density at fixed $t$ and the tangential density\n$\\mathcal D=\\Pi_PP_\\theta+\\Pi_QQ_\\theta$ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make $\\mathcal H_t$ a vanishing gravitational constraint. Areal time is already a gauge choice.","ap Hamiltonian is\n\\begin{equation}\n \\mathcal H_t=\\frac{\\Pi_P^2+e^{-2P}\\Pi_Q^2}{2t}\n                   +\\frac t2(P_\\theta^2+e^{2P}Q_\\theta^2).\n\\end{equation}\nThe physical Hamiltonian density at fixed $t$ and the tangential density\n$\\mathcal D=\\Pi_PP_\\theta+\\Pi_QQ_\\theta$ have the continuum bracket pattern of two deformation generators with densitized smearings. This algebraic target does not make $\\mathcal H_t$ a vanishing gravitational constraint. Areal time is already a gauge choice.","\\subsection{Fields, action, and polynomial structure}\nFix $\\eta_{IJ}=\\operatorname{diag}(-1,1,1,1)$ and $\\epsilon_{0123}=1$. The independent fields are a real coframe $e^I$, a real antisymmetric Lorentz connection $\\omega^{IJ}$, an Abelian connection $a$, an antisymmetric auxiliary scalar $\\varphi_{IJ}$, and Dirac fields $\\psi,\\bar\\psi$. The spinors are Grassmann odd. Define\n\\begin{align}\n R^{IJ}&=\\mathrm d\\omega^{IJ}+\\omega^I{}_K\\wedge\\omega^{KJ},\n &F&=\\mathrm da,","\\subsection{Fields, action, and polynomial structure}\nFix $\\eta_{IJ}=\\operatorname{diag}(-1,1,1,1)$ and $\\epsilon_{0123}=1$. The independent fields are a real coframe $e^I$, a real antisymmetric Lorentz connection $\\omega^{IJ}$, an Abelian connection $a$, an antisymmetric auxiliary scalar $\\varphi_{IJ}$, and Dirac fields $\\psi,\\bar\\psi$. The spinors are Grassmann odd. Define\n\\begin{align}\n R^{IJ}&=\\mathrm d\\omega^{IJ}+\\omega^I{}_K\\wedge\\omega^{KJ},\n &F&=\\mathrm da,\\\\\n \\operatorname{vol}_e&=\\tfrac1{4!}\\epsilon_{IJKL}\n                         e^I\\wedge e^J\\wedge e^K\\wedge e^L,\n &\\Sigma_I&=\\tfrac1{3!}\\epsilon_{IJKL}e^J\\wedge e^K\\wedge e^L.\n\\end{align}\nUse $\\{\\gamma^I,\\gamma^J\\}=2\\eta^{IJ}$ and\n$\\gamma_{IJ}=\\tfrac12[\\gamma_I,\\gamma_J]$. The covariant derivatives are\n\\begin{align}\n \\mathcal D\\psi&=\\mathrm d\\psi+\\tfrac14\\omega^{IJ}\\gamma_{IJ}\\psi-iq a\\psi,","\\operatorname{vol}_e&=\\tfrac1{4!}\\epsilon_{IJKL}\n                         e^I\\wedge e^J\\wedge e^K\\wedge e^L,\n &\\Sigma_I&=\\tfrac1{3!}\\epsilon_{IJKL}e^J\\wedge e^K\\wedge e^L.\n\\end{align}\nUse $\\{\\gamma^I,\\gamma^J\\}=2\\eta^{IJ}$ and\n$\\gamma_{IJ}=\\tfrac12[\\gamma_I,\\gamma_J]$. The covariant derivatives are\n\\begin{align}\n \\mathcal D\\psi&=\\mathrm d\\psi+\\tfrac14\\omega^{IJ}\\gamma_{IJ}\\psi-iq a\\psi,","\\operatorname{vol}_e&=\\tfrac1{4!}\\epsilon_{IJKL}\n                         e^I\\wedge e^J\\wedge e^K\\wedge e^L,\n &\\Sigma_I&=\\tfrac1{3!}\\epsilon_{IJKL}e^J\\wedge e^K\\wedge e^L.\n\\end{align}\nUse $\\{\\gamma^I,\\gamma^J\\}=2\\eta^{IJ}$ and\n$\\gamma_{IJ}=\\tfrac12[\\gamma_I,\\gamma_J]$. The covariant derivatives are\n\\begin{align}\n \\mathcal D\\psi&=\\mathrm d\\psi+\\tfrac14\\omega^{IJ}\\gamma_{IJ}\\psi-iq a\\psi,\\\\\n \\mathcal D\\bar\\psi&=\\mathrm d\\bar\\psi-\\tfrac14\\bar\\psi\\omega^{IJ}\\gamma_{IJ}+iq a\\bar\\psi.\n\\end{align}\nThe parameter $q$ here is electric charge. The determinant of a spatial metric below will be written $\\det(q_{ab})$ to avoid confusing the two uses.","\\mathcal D\\bar\\psi&=\\mathrm d\\bar\\psi-\\tfrac14\\bar\\psi\\omega^{IJ}\\gamma_{IJ}+iq a\\bar\\psi.\n\\end{align}\nThe parameter $q$ here is electric charge. The determinant of a spatial metric below will be written $\\det(q_{ab})$ to avoid confusing the two uses.","The parent action is $S_0=S_{\\mathrm G}+S_{\\mathrm M}^{(1)}+S_{\\mathrm D}$, with\n\\begin{align}\n S_{\\mathrm G}\n &=\\frac1{4\\kappa}\\int\\epsilon_{IJKL}e^I\\wedge e^J\\wedge R^{KL}\n                  -\\frac\\Lambda\\kappa\\int\\operatorname{vol}_e,","The parent action is $S_0=S_{\\mathrm G}+S_{\\mathrm M}^{(1)}+S_{\\mathrm D}$, with\n\\begin{align}\n S_{\\mathrm G}\n &=\\frac1{4\\kappa}\\int\\epsilon_{IJKL}e^I\\wedge e^J\\wedge R^{KL}\n                  -\\frac\\Lambda\\kappa\\int\\operatorname{vol}_e,\\\\\n S_{\\mathrm M}^{(1)}\n &=\\int\\left(\\tfrac12\\varphi_{IJ}F\\wedge e^I\\wedge e^J\n                    -\\tfrac14\\varphi_{IJ}\\varphi^{IJ}\\operatorname{vol}_e\\right),\\\\\n S_{\\mathrm D}\n &=\\frac i2\\int\\Sigma_I\\wedge\n       \\left(\\bar\\psi\\gamma^I\\mathcal D\\psi\n               -(\\mathcal D\\bar\\psi)\\gamma^I\\psi\\right)\n       -m\\int\\operatorname{vol}_e\\bar\\psi\\psi.\n \\label{c:constraints:parent-action}\n\\end{align}\nThis formulation uses no inverse tetrad in its defining action. Its largest monomials have six field factors. Nondegeneracy becomes necessary when the auxiliary variables are eliminated or a metric interpretation is imposed.","The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor $i$, the flat-space operators $i\\gamma^I\\partial_I\\pm m$ have product\n\\begin{equation}\n (i\\gamma^I\\partial_I+m)(i\\gamma^J\\partial_J-m)\n       =-\\Box-m^2,\n \\qquad \\Box=\\eta^{IJ}\\partial_I\\partial_J.\n\\end{equation}\nThus its squared field equation is $\\Box+m^2=0$, whereas the conventional real massive dispersion in this mostly-plus metric uses $\\Box-m^2=0$. A consistent translation of the Clifford, kinetic, mass, and adjoint conventions is required before identifying this sector with conventional physical massive Dirac matter. Gauge invariance and the graded master-equation argument do not require that physical identification, and they do not resolve it. The massless algebraic sector does not hav","The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor $i$, the flat-space operators $i\\gamma^I\\partial_I\\pm m$ have product\n\\begin{equation}\n (i\\gamma^I\\partial_I+m)(i\\gamma^J\\partial_J-m)\n       =-\\Box-m^2,\n \\qquad \\Box=\\eta^{IJ}\\partial_I\\partial_J.\n\\end{equation}\nThus its squared field equation is $\\Box+m^2=0$, whereas the conventional real massive dispersion in this mostly-plus metric uses $\\Box-m^2=0$. A consistent translation of the Clifford, kinetic, mass, and adjoint conventions is required before identifying this sector with conventional physical massive Dirac matter. Gauge invariance and the graded master-equation argument do not require that physical identification, and they do not resolve it. The massless algebraic sector does not have this squared-mass discrepancy.","needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor $i$, the flat-space operators $i\\gamma^I\\partial_I\\pm m$ have product\n\\begin{equation}\n (i\\gamma^I\\partial_I+m)(i\\gamma^J\\partial_J-m)\n       =-\\Box-m^2,\n \\qquad \\Box=\\eta^{IJ}\\partial_I\\partial_J.\n\\end{equation}\nThus its squared field equation is $\\Box+m^2=0$, whereas the conventional real massive dispersion in this mostly-plus metric uses $\\Box-m^2=0$. A consistent translation of the Clifford, kinetic, mass, and adjoint conventions is required before identifying this sector with conventional physical massive Dirac matter. Gauge invariance and the graded master-equation argument do not require that physical identification, and they do not resolve it. The massless algebraic sector does not have this squared-mass discrepancy.","model. With the stated Clifford sign and kinetic factor $i$, the flat-space operators $i\\gamma^I\\partial_I\\pm m$ have product\n\\begin{equation}\n (i\\gamma^I\\partial_I+m)(i\\gamma^J\\partial_J-m)\n       =-\\Box-m^2,\n \\qquad \\Box=\\eta^{IJ}\\partial_I\\partial_J.\n\\end{equation}\nThus its squared field equation is $\\Box+m^2=0$, whereas the conventional real massive dispersion in this mostly-plus metric uses $\\Box-m^2=0$. A consistent translation of the Clifford, kinetic, mass, and adjoint conventions is required before identifying this sector with conventional physical massive Dirac matter. Gauge invariance and the graded master-equation argument do not require that physical identification, and they do not resolve it. The massless algebraic sector does not have this squared-mass discrepancy.","Variation of $\\varphi$ gives the unambiguous wedge equation\n\\begin{equation}\n F\\wedge e^I\\wedge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}","Variation of $\\varphi$ gives the unambiguous wedge equation\n\\begin{equation}\n F\\wedge e^I\\wedge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","n of $\\varphi$ gives the unambiguous wedge equation\n\\begin{equation}\n F\\wedge e^I\\wedge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","unambiguous wedge equation\n\\begin{equation}\n F\\wedge e^I\\wedge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","ge e^J=\\varphi^{IJ}\\operatorname{vol}_e.\n \\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","\\label{c:constraints:maxwell-auxiliary}\n\\end{equation}\nUnder the standard Lorentzian Hodge convention\n$\\alpha\\wedge\\star\\beta=\\langle\\alpha,\\beta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","ta\\rangle\\operatorname{vol}_e$,\nthis equation identifies $\\varphi$ with minus the internal components of $\\star F$. For example, $F=E e^0\\wedge e^1$ has\n$\\star F=-E e^2\\wedge e^3$, whereas the wedge equation gives $\\varphi^{23}=E$. The inherited shorthand $\\varphi=\\star F$ needs this sign translation. Substitution of the wedge equation into the action gives the conventional result\n\\begin{equation}\n S_{\\mathrm M}^{(1)}\\big|_{\\varphi}\n      =-\\tfrac14\\int F_{IJ}F^{IJ}\\operatorname{vol}_e.\n\\end{equation}\nThe substitution repairs the Hodge shorthand while retaining the on-shell Maxwell action.","\\subsection{Gauge transformations and graded conventions}\nFor even parameters $(\\xi,\\lambda,u)$, the coframe transforms by its Lie derivative and Lorentz rotation, the connection by\n$\\mathcal L_\\xi\\omega-D_\\omega\\lambda$, and the Abelian potential by $\\mathcal L_\\xi a+\\mathrm du$. The spinor transforms by\n$\\mathcal L_\\xi\\psi+\\tfrac14\\lambda^{IJ}\\gamma_{IJ}\\psi+iqu\\psi$,\nwith the dual action on $\\bar\\psi$. All internal contractions in~\\eqref{c:constraints:parent-action} are invariant. A diffeomorphism changes its four-form density by a Lie derivative, which integrates to zero on a periodic domain or under stated boundary conditions.","The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention,\n\\begin{equation}\n [\\delta_U,\\delta_V]_{\\mathrm{fld}}\n       =-\\delta_{\\left[U,V\\right]_{\\mathrm{par}}},\n\\end{equation}\nwhere\n\\begin{equation}\n \\left[(\\xi,\\lambda,u),(\\eta,\\mu,v)\\right]_{\\mathrm{par}}\n =\\left([\\xi,\\eta],\\mathcal L_\\xi\\mu-\\mathcal L_\\eta\\lambda+[\\lambda,\\mu],\n                  \\mathcal L_\\xi v-\\mathcal L_\\eta u\\right).\n\\end{equation}\nThe minus sign is a convention for these field-space generators. It fixes the plus sign in the ghost differential below.","Let $c,\\chi,\\upsilon$ be odd ghosts and let $s$ be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set\n\\begin{equation}\n sc^\\mu=c^\\nu\\partial_\\nu c^\\mu,\\qquad\n s\\chi=\\mathcal L_c\\chi+\\chi^2,\\qquad\n s\\upsilon=\\mathcal L_c\\upsilon.\n \\label{c:constraints:ghost-differential}\n\\end{equation}\nOn the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular,\n\\begin{align}\n s\\psi&=\\mathcal L_c\\psi+\\tfrac14\\chi^{IJ}\\gamma_{IJ}\\psi+iq\\upsilon\\psi,","ultiplying component fields are placed on the left. Set\n\\begin{equation}\n sc^\\mu=c^\\nu\\partial_\\nu c^\\mu,\\qquad\n s\\chi=\\mathcal L_c\\chi+\\chi^2,\\qquad\n s\\upsilon=\\mathcal L_c\\upsilon.\n \\label{c:constraints:ghost-differential}\n\\end{equation}\nOn the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular,\n\\begin{align}\n s\\psi&=\\mathcal L_c\\psi+\\tfrac14\\chi^{IJ}\\gamma_{IJ}\\psi+iq\\upsilon\\psi,\\\\\n s\\bar\\psi&=\\mathcal L_c\\bar\\psi\n                -\\tfrac14\\chi^{IJ}\\bar\\psi\\gamma_{IJ}-iq\\upsilon\\bar\\psi.\n \\label{c:constraints:dual-ghost-action}\n\\end{align}\nThe ordering in the second line matters: both $\\chi$ and $\\bar\\psi$ are odd. The expression $-\\bar\\psi\\chi$ is not the same expression. Moving a ghost through an odd dual field changes its sign. The inherited formula with right-placed ghosts is not used without this correction.","s\\bar\\psi&=\\mathcal L_c\\bar\\psi\n                -\\tfrac14\\chi^{IJ}\\bar\\psi\\gamma_{IJ}-iq\\upsilon\\bar\\psi.\n \\label{c:constraints:dual-ghost-action}\n\\end{align}\nThe ordering in the second line matters: both $\\chi$ and $\\bar\\psi$ are odd. The expression $-\\bar\\psi\\chi$ is not the same expression. Moving a ghost through an odd dual field changes its sign. The inherited formula with right-placed ghosts is not used without this correction.","Nilpotency follows from the graded Jacobi identity for the parameter algebra and its action on each representation. A simple internal check makes the sign visible. Write the Lorentz representation-valued ghost as $g=\\tfrac14\\chi^{IJ}\\gamma_{IJ}$ and suppress diffeomorphisms and the Abelian factor. Then $sg=g^2$, $s\\psi=g\\psi$, and the dual rule is $s\\bar\\psi=\\bar\\psi g$ when the odd ghost has been moved to the right. Thus\n\\begin{equation}\n s^2\\psi=g^2\\psi-g(g\\psi)=0,\\qquad\n s^2\\bar\\psi=(\\bar\\psi g)g-\\bar\\psi g^2=0.\n\\end{equation}\nUsing $s\\bar\\psi=-\\bar\\psi g$ with the same left-derivation convention instead gives $2\\bar\\psi g^2$. A commuting-parameter covariance test does not detect this Grassmann sign error.","identity for the parameter algebra and its action on each representation. A simple internal check makes the sign visible. Write the Lorentz representation-valued ghost as $g=\\tfrac14\\chi^{IJ}\\gamma_{IJ}$ and suppress diffeomorphisms and the Abelian factor. Then $sg=g^2$, $s\\psi=g\\psi$, and the dual rule is $s\\bar\\psi=\\bar\\psi g$ when the odd ghost has been moved to the right. Thus\n\\begin{equation}\n s^2\\psi=g^2\\psi-g(g\\psi)=0,\\qquad\n s^2\\bar\\psi=(\\bar\\psi g)g-\\bar\\psi g^2=0.\n\\end{equation}\nUsing $s\\bar\\psi=-\\bar\\psi g$ with the same left-derivation convention instead gives $2\\bar\\psi g^2$. A commuting-parameter covariance test does not detect this Grassmann sign error.","es the sign visible. Write the Lorentz representation-valued ghost as $g=\\tfrac14\\chi^{IJ}\\gamma_{IJ}$ and suppress diffeomorphisms and the Abelian factor. Then $sg=g^2$, $s\\psi=g\\psi$, and the dual rule is $s\\bar\\psi=\\bar\\psi g$ when the odd ghost has been moved to the right. Thus\n\\begin{equation}\n s^2\\psi=g^2\\psi-g(g\\psi)=0,\\qquad\n s^2\\bar\\psi=(\\bar\\psi g)g-\\bar\\psi g^2=0.\n\\end{equation}\nUsing $s\\bar\\psi=-\\bar\\psi g$ with the same left-derivation convention instead gives $2\\bar\\psi g^2$. A commuting-parameter covariance test does not detect this Grassmann sign error.","14\\chi^{IJ}\\gamma_{IJ}$ and suppress diffeomorphisms and the Abelian factor. Then $sg=g^2$, $s\\psi=g\\psi$, and the dual rule is $s\\bar\\psi=\\bar\\psi g$ when the odd ghost has been moved to the right. Thus\n\\begin{equation}\n s^2\\psi=g^2\\psi-g(g\\psi)=0,\\qquad\n s^2\\bar\\psi=(\\bar\\psi g)g-\\bar\\psi g^2=0.\n\\end{equation}\nUsing $s\\bar\\psi=-\\bar\\psi g$ with the same left-derivation convention instead gives $2\\bar\\psi g^2$. A commuting-parameter covariance test does not detect this Grassmann sign error.","\\subsection{The classical BV action}\nFor all fields and ghosts $\\Xi$, introduce antifields $\\Xi^\\ddagger$ with opposite parity and ghost number $-1-\\operatorname{gh}(\\Xi)$. Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written\n\\begin{equation}\n S_{\\mathrm{BV}}=S_0+\n     \\int\\sum_\\Xi\\langle\\Xi^\\ddagger,s\\Xi\\rangle.\n \\label{c:constraints:bv-action}\n\\end{equation}\nAntisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor $1/2$. No higher-antifield terms are needed for this off-shell closed representation algebra.","orname{gh}(\\Xi)$. Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written\n\\begin{equation}\n S_{\\mathrm{BV}}=S_0+\n     \\int\\sum_\\Xi\\langle\\Xi^\\ddagger,s\\Xi\\rangle.\n \\label{c:constraints:bv-action}\n\\end{equation}\nAntisymmetric Lorentz indices are counted once, or equivalently their redundant sum is given its usual factor $1/2$. No higher-antifield terms are needed for this off-shell closed representation algebra.","\\begin{proposition}[Classical master equation]\nWith the declared graded conventions, $sS_0=0$ and $s^2=0$ imply\n\\begin{equation}\n (S_{\\mathrm{BV}},S_{\\mathrm{BV}})=0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe bracket of $S_0$ with the antifield-linear terms is twice its gauge variation. The bracket among antifield-linear terms has coefficient $s^2\\Xi$ at each antifield, including the ghost antifields. These coefficients vanish. There is no bracket contribution from $S_0$ with itself. This exhausts the antifield expansion.\n\\end{proof}","ons, $sS_0=0$ and $s^2=0$ imply\n\\begin{equation}\n (S_{\\mathrm{BV}},S_{\\mathrm{BV}})=0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe bracket of $S_0$ with the antifield-linear terms is twice its gauge variation. The bracket among antifield-linear terms has coefficient $s^2\\Xi$ at each antifield, including the ghost antifields. These coefficients vanish. There is no bracket contribution from $S_0$ with itself. This exhausts the antifield expansion.\n\\end{proof}","For coefficients in $\\mathcal T_K^{(4)}$ and a finite Grassmann algebra, the parent action has degree at most six, and the displayed antifield couplings have degree at most three. The master-equation and invariance coefficients have at most seven band-$K$ factors. The sufficient rule $16K<N$ places them below Nyquist and transfers the classical identities coefficientwise. The rule concerns the polynomial parent. It does not automatically survive elimination of inverse tetrads or density square roots.","Reality is built into $e$ and $\\omega$. For real Lorentz bivectors, the internal dual obeys $\\star^2=-\\mathrm{id}$ and the projectors\n\\begin{equation}\n P_\\pm=\\tfrac12(\\mathrm{id}\\mp i\\star)\n\\end{equation}\nsatisfy $\\overline{P_+X}=P_-X$. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical starting point. It does not prove a missing finite reality-preserving step for the earlier isolated self-dual coordinates.","\\subsection{An exact time--space split}\nOn $M=I\\times\\Sigma$ with orientation $\\mathrm dt\\wedge\\operatorname{or}(\\Sigma)$, write\n\\begin{equation}\n e^I=e_\\perp^I\\mathrm dt+E^I,\\qquad\n \\omega^{IJ}=\\omega_\\perp^{IJ}\\mathrm dt+\\Omega^{IJ},\\qquad\n a=a_\\perp\\mathrm dt+A.\n\\end{equation}\nLet $\\mathcal R=\\mathrm d_\\Sigma\\Omega+\\Omega\\wedge\\Omega$ and $\\mathcal F=\\mathrm d_\\Sigma A$. The curvature and field strength split as\n\\begin{equation}\n R=\\mathrm dt\\wedge(\\dot\\Omega-D_\\Omega\\omega_\\perp)+\\mathcal R,\n \\qquad\n F=\\mathrm dt\\wedge(\\dot A-\\mathrm d_\\Sigma a_\\perp)+\\mathcal F.\n\\end{equation}\nDefine\n\\begin{align}\n B_{IJ}&=\\tfrac1{4\\kappa}\\epsilon_{KLIJ}E^K\\wedge E^L,","lit}\nOn $M=I\\times\\Sigma$ with orientation $\\mathrm dt\\wedge\\operatorname{or}(\\Sigma)$, write\n\\begin{equation}\n e^I=e_\\perp^I\\mathrm dt+E^I,\\qquad\n \\omega^{IJ}=\\omega_\\perp^{IJ}\\mathrm dt+\\Omega^{IJ},\\qquad\n a=a_\\perp\\mathrm dt+A.\n\\end{equation}\nLet $\\mathcal R=\\mathrm d_\\Sigma\\Omega+\\Omega\\wedge\\Omega$ and $\\mathcal F=\\mathrm d_\\Sigma A$. The curvature and field strength split as\n\\begin{equation}\n R=\\mathrm dt\\wedge(\\dot\\Omega-D_\\Omega\\omega_\\perp)+\\mathcal R,\n \\qquad\n F=\\mathrm dt\\wedge(\\dot A-\\mathrm d_\\Sigma a_\\perp)+\\mathcal F.\n\\end{equation}\nDefine\n\\begin{align}\n B_{IJ}&=\\tfrac1{4\\kappa}\\epsilon_{KLIJ}E^K\\wedge E^L,","IJ}\\mathrm dt+\\Omega^{IJ},\\qquad\n a=a_\\perp\\mathrm dt+A.\n\\end{equation}\nLet $\\mathcal R=\\mathrm d_\\Sigma\\Omega+\\Omega\\wedge\\Omega$ and $\\mathcal F=\\mathrm d_\\Sigma A$. The curvature and field strength split as\n\\begin{equation}\n R=\\mathrm dt\\wedge(\\dot\\Omega-D_\\Omega\\omega_\\perp)+\\mathcal R,\n \\qquad\n F=\\mathrm dt\\wedge(\\dot A-\\mathrm d_\\Sigma a_\\perp)+\\mathcal F.\n\\end{equation}\nDefine\n\\begin{align}\n B_{IJ}&=\\tfrac1{4\\kappa}\\epsilon_{KLIJ}E^K\\wedge E^L,\\\\\n \\Pi&=\\tfrac12\\varphi_{IJ}E^I\\wedge E^J,\\qquad\n \\sigma_I=\\tfrac1{3!}\\epsilon_{IJKL}E^J\\wedge E^K\\wedge E^L.\n\\end{align}\nDirect exterior multiplication gives the spatial symplectic potential\n\\begin{equation}\n \\begin{aligned}\n \\Theta_\\Sigma=\\int_\\Sigma\\bigl[&B_{IJ}\\wedge\\delta\\Omega^{IJ}+\\Pi\\wedge\\delta A","\\Pi&=\\tfrac12\\varphi_{IJ}E^I\\wedge E^J,\\qquad\n \\sigma_I=\\tfrac1{3!}\\epsilon_{IJKL}E^J\\wedge E^K\\wedge E^L.\n\\end{align}\nDirect exterior multiplication gives the spatial symplectic potential\n\\begin{equation}\n \\begin{aligned}\n \\Theta_\\Sigma=\\int_\\Sigma\\bigl[&B_{IJ}\\wedge\\delta\\Omega^{IJ}+\\Pi\\wedge\\delta A\\\\\n &-\\tfrac i2\\sigma_I(\\bar\\psi\\gamma^I\\delta\\psi\n                         -\\delta\\bar\\psi\\gamma^I\\psi)\\bigr].\n \\end{aligned}\n \\label{c:constraints:boundary-potential}\n\\end{equation}\nThe minus sign in the spinor term is fixed by the time--space orientation and by moving $\\mathrm dt$ past the spatial three-form. It must be retained when comparing the spacetime action and its canonical chart.","&-\\tfrac i2\\sigma_I(\\bar\\psi\\gamma^I\\delta\\psi\n                         -\\delta\\bar\\psi\\gamma^I\\psi)\\bigr].\n \\end{aligned}\n \\label{c:constraints:boundary-potential}\n\\end{equation}\nThe minus sign in the spinor term is fixed by the time--space orientation and by moving $\\mathrm dt$ past the spatial three-form. It must be retained when comparing the spacetime action and its canonical chart.","The temporal coframe multiplies the Einstein--matter three-forms\n$\\mathcal C_I=\\mathcal C_I^{\\mathrm G}+\\mathcal C_I^{\\mathrm M}+\\mathcal C_I^{\\mathrm D}$, where\n\\begin{align}\n \\mathcal C_I^{\\mathrm G}\n &=\\tfrac1{2\\kappa}\\epsilon_{IJKL}E^J\\wedge\\mathcal R^{KL}\n                  -\\tfrac\\Lambda\\kappa\\sigma_I,","The temporal coframe multiplies the Einstein--matter three-forms\n$\\mathcal C_I=\\mathcal C_I^{\\mathrm G}+\\mathcal C_I^{\\mathrm M}+\\mathcal C_I^{\\mathrm D}$, where\n\\begin{align}\n \\mathcal C_I^{\\mathrm G}\n &=\\tfrac1{2\\kappa}\\epsilon_{IJKL}E^J\\wedge\\mathcal R^{KL}\n                  -\\tfrac\\Lambda\\kappa\\sigma_I,\\\\\n \\mathcal C_I^{\\mathrm M}\n &=\\varphi_{IJ}\\mathcal F\\wedge E^J-\\tfrac14\\varphi_{KL}\\varphi^{KL}\\sigma_I,\\\\\n \\mathcal C_I^{\\mathrm D}\n &=\\tfrac i4\\epsilon_{JIKL}E^K\\wedge E^L\\wedge\n       \\bigl(\\bar\\psi\\gamma^J\\mathcal D_\\Sigma\\psi\n                  -(\\mathcal D_\\Sigma\\bar\\psi)\\gamma^J\\psi\\bigr)\n                      -m\\sigma_I\\bar\\psi\\psi.\n\\end{align}\nSpatial integration by parts identifies the Lorentz and electric Gauss forms\n\\begin{align}\n \\mathcal G_{IJ}\n &=D_\\Omega B_{IJ}\n       -\\tfrac i8\\sigma_K\\bar\\psi\\{\\gamma^K,\\gamma_{IJ}\\}\\psi,","\\mathcal C_I^{\\mathrm D}\n &=\\tfrac i4\\epsilon_{JIKL}E^K\\wedge E^L\\wedge\n       \\bigl(\\bar\\psi\\gamma^J\\mathcal D_\\Sigma\\psi\n                  -(\\mathcal D_\\Sigma\\bar\\psi)\\gamma^J\\psi\\bigr)\n                      -m\\sigma_I\\bar\\psi\\psi.\n\\end{align}\nSpatial integration by parts identifies the Lorentz and electric Gauss forms\n\\begin{align}\n \\mathcal G_{IJ}\n &=D_\\Omega B_{IJ}\n       -\\tfrac i8\\sigma_K\\bar\\psi\\{\\gamma^K,\\gamma_{IJ}\\}\\psi,\\\\\n \\mathcal G_{\\mathrm U}\n &=\\mathrm d_\\Sigma\\Pi-q\\sigma_I\\bar\\psi\\gamma^I\\psi.\n\\end{align}\nConsequently the multiplier equations are\n$\\mathcal C_I=\\mathcal G_{IJ}=\\mathcal G_{\\mathrm U}=0$.\nThese are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.","\\mathcal G_{\\mathrm U}\n &=\\mathrm d_\\Sigma\\Pi-q\\sigma_I\\bar\\psi\\gamma^I\\psi.\n\\end{align}\nConsequently the multiplier equations are\n$\\mathcal C_I=\\mathcal G_{IJ}=\\mathcal G_{\\mathrm U}=0$.\nThese are exact formulas for the split action. The auxiliary constitutive equations and the simplicity relations are additional primary relations; they are not extra first-class gauge generators.","\\subsection{What is explicit in the internal BFV sector}\nOn a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian:\n\\begin{equation}\n \\{G_{\\mathrm L}[\\lambda],G_{\\mathrm L}[\\mu]\\}\n       =G_{\\mathrm L}[[\\lambda,\\mu]],\\qquad\n \\{G_{\\mathrm L},G_{\\mathrm U}\\}=\\{G_{\\mathrm U},G_{\\mathrm U}\\}=0.\n\\end{equation}\nIts canonical BFV charge is explicit:\n\\begin{equation}\n \\Omega_{\\mathrm{int}}=G_{\\mathrm L}[\\chi]+G_{\\mathrm U}[\\upsilon]\n                   -\\tfrac12\\langle\\pi_\\chi,[\\chi,\\chi]\\rangle.\n\\end{equation}\nThe moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus\n$\\{\\Omega_{\\mathrm{int}},\\Omega_{\\mathrm{int}}\\}=0$ under that regular moment-map hypothesis. This finite Lie-algebra mechanism is more explicit than a claim about a complete gravitational boundary charge.","nal BFV sector}\nOn a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian:\n\\begin{equation}\n \\{G_{\\mathrm L}[\\lambda],G_{\\mathrm L}[\\mu]\\}\n       =G_{\\mathrm L}[[\\lambda,\\mu]],\\qquad\n \\{G_{\\mathrm L},G_{\\mathrm U}\\}=\\{G_{\\mathrm U},G_{\\mathrm U}\\}=0.\n\\end{equation}\nIts canonical BFV charge is explicit:\n\\begin{equation}\n \\Omega_{\\mathrm{int}}=G_{\\mathrm L}[\\chi]+G_{\\mathrm U}[\\upsilon]\n                   -\\tfrac12\\langle\\pi_\\chi,[\\chi,\\chi]\\rangle.\n\\end{equation}\nThe moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus\n$\\{\\Omega_{\\mathrm{int}},\\Omega_{\\mathrm{int}}\\}=0$ under that regular moment-map hypothesis. This finite Lie-algebra mechanism is more explicit than a claim about a complete gravitational boundary charge.","hrm L},G_{\\mathrm U}\\}=\\{G_{\\mathrm U},G_{\\mathrm U}\\}=0.\n\\end{equation}\nIts canonical BFV charge is explicit:\n\\begin{equation}\n \\Omega_{\\mathrm{int}}=G_{\\mathrm L}[\\chi]+G_{\\mathrm U}[\\upsilon]\n                   -\\tfrac12\\langle\\pi_\\chi,[\\chi,\\chi]\\rangle.\n\\end{equation}\nThe moment-map relation cancels the terms quadratic in ghosts, and the Lorentz Jacobi identity cancels the cubic terms. Thus\n$\\{\\Omega_{\\mathrm{int}},\\Omega_{\\mathrm{int}}\\}=0$ under that regular moment-map hypothesis. This finite Lie-algebra mechanism is more explicit than a claim about a complete gravitational boundary charge.","\\subsection{Normal deformations require more boundary data}\nFor a spatial vector field $\\eta$, the Cartan identities give\n\\begin{equation}\n \\mathcal L_\\eta\\Omega=\\iota_\\eta\\mathcal R+D_\\Omega(\\iota_\\eta\\Omega),\n \\qquad\n \\mathcal L_\\eta A=\\iota_\\eta\\mathcal F+\\mathrm d_\\Sigma(\\iota_\\eta A).\n\\end{equation}\nAccordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression\n\\begin{equation}\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n\\end{equation}\nFor tangential $c$, $\\zeta(c)=\\iota_cE$ is a spatial contraction. For a normal deformation, a spatial one-form does not b","\\subsection{Normal deformations require more boundary data}\nFor a spatial vector field $\\eta$, the Cartan identities give\n\\begin{equation}\n \\mathcal L_\\eta\\Omega=\\iota_\\eta\\mathcal R+D_\\Omega(\\iota_\\eta\\Omega),\n \\qquad\n \\mathcal L_\\eta A=\\iota_\\eta\\mathcal F+\\mathrm d_\\Sigma(\\iota_\\eta A).\n\\end{equation}\nAccordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression\n\\begin{equation}\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n\\end{equation}\nFor tangential $c$, $\\zeta(c)=\\iota_cE$ is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta, their symplectic extension, and the relevant normal field data. One must then quotient the presymplec","$, the Cartan identities give\n\\begin{equation}\n \\mathcal L_\\eta\\Omega=\\iota_\\eta\\mathcal R+D_\\Omega(\\iota_\\eta\\Omega),\n \\qquad\n \\mathcal L_\\eta A=\\iota_\\eta\\mathcal F+\\mathrm d_\\Sigma(\\iota_\\eta A).\n\\end{equation}\nAccordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression\n\\begin{equation}\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n\\end{equation}\nFor tangential $c$, $\\zeta(c)=\\iota_cE$ is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta, their symplectic extension, and the relevant normal field data. One must then quotient the presymplectic kernel and prove that the gauge action and its Hamiltonian descend.","A).\n\\end{equation}\nAccordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression\n\\begin{equation}\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n\\end{equation}\nFor tangential $c$, $\\zeta(c)=\\iota_cE$ is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta, their symplectic extension, and the relevant normal field data. One must then quotient the presymplectic kernel and prove that the gauge action and its Hamiltonian descend.","end{equation}\nAccordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression\n\\begin{equation}\n J_\\Sigma[c,\\chi,u]=C[\\zeta(c)]\n             +G_{\\mathrm L}[\\chi-\\iota_c\\Omega]\n             +G_{\\mathrm U}[u+\\iota_cA].\n \\label{c:constraints:boundary-candidate}\n\\end{equation}\nFor tangential $c$, $\\zeta(c)=\\iota_cE$ is a spatial contraction. For a normal deformation, a spatial one-form does not by itself supply that contraction. One must specify the embedding variables and conjugate momenta, their symplectic extension, and the relevant normal field data. One must then quotient the presymplectic kernel and prove that the gauge action and its Hamiltonian descend.","\\begin{proposition}[Conditional boundary BFV construction]\nSuppose a regular derived boundary symplectic space, including the required embedding data, has been constructed. Suppose its full normal and tangential gauge action has an equivariant moment map $J_\\Sigma$ for the declared parameter algebra. Add canonical ghosts and their momenta, and let the ghost Hamiltonian generate the Chevalley--Eilenberg differential in the same conventions. Then their sum $\\Omega_\\Sigma$ obeys\n\\begin{equation}\n \\{\\Omega_\\Sigma,\\Omega_\\Sigma\\}=0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe bracket of the two moment-map terms reproduces the parameter bracket. The cross bracket with the ghost Hamiltonian cancels that term. The bracket of the ghost Hamiltonian with itself vanishes by the graded Jacobi identity. Equivariance excludes a central defect in the moment-map relation.\n\\end{proof}","uppose a regular derived boundary symplectic space, including the required embedding data, has been constructed. Suppose its full normal and tangential gauge action has an equivariant moment map $J_\\Sigma$ for the declared parameter algebra. Add canonical ghosts and their momenta, and let the ghost Hamiltonian generate the Chevalley--Eilenberg differential in the same conventions. Then their sum $\\Omega_\\Sigma$ obeys\n\\begin{equation}\n \\{\\Omega_\\Sigma,\\Omega_\\Sigma\\}=0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe bracket of the two moment-map terms reproduces the parameter bracket. The cross bracket with the ghost Hamiltonian cancels that term. The bracket of the ghost Hamiltonian with itself vanishes by the graded Jacobi identity. Equivariance excludes a central defect in the moment-map relation.\n\\end{proof}","plectic space, including the required embedding data, has been constructed. Suppose its full normal and tangential gauge action has an equivariant moment map $J_\\Sigma$ for the declared parameter algebra. Add canonical ghosts and their momenta, and let the ghost Hamiltonian generate the Chevalley--Eilenberg differential in the same conventions. Then their sum $\\Omega_\\Sigma$ obeys\n\\begin{equation}\n \\{\\Omega_\\Sigma,\\Omega_\\Sigma\\}=0.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe bracket of the two moment-map terms reproduces the parameter bracket. The cross bracket with the ghost Hamiltonian cancels that term. The bracket of the ghost Hamiltonian with itself vanishes by the graded Jacobi identity. Equivariance excludes a central defect in the moment-map relation.\n\\end{proof}","\\subsection{ADM ghosts and spectral headroom}\nOn a nondegenerate spacelike slice, let\n$q_{ab}=\\eta_{IJ}E_a^IE_b^J$ and define the densitized normal\n\\begin{equation}\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n\\end{equation}\nIt satisfies $\\widetilde n_IE_a^I=0$ and\n$-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})$.\nA parameter $\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I$ yields a densitized lapse $\\nu$ and shift $\\eta$. On a correctly reduced boundary space the expected normal bracket has structure shift\n\\begin{equation}\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n\\end{equation}\nThe improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal br","\\subsection{ADM ghosts and spectral headroom}\nOn a nondegenerate spacelike slice, let\n$q_{ab}=\\eta_{IJ}E_a^IE_b^J$ and define the densitized normal\n\\begin{equation}\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n\\end{equation}\nIt satisfies $\\widetilde n_IE_a^I=0$ and\n$-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})$.\nA parameter $\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I$ yields a densitized lapse $\\nu$ and shift $\\eta$. On a correctly reduced boundary space the expected normal bracket has structure shift\n\\begin{equation}\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n\\end{equation}\nThe improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket may contain both spatial and Gauss terms in the first-order chart.","pectral headroom}\nOn a nondegenerate spacelike slice, let\n$q_{ab}=\\eta_{IJ}E_a^IE_b^J$ and define the densitized normal\n\\begin{equation}\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n\\end{equation}\nIt satisfies $\\widetilde n_IE_a^I=0$ and\n$-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})$.\nA parameter $\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I$ yields a densitized lapse $\\nu$ and shift $\\eta$. On a correctly reduced boundary space the expected normal bracket has structure shift\n\\begin{equation}\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n\\end{equation}\nThe improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket may contain both spatial and Gauss terms in the first-order chart.","}\nOn a nondegenerate spacelike slice, let\n$q_{ab}=\\eta_{IJ}E_a^IE_b^J$ and define the densitized normal\n\\begin{equation}\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n\\end{equation}\nIt satisfies $\\widetilde n_IE_a^I=0$ and\n$-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})$.\nA parameter $\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I$ yields a densitized lapse $\\nu$ and shift $\\eta$. On a correctly reduced boundary space the expected normal bracket has structure shift\n\\begin{equation}\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n\\end{equation}\nThe improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket may contain both spatial and Gauss terms in the first-order chart.","n{equation}\n \\widetilde n_I=\\tfrac1{3!}\\epsilon_{IJKL}\\epsilon^{abc}E_a^JE_b^KE_c^L.\n\\end{equation}\nIt satisfies $\\widetilde n_IE_a^I=0$ and\n$-\\eta^{IJ}\\widetilde n_I\\widetilde n_J=\\det(q_{ab})$.\nA parameter $\\zeta^I=\\nu\\widetilde n^I+\\eta^aE_a^I$ yields a densitized lapse $\\nu$ and shift $\\eta$. On a correctly reduced boundary space the expected normal bracket has structure shift\n\\begin{equation}\n S^a(\\nu,\\mu)=\\det(q_{cd})q^{ab}\n                          (\\nu\\partial_b\\mu-\\mu\\partial_b\\nu).\n\\end{equation}\nThe improved spatial generator differs from the covariant vector generator by its internal compensators. This is the usual reason that the normal bracket may contain both spatial and Gauss terms in the first-order chart.","The map from covariant ghosts to lapse, shift, and internal ghosts depends on $E,\\Omega,A$. A canonical lift must transform ghost momenta and, generally, physical momenta as well. It may generate higher ghost terms. Merely replacing the ghosts in a quadratic formula does not prove that the new formula is canonical or nilpotent. The complete charge must be carried through that lift, rather than inferred from its ghost-linear coefficients.","The source estimates at most nine band-$K$ factors in its polynomial ADM representation, so at most sixteen in a bracket. Its sufficient rule is $32K<N$. This explains the additional spectral room required by the densitized normal. It does not discharge the boundary construction, the canonical ghost lift, or rational inverses created after reduction. Those prerequisites are independent of a degree count.","\\subsection{Time gauge and simplicity pullback}\nChoose an oriented time-gauge patch\n\\begin{equation}\n E^0=0,\\qquad E^i=e_a{}^i\\,\\mathrm dx^a,\n \\qquad e:=\\det(e_a{}^i)>0.\n \\label{c:constraints:regular-triad}\n\\end{equation}\nThe positive determinant fixes the orientation and ensures invertibility. Define\n\\begin{equation}\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n\\end{equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the facto","\\subsection{Time gauge and simplicity pullback}\nChoose an oriented time-gauge patch\n\\begin{equation}\n E^0=0,\\qquad E^i=e_a{}^i\\,\\mathrm dx^a,\n \\qquad e:=\\det(e_a{}^i)>0.\n \\label{c:constraints:regular-triad}\n\\end{equation}\nThe positive determinant fixes the orientation and ensures invertibility. Define\n\\begin{equation}\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n\\end{equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equatio…","ented time-gauge patch\n\\begin{equation}\n E^0=0,\\qquad E^i=e_a{}^i\\,\\mathrm dx^a,\n \\qquad e:=\\det(e_a{}^i)>0.\n \\label{c:constraints:regular-triad}\n\\end{equation}\nThe positive determinant fixes the orientation and ensures invertibility. Define\n\\begin{equation}\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n\\end{equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","\\begin{equation}\n E^0=0,\\qquad E^i=e_a{}^i\\,\\mathrm dx^a,\n \\qquad e:=\\det(e_a{}^i)>0.\n \\label{c:constraints:regular-triad}\n\\end{equation}\nThe positive determinant fixes the orientation and ensures invertibility. Define\n\\begin{equation}\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n\\end{equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","c:constraints:regular-triad}\n\\end{equation}\nThe positive determinant fixes the orientation and ensures invertibility. Define\n\\begin{equation}\n \\widetilde E^a{}_i=\\tfrac12\\epsilon^{abc}\\epsilon_{ijk}e_b{}^je_c{}^k,\n \\qquad \\widetilde E^a{}_ie_a{}^j=e\\delta_i{}^j.\n\\end{equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","equation}\nLet $K^i=\\Omega^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","ga^{0i}$ and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","and\n$\\Gamma^i=-\\tfrac12\\epsilon^i{}_{jk}\\Omega^{jk}$.\nThe simplicity pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","ty pullback gives\n\\begin{equation}\n B_{ij}=0,\\qquad\n B_{0i}=\\tfrac1{4\\kappa}\\epsilon_{ijk}E^j\\wedge E^k,\n \\qquad\n \\int_\\Sigma B_{IJ}\\wedge\\delta\\Omega^{IJ}\n      =\\frac1\\kappa\\int_\\Sigma\\widetilde E^a{}_i\\delta K_a{}^i\\,\\mathrm d^3x.\n \\label{c:constraints:simplicity}\n\\end{equation}\nThe antisymmetric $IJ$ sum includes both $0i$ and $i0$, accounting for the factor in the last identity. There is no $\\delta\\Gamma$ term. Those directions lie in the kernel of the pulled-back gravitational presymplectic form and are fixed by the algebraic connection equation on the regular sector.","Similarly, with $\\mathcal E^a=\\tfrac12\\epsilon^{abc}\\Pi_{bc}$,\n\\begin{equation}\n \\int_\\Sigma\\Pi\\wedge\\delta A\n        =\\int_\\Sigma\\mathcal E^a\\delta A_a\\,\\mathrm d^3x.\n\\end{equation}\nThe Maxwell auxiliary equation then gives its metric Hamiltonian\n\\begin{equation}\n \\mathcal H_{\\mathrm M}=\\frac1{2e}q_{ab}\n       (\\mathcal E^a\\mathcal E^b+\\mathcal B^a\\mathcal B^b),\n \\qquad \\mathcal B^a=\\tfrac12\\epsilon^{abc}\\mathcal F_{bc}.\n\\end{equation}\nThe inverse $1/e$ is well defined in the patch and singular at its boundary.","i_{bc}$,\n\\begin{equation}\n \\int_\\Sigma\\Pi\\wedge\\delta A\n        =\\int_\\Sigma\\mathcal E^a\\delta A_a\\,\\mathrm d^3x.\n\\end{equation}\nThe Maxwell auxiliary equation then gives its metric Hamiltonian\n\\begin{equation}\n \\mathcal H_{\\mathrm M}=\\frac1{2e}q_{ab}\n       (\\mathcal E^a\\mathcal E^b+\\mathcal B^a\\mathcal B^b),\n \\qquad \\mathcal B^a=\\tfrac12\\epsilon^{abc}\\mathcal F_{bc}.\n\\end{equation}\nThe inverse $1/e$ is well defined in the patch and singular at its boundary.","\\subsection{Fermionic primary constraints and half densities}\nImpose the inherited algebraic spinor relation $\\bar\\psi=\\psi^\\dagger\\gamma^0$, where $(\\gamma^0)^2=-\\mathrm{id}$. Its phase convention must be included in the physical translation just described. In time gauge $\\sigma_0=e\\,\\mathrm d^3x$ and $\\sigma_i=0$. Equation~\\eqref{c:constraints:boundary-potential} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation g","\\subsection{Fermionic primary constraints and half densities}\nImpose the inherited algebraic spinor relation $\\bar\\psi=\\psi^\\dagger\\gamma^0$, where $(\\gamma^0)^2=-\\mathrm{id}$. Its phase convention must be included in the physical translation just described. In time gauge $\\sigma_0=e\\,\\mathrm d^3x$ and $\\sigma_i=0$. Equation~\\eqref{c:constraints:boundary-potential} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\de","\\subsection{Fermionic primary constraints and half densities}\nImpose the inherited algebraic spinor relation $\\bar\\psi=\\psi^\\dagger\\gamma^0$, where $(\\gamma^0)^2=-\\mathrm{id}$. Its phase convention must be included in the physical translation just described. In time gauge $\\sigma_0=e\\,\\mathrm d^3x$ and $\\sigma_i=0$. Equation~\\eqref{c:constraints:boundary-potential} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,","the physical translation just described. In time gauge $\\sigma_0=e\\,\\mathrm d^3x$ and $\\sigma_i=0$. Equation~\\eqref{c:constraints:boundary-potential} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,","ndary-potential} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,","al} becomes\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,","tion}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma e\n        (\\psi^\\dagger\\delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,","delta\\psi-\\delta\\psi^\\dagger\\psi)\\,\\mathrm d^3x.\n\\end{equation}\nIn the inherited right--left graded canonical convention, the primary matrix is\n\\begin{equation}\n \\{\\chi_A(x),\\bar\\chi_B(y)\\}_{\\mathrm P}\n            =-ie(x)\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nIt is invertible exactly where $e\\ne0$. Set\n$\\Psi=e^{1/2}\\psi$ and $\\Psi^\\dagger=e^{1/2}\\psi^\\dagger$.\nDifferentiation gives\n\\begin{align}\n \\Psi^\\dagger\\delta\\Psi\n   &=e\\psi^\\dagger\\delta\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e,\\\\\n \\delta\\Psi^\\dagger\\Psi\n   &=e\\delta\\psi^\\dagger\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e.\n\\end{align}\nThe determinant terms cancel, so\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma\n       (\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\,\\mathrm d^3x,\n \\qquad\n \\{\\Psi_A(x),\\Psi_B^\\dagger(y)\\}_{\\mathrm D}\n             =-i\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nThe half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of $e$, so it is a local coordinate choice on~\\eqref{…","\\delta\\Psi^\\dagger\\Psi\n   &=e\\delta\\psi^\\dagger\\psi+\\tfrac12\\psi^\\dagger\\psi\\,\\delta e.\n\\end{align}\nThe determinant terms cancel, so\n\\begin{equation}\n \\Theta_{\\mathrm D}=\\frac i2\\int_\\Sigma\n       (\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\,\\mathrm d^3x,\n \\qquad\n \\{\\Psi_A(x),\\Psi_B^\\dagger(y)\\}_{\\mathrm D}\n             =-i\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nThe half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of $e$, so it is a local coordinate choice on~\\eqref{c:constraints:regular-triad}.","{\\mathrm D}=\\frac i2\\int_\\Sigma\n       (\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\,\\mathrm d^3x,\n \\qquad\n \\{\\Psi_A(x),\\Psi_B^\\dagger(y)\\}_{\\mathrm D}\n             =-i\\delta_{AB}\\delta^3(x-y).\n\\end{equation}\nThe half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of $e$, so it is a local coordinate choice on~\\eqref{c:constraints:regular-triad}.","\\subsection{Algebraic torsion and the reduced phase space}\nFor the symmetric Dirac action, define\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and\n$J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi$.\nIn the parent convention, the connection equation has the algebraic solution\n\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contr","\\subsection{Algebraic torsion and the reduced phase space}\nFor the symmetric Dirac action, define\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and\n$J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi$.\nIn the parent convention, the connection equation has the algebraic solution\n\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5","\\subsection{Algebraic torsion and the reduced phase space}\nFor the symmetric Dirac action, define\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and\n$J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi$.\nIn the parent convention, the connection equation has the algebraic solution\n\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e$.\nIts sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.","ebraic torsion and the reduced phase space}\nFor the symmetric Dirac action, define\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and\n$J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi$.\nIn the parent convention, the connection equation has the algebraic solution\n\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e$.\nIts sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.","a^2\\gamma^3$ and\n$J_5^I=\\bar\\psi\\gamma^I\\gamma_5\\psi$.\nIn the parent convention, the connection equation has the algebraic solution\n\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e$.\nIts sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.","\\begin{equation}\n T^I=\\tfrac\\kappa4\\epsilon^I{}_{JKL}J_5^Je^K\\wedge e^L,\n \\qquad\n C^{IJ}=\\tfrac\\kappa4\\epsilon^{IJ}{}_{KL}J_5^Ke^L,\n\\end{equation}\nwith $\\omega=\\mathring\\omega(e)+C$.\nThe solution is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e$.\nIts sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.","tion is unique on the nondegenerate coframe sector. It fixes\n\\begin{equation}\n \\Gamma_a{}^i=\\mathring\\Gamma_a{}^i(E)\n                     -\\tfrac\\kappa4J_5^0e_a{}^i.\n\\end{equation}\nOne may retain torsion in $K$ and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is\n$-\\tfrac{3\\kappa}{16}\\int J_{5I}J_5^I\\operatorname{vol}_e$.\nIts sign belongs to that convention; changing the Clifford, orientation, or Dirac-action conventions requires translating it.","The resulting real canonical superphase space has coordinates\n\\begin{equation}\n (K_a{}^i,\\widetilde E^a{}_i),\\qquad\n (A_a,\\mathcal E^a),\\qquad (\\Psi,\\Psi^\\dagger),\n\\end{equation}\nand one-form\n\\begin{equation}\n \\Theta_{\\mathrm{red}}=\\int_\\Sigma\\mathrm d^3x\\,\n \\left[\\frac1\\kappa\\widetilde E^a{}_i\\delta K_a{}^i\n       +\\mathcal E^a\\delta A_a\n       +\\frac i2(\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\right].\n \\label{c:constraints:reduced-one-form}\n\\end{equation}\nIts even canonical dimension is $24$ per spatial point: eighteen gravitational and six electromagnetic coordinates. The odd primary block has eight components in the stated complexified graded presentation, with the real spinor relation imposed. These counts are local phase-space counts, not a count of globally gauge-quotiented physical degrees of freedom. Rotation, electric, spatial, and Hami","The resulting real canonical superphase space has coordinates\n\\begin{equation}\n (K_a{}^i,\\widetilde E^a{}_i),\\qquad\n (A_a,\\mathcal E^a),\\qquad (\\Psi,\\Psi^\\dagger),\n\\end{equation}\nand one-form\n\\begin{equation}\n \\Theta_{\\mathrm{red}}=\\int_\\Sigma\\mathrm d^3x\\,\n \\left[\\frac1\\kappa\\widetilde E^a{}_i\\delta K_a{}^i\n       +\\mathcal E^a\\delta A_a\n       +\\frac i2(\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\right].\n \\label{c:constraints:reduced-one-form}\n\\end{equation}\nIts even canonical dimension is $24$ per spatial point: eighteen gravitational and six electromagnetic coordinates. The odd primary block has eight components in the stated complexified graded presentation, with the real spinor relation imposed. These counts are local phase-space counts, not a count of globally gauge-quotiented physical degrees of freedom. Rotation, electric, spatial, and Hamiltonian constraints are still first class.","^a{}_i),\\qquad\n (A_a,\\mathcal E^a),\\qquad (\\Psi,\\Psi^\\dagger),\n\\end{equation}\nand one-form\n\\begin{equation}\n \\Theta_{\\mathrm{red}}=\\int_\\Sigma\\mathrm d^3x\\,\n \\left[\\frac1\\kappa\\widetilde E^a{}_i\\delta K_a{}^i\n       +\\mathcal E^a\\delta A_a\n       +\\frac i2(\\Psi^\\dagger\\delta\\Psi-\\delta\\Psi^\\dagger\\Psi)\\right].\n \\label{c:constraints:reduced-one-form}\n\\end{equation}\nIts even canonical dimension is $24$ per spatial point: eighteen gravitational and six electromagnetic coordinates. The odd primary block has eight components in the stated complexified graded presentation, with the real spinor relation imposed. These counts are local phase-space counts, not a count of globally gauge-quotiented physical degrees of freedom. Rotation, electric, spatial, and Hamiltonian constraints are still first class.","For example, the residual electric constraint is\n\\begin{equation}\n \\mathcal G_{\\mathrm U}^{\\mathrm{red}}=\\partial_a\\mathcal E^a+q\\Psi^\\dagger\\Psi,\n \\qquad Q[u]=-\\int_\\Sigma u\\mathcal G_{\\mathrm U}^{\\mathrm{red}}\\,\\mathrm d^3x.\n\\end{equation}\nIt generates $\\delta_uA_a=\\partial_au$ and $\\delta_u\\Psi=iqu\\Psi$. The half-density Lie derivative in the spatial generator is\n\\begin{equation}\n \\mathcal L_\\eta^{(1/2)}\\Psi\n       =\\eta^a\\partial_a\\Psi+\\tfrac12(\\partial_a\\eta^a)\\Psi.\n\\end{equation}\nThis term is forced by the density weight. Treating $\\Psi$ as a scalar would give a different moment map.","i,\n \\qquad Q[u]=-\\int_\\Sigma u\\mathcal G_{\\mathrm U}^{\\mathrm{red}}\\,\\mathrm d^3x.\n\\end{equation}\nIt generates $\\delta_uA_a=\\partial_au$ and $\\delta_u\\Psi=iqu\\Psi$. The half-density Lie derivative in the spatial generator is\n\\begin{equation}\n \\mathcal L_\\eta^{(1/2)}\\Psi\n       =\\eta^a\\partial_a\\Psi+\\tfrac12(\\partial_a\\eta^a)\\Psi.\n\\end{equation}\nThis term is forced by the density weight. Treating $\\Psi$ as a scalar would give a different moment map.","\\subsection{Descent and the boundary of the chart}\nLet $\\iota$ denote the local solution map for the second-class relations and let $\\{\\cdot,\\cdot\\}_{\\mathrm D}$ be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then\n\\begin{equation}\n \\Omega_{\\mathrm{red}}=\\iota^*\\Omega_\\Sigma,\n \\qquad\n \\{\\Omega_{\\mathrm{red}},\\Omega_{\\mathrm{red}}\\}_{\\mathrm D}=0.\n\\end{equation}\nThe implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missing full boundary charge. Fixing time gauge also removes boost gauge directions, so the tangent action must","\\subsection{Descent and the boundary of the chart}\nLet $\\iota$ denote the local solution map for the second-class relations and let $\\{\\cdot,\\cdot\\}_{\\mathrm D}$ be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then\n\\begin{equation}\n \\Omega_{\\mathrm{red}}=\\iota^*\\Omega_\\Sigma,\n \\qquad\n \\{\\Omega_{\\mathrm{red}},\\Omega_{\\mathrm{red}}\\}_{\\mathrm D}=0.\n\\end{equation}\nThe implication follows from compatibility of the reduced bracket with the Dirac bracket and the vanishing fine master equation. It does not construct the missing full boundary charge. Fixing time gauge also removes boost gauge directions, so the tangent action must be the residual action or a correctly compensated one.","Degenerate triads, changes of orientation, and singular gauge patches are outside this chart. At $e=0$ the primary fermionic matrix collapses and the coordinate $e^{1/2}\\psi$ ceases to be an invertible change of variables. This is a stratified boundary of the reduction, not a small numerical error. A global reduction would need compatible charts and a treatment of these singular strata.","There is also a spectral boundary. Even when $e_a{}^i$ are trigonometric polynomials, $e^{-1}$, $e^{1/2}$, and the torsion-free connection are generally not finite Fourier polynomials. The unreduced degree rules $16K<N$ and $32K<N$ therefore cannot simply be reapplied to the reduced chart. One may evaluate its finite-dimensional regular functions directly, or retain auxiliary polynomial variables, but either choice needs its own approximation and support analysis.","\\subsection{Endpoint Dirac projection}\nConsider a finite-dimensional real canonical chart $z\\in\\mathbb R^{2n}$ with constant skew matrix $J$. Let $\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})$ collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have r","\\subsection{Endpoint Dirac projection}\nConsider a finite-dimensional real canonical chart $z\\in\\mathbb R^{2n}$ with constant skew matrix $J$. Let $\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})$ collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$.","\\subsection{Endpoint Dirac projection}\nConsider a finite-dimensional real canonical chart $z\\in\\mathbb R^{2n}$ with constant skew matrix $J$. Let $\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})$ collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\D","rojection}\nConsider a finite-dimensional real canonical chart $z\\in\\mathbb R^{2n}$ with constant skew matrix $J$. Let $\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})$ collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta…","hart $z\\in\\mathbb R^{2n}$ with constant skew matrix $J$. Let $\\Phi=(\\Phi_1,\\ldots,\\Phi_{2m})$ collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, tra…","econd-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need …","smooth autonomous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary …","ous function $F$, define the segment discrete gradient\n\\begin{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preser…","{equation}\n \\overline\\nabla F(z_0,z_1)=\\int_0^1\\nabla F((1-s)z_0+sz_1)\\,\\mathrm ds.\n \\label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preservation argument.","label{c:constraints:discrete-gradient}\n\\end{equation}\nThe ordinary fundamental theorem of calculus gives\n\\begin{equation}\n F(z_1)-F(z_0)=\\overline\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preservation argument.","e\\nabla F(z_0,z_1)^T(z_1-z_0).\n \\label{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preservation argument.","bel{c:constraints:discrete-chain-rule}\n\\end{equation}\nLet $G$ have rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preservation argument.","ave rows $\\overline\\nabla\\Phi_j^T$. The endpoint constraint matrix is $\\Delta_\\Delta=GJG^T$. Require this matrix to be invertible and define\n\\begin{equation}\n J_{\\mathrm D}^{\\Delta}\n       =J-JG^T\\Delta_\\Delta^{-1}GJ.\n \\label{c:constraints:endpoint-tensor}\n\\end{equation}\nDirect multiplication proves $GJ_{\\mathrm D}^{\\Delta}=0$. Since $\\Delta_\\Delta$ and its inverse are skew, transposition proves\n$(J_{\\mathrm D}^{\\Delta})^T=-J_{\\mathrm D}^{\\Delta}$.\nThe endpoint tensor is a projection of the canonical direction into the kernel of all discrete constraint gradients. It need not define a global Poisson bracket or satisfy a Jacobi identity away from the step. Those properties are unnecessary for the preservation argument.","\\Needspace{22\\baselineskip}\n\\begin{theorem}[Exact finite-step preservation]\nLet $H$ be a smooth autonomous Hamiltonian. Every exact solution of\n\\begin{equation}\n z_1-z_0=\\Delta t\\,J_{\\mathrm D}^{\\Delta}(z_0,z_1)\n                     \\overline\\nabla H(z_0,z_1)\n \\label{c:constraints:implicit-step}\n\\end{equation}\nwith invertible $\\Delta_\\Delta$ satisfies\n\\begin{equation}\n \\Phi(z_1)=\\Phi(z_0),\\qquad H(z_1)=H(z_0).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe componentwise chain rule gives\n$\\Phi(z_1)-\\Phi(z_0)=G(z_1-z_0)$, which vanishes because $GJ_{\\mathrm D}^{\\Delta}=0$.\nFor the Hamiltonian,\n\\begin{equation}\n H(z_1)-H(z_0)\n =\\Delta t\\,\\overline\\nabla H^T J_{\\mathrm D}^{\\Delta}\\overline\\nabla H=0\n\\end{equation}\nby skewness.\n\\end{proof}","th autonomous Hamiltonian. Every exact solution of\n\\begin{equation}\n z_1-z_0=\\Delta t\\,J_{\\mathrm D}^{\\Delta}(z_0,z_1)\n                     \\overline\\nabla H(z_0,z_1)\n \\label{c:constraints:implicit-step}\n\\end{equation}\nwith invertible $\\Delta_\\Delta$ satisfies\n\\begin{equation}\n \\Phi(z_1)=\\Phi(z_0),\\qquad H(z_1)=H(z_0).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe componentwise chain rule gives\n$\\Phi(z_1)-\\Phi(z_0)=G(z_1-z_0)$, which vanishes because $GJ_{\\mathrm D}^{\\Delta}=0$.\nFor the Hamiltonian,\n\\begin{equation}\n H(z_1)-H(z_0)\n =\\Delta t\\,\\overline\\nabla H^T J_{\\mathrm D}^{\\Delta}\\overline\\nabla H=0\n\\end{equation}\nby skewness.\n\\end{proof}","{equation}\n z_1-z_0=\\Delta t\\,J_{\\mathrm D}^{\\Delta}(z_0,z_1)\n                     \\overline\\nabla H(z_0,z_1)\n \\label{c:constraints:implicit-step}\n\\end{equation}\nwith invertible $\\Delta_\\Delta$ satisfies\n\\begin{equation}\n \\Phi(z_1)=\\Phi(z_0),\\qquad H(z_1)=H(z_0).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe componentwise chain rule gives\n$\\Phi(z_1)-\\Phi(z_0)=G(z_1-z_0)$, which vanishes because $GJ_{\\mathrm D}^{\\Delta}=0$.\nFor the Hamiltonian,\n\\begin{equation}\n H(z_1)-H(z_0)\n =\\Delta t\\,\\overline\\nabla H^T J_{\\mathrm D}^{\\Delta}\\overline\\nabla H=0\n\\end{equation}\nby skewness.\n\\end{proof}","\\overline\\nabla H(z_0,z_1)\n \\label{c:constraints:implicit-step}\n\\end{equation}\nwith invertible $\\Delta_\\Delta$ satisfies\n\\begin{equation}\n \\Phi(z_1)=\\Phi(z_0),\\qquad H(z_1)=H(z_0).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe componentwise chain rule gives\n$\\Phi(z_1)-\\Phi(z_0)=G(z_1-z_0)$, which vanishes because $GJ_{\\mathrm D}^{\\Delta}=0$.\nFor the Hamiltonian,\n\\begin{equation}\n H(z_1)-H(z_0)\n =\\Delta t\\,\\overline\\nabla H^T J_{\\mathrm D}^{\\Delta}\\overline\\nabla H=0\n\\end{equation}\nby skewness.\n\\end{proof}","The theorem preserves the initial values of the constraints. It does not repair inconsistent initial data. For explicitly time-dependent gauges or $H(z,t)$, ordinary energy is not a conserved quantity. One must formulate the desired balance law or enlarge the phase space by a clock and its conjugate momentum before applying an autonomous preservation theorem.","\\subsection{A worked particle constraint}\nIn $\\mathbb R^6$ with $z=(q,p)$ and canonical $J$, set\n\\begin{equation}\n \\Phi_1=q\\cdot q-1,\\qquad \\Phi_2=q\\cdot p.\n\\end{equation}\nAt one point the ordinary Dirac matrix is\n\\begin{equation}\n \\Delta(z)=\\begin{pmatrix}0&2\\|q\\|^2","\\subsection{A worked particle constraint}\nIn $\\mathbb R^6$ with $z=(q,p)$ and canonical $J$, set\n\\begin{equation}\n \\Phi_1=q\\cdot q-1,\\qquad \\Phi_2=q\\cdot p.\n\\end{equation}\nAt one point the ordinary Dirac matrix is\n\\begin{equation}\n \\Delta(z)=\\begin{pmatrix}0&2\\|q\\|^2\\\\-2\\|q\\|^2&0\\end{pmatrix}.\n\\end{equation}\nIt is invertible whenever $q\\ne0$. On the intended constraint surface, $q$ lies on the unit sphere and $p$ is tangent to that sphere.\nWriting $\\bar q=(q_0+q_1)/2$ and $\\bar p=(p_0+p_1)/2$, the exact discrete gradients are\n\\begin{equation}\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n\\end{equation}\nConsequently $\\Delta_\\Delta$ has the same displayed form with $q$ replaced by $\\bar q$. Endpoints on the unit sphere can have $\\bar q=0$ if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an arbitrary large step. Small local steps avoid this singular pair under the regularity hypotheses below.","-2\\|q\\|^2&0\\end{pmatrix}.\n\\end{equation}\nIt is invertible whenever $q\\ne0$. On the intended constraint surface, $q$ lies on the unit sphere and $p$ is tangent to that sphere.\nWriting $\\bar q=(q_0+q_1)/2$ and $\\bar p=(p_0+p_1)/2$, the exact discrete gradients are\n\\begin{equation}\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n\\end{equation}\nConsequently $\\Delta_\\Delta$ has the same displayed form with $q$ replaced by $\\bar q$. Endpoints on the unit sphere can have $\\bar q=0$ if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an arbitrary large step. Small local steps avoid this singular pair under the regularity hypotheses below.","It is invertible whenever $q\\ne0$. On the intended constraint surface, $q$ lies on the unit sphere and $p$ is tangent to that sphere.\nWriting $\\bar q=(q_0+q_1)/2$ and $\\bar p=(p_0+p_1)/2$, the exact discrete gradients are\n\\begin{equation}\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n\\end{equation}\nConsequently $\\Delta_\\Delta$ has the same displayed form with $q$ replaced by $\\bar q$. Endpoints on the unit sphere can have $\\bar q=0$ if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an arbitrary large step. Small local steps avoid this singular pair under the regularity hypotheses below.","e whenever $q\\ne0$. On the intended constraint surface, $q$ lies on the unit sphere and $p$ is tangent to that sphere.\nWriting $\\bar q=(q_0+q_1)/2$ and $\\bar p=(p_0+p_1)/2$, the exact discrete gradients are\n\\begin{equation}\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n\\end{equation}\nConsequently $\\Delta_\\Delta$ has the same displayed form with $q$ replaced by $\\bar q$. Endpoints on the unit sphere can have $\\bar q=0$ if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an arbitrary large step. Small local steps avoid this singular pair under the regularity hypotheses below.","urface, $q$ lies on the unit sphere and $p$ is tangent to that sphere.\nWriting $\\bar q=(q_0+q_1)/2$ and $\\bar p=(p_0+p_1)/2$, the exact discrete gradients are\n\\begin{equation}\n \\overline\\nabla\\Phi_1=(2\\bar q,0),\\qquad\n \\overline\\nabla\\Phi_2=(\\bar p,\\bar q).\n\\end{equation}\nConsequently $\\Delta_\\Delta$ has the same displayed form with $q$ replaced by $\\bar q$. Endpoints on the unit sphere can have $\\bar q=0$ if they are antipodal. Endpoint admissibility therefore does not imply endpoint-matrix invertibility for an arbitrary large step. Small local steps avoid this singular pair under the regularity hypotheses below.","For $H(q,p)=\\tfrac12\\|p\\|^2+V(q)$, the segment formula gives an exact discrete gradient of the potential. Polynomial $V$ makes the segment integral a finite algebraic expression. The resulting implicit equations preserve both sphere constraints and $H$ when solved exactly. This is a genuine finite constrained mechanical model. It is a diagnostic for the step mechanism, not an Einstein--Maxwell--Dirac simulation.","\\subsection{Solver residuals have a precise effect}\nFor an approximate endpoint, define its equation residual by\n\\begin{equation}\n r=(z_1-z_0)-\\Delta t\\,J_{\\mathrm D}^{\\Delta}\\overline\\nabla H.\n\\end{equation}\nIf the discrete gradients and tensor are evaluated exactly, the same proof gives\n\\begin{equation}\n \\Phi(z_1)-\\Phi(z_0)=Gr,\\qquad\n H(z_1)-H(z_0)=\\overline\\nabla H^Tr.\n \\label{c:constraints:residual-bound}\n\\end{equation}\nThus $\\|\\Phi(z_1)-\\Phi(z_0)\\|\\le\\|G\\|\\,\\|r\\|$ and\n$|H(z_1)-H(z_0)|\\le\\|\\overline\\nabla H\\|\\,\\|r\\|$.\nThese estimates explain why an incompletely solved step need not preserve the invariants. Numerical quadrature of nonpolynomial segment integrals and floating-point linear algebra contribute additional errors and must also be measured. A printed solver tolerance is not the same quantity as the actual equation residual.","a precise effect}\nFor an approximate endpoint, define its equation residual by\n\\begin{equation}\n r=(z_1-z_0)-\\Delta t\\,J_{\\mathrm D}^{\\Delta}\\overline\\nabla H.\n\\end{equation}\nIf the discrete gradients and tensor are evaluated exactly, the same proof gives\n\\begin{equation}\n \\Phi(z_1)-\\Phi(z_0)=Gr,\\qquad\n H(z_1)-H(z_0)=\\overline\\nabla H^Tr.\n \\label{c:constraints:residual-bound}\n\\end{equation}\nThus $\\|\\Phi(z_1)-\\Phi(z_0)\\|\\le\\|G\\|\\,\\|r\\|$ and\n$|H(z_1)-H(z_0)|\\le\\|\\overline\\nabla H\\|\\,\\|r\\|$.\nThese estimates explain why an incompletely solved step need not preserve the invariants. Numerical quadrature of nonpolynomial segment integrals and floating-point linear algebra contribute additional errors and must also be measured. A printed solver tolerance is not the same quantity as the actual equation residual.",", define its equation residual by\n\\begin{equation}\n r=(z_1-z_0)-\\Delta t\\,J_{\\mathrm D}^{\\Delta}\\overline\\nabla H.\n\\end{equation}\nIf the discrete gradients and tensor are evaluated exactly, the same proof gives\n\\begin{equation}\n \\Phi(z_1)-\\Phi(z_0)=Gr,\\qquad\n H(z_1)-H(z_0)=\\overline\\nabla H^Tr.\n \\label{c:constraints:residual-bound}\n\\end{equation}\nThus $\\|\\Phi(z_1)-\\Phi(z_0)\\|\\le\\|G\\|\\,\\|r\\|$ and\n$|H(z_1)-H(z_0)|\\le\\|\\overline\\nabla H\\|\\,\\|r\\|$.\nThese estimates explain why an incompletely solved step need not preserve the invariants. Numerical quadrature of nonpolynomial segment integrals and floating-point linear algebra contribute additional errors and must also be measured. A printed solver tolerance is not the same quantity as the actual equation residual.","The particle certificate records $1000$ steps at $\\Delta t=0.1$, with energy error about $7.8\\times10^{-16}$, constraint error about $3.9\\times10^{-15}$, and equation residual about $1.1\\times10^{-16}$. A Runge--Kutta negative control on the continuous Dirac vector field has errors around $10^{-6}$. Those values concern that fixed test. They do not establish exact floating-point preservation or a uniform gravity stability theorem. The accompanying atlas reruns a declared finite particle example and also tests incomplete implicit solves.","\\subsection{Local existence, boundedness, and refinement}\nAt $\\Delta t=0$ and $z_1=z_0$, the endpoint tensor reduces to the ordinary Dirac tensor if the point constraint matrix is invertible. The derivative with respect to $z_1$ of the left side of~\\eqref{c:constraints:implicit-step} minus its right side is then the identity. If the segment data are smooth in a neighborhood and the inverse matrix stays bounded, the implicit-function theorem gives a unique nearby step for sufficiently small $\\Delta t$. It does not prove existence of a unique endpoint at arbitrary step size or after leaving that neighborhood.","point tensor reduces to the ordinary Dirac tensor if the point constraint matrix is invertible. The derivative with respect to $z_1$ of the left side of~\\eqref{c:constraints:implicit-step} minus its right side is then the identity. If the segment data are smooth in a neighborhood and the inverse matrix stays bounded, the implicit-function theorem gives a unique nearby step for sufficiently small $\\Delta t$. It does not prove existence of a unique endpoint at arbitrary step size or after leaving that neighborhood.","If the gauge-fixed autonomous Hamiltonian satisfies\n\\begin{equation}\n H(z)\\ge a\\|z\\|^p-b,\\qquad a>0,\\quad p>0,\n\\end{equation}\nexact preservation yields\n\\begin{equation}\n \\|z_n\\|\\le\\left(\\frac{H(z_0)+b}{a}\\right)^{1/p}\n\\end{equation}\nfor every continued step in the regular patch. This coercivity is a separate assumption. It is not automatic for a gravitational constraint or a Hamiltonian that vanishes on its constraint surface. Bounded coordinates do not ensure that the gauge matrix stays nonsingular or that the triad determinant stays away from zero.","For an embedding $I_{K\\to L}$ and a level-$L$ step $U_L$, the adaptive map is $U_L\\circ I_{K\\to L}$. The level-$L$ step preserves the invariants of its embedded initial state. To preserve the intended old data, the embedding must also satisfy the appropriate pullback relations for the Hamiltonian, constraints, gauge conditions, and symplectic form. A generic zero padding or a generic change of gauge does not prove all those relations.","\\subsection{A nonlinear Gaussian reduction in closed form}\nThe small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let $m,n>0$ and\n\\begin{equation}\n S_F(x_1,x_2,y,z)=\\tfrac12(x_2-x_1y)^2\n                         +\\tfrac m2(y-z)^2+\\tfrac n2z^2.\n\\end{equation}\nThe fine gauge vector $R_F=\\partial_{x_1}+y\\partial_{x_2}$ leaves $S_F$ invariant and has zero ordinary divergence. The detail Hessian is\n\\begin{equation}\n M_D=\\begin{pmatrix}x_1^2+m&-m","n{A nonlinear Gaussian reduction in closed form}\nThe small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let $m,n>0$ and\n\\begin{equation}\n S_F(x_1,x_2,y,z)=\\tfrac12(x_2-x_1y)^2\n                         +\\tfrac m2(y-z)^2+\\tfrac n2z^2.\n\\end{equation}\nThe fine gauge vector $R_F=\\partial_{x_1}+y\\partial_{x_2}$ leaves $S_F$ invariant and has zero ordinary divergence. The detail Hessian is\n\\begin{equation}\n M_D=\\begin{pmatrix}x_1^2+m&-m","nsfer chapter explains why an effective action and an effective measure must be transported together. Let $m,n>0$ and\n\\begin{equation}\n S_F(x_1,x_2,y,z)=\\tfrac12(x_2-x_1y)^2\n                         +\\tfrac m2(y-z)^2+\\tfrac n2z^2.\n\\end{equation}\nThe fine gauge vector $R_F=\\partial_{x_1}+y\\partial_{x_2}$ leaves $S_F$ invariant and has zero ordinary divergence. The detail Hessian is\n\\begin{equation}\n M_D=\\begin{pmatrix}x_1^2+m&-m\\\\-m&m+n\\end{pmatrix},\n \\qquad D=(m+n)x_1^2+mn>0.\n\\end{equation}\nWriting $\\mu=mn/(m+n)$ and $d=x_1^2+\\mu$, the stationary detail values are\n$y_*=x_1x_2/d$ and $z_*=my_* /(m+n)$. Substitution gives\n\\begin{equation}\n S_C=\\frac{\\mu x_2^2}{2d},\\qquad\n R_C=\\partial_{x_1}+\\frac{x_1x_2}{d}\\partial_{x_2}.\n\\end{equation}\nDifferentiation proves $R_CS_C=0$. Gaussian integration, rather than stationary substitution alone, also gives the density $\\rho_C\\propto D^{-1/2}$, up to a constant. It satisfies\n\\begin{equation}\n \\operatorname{div}R_C=\\frac{x_1}{d},\\qquad\n R_C\\log\\rho_C=-\\frac{x_1}{d}.\n\\end{equation}\nTheir sum vanishes. The effective measure therefore restores the div…","-m&m+n\\end{pmatrix},\n \\qquad D=(m+n)x_1^2+mn>0.\n\\end{equation}\nWriting $\\mu=mn/(m+n)$ and $d=x_1^2+\\mu$, the stationary detail values are\n$y_*=x_1x_2/d$ and $z_*=my_* /(m+n)$. Substitution gives\n\\begin{equation}\n S_C=\\frac{\\mu x_2^2}{2d},\\qquad\n R_C=\\partial_{x_1}+\\frac{x_1x_2}{d}\\partial_{x_2}.\n\\end{equation}\nDifferentiation proves $R_CS_C=0$. Gaussian integration, rather than stationary substitution alone, also gives the density $\\rho_C\\propto D^{-1/2}$, up to a constant. It satisfies\n\\begin{equation}\n \\operatorname{div}R_C=\\frac{x_1}{d},\\qquad\n R_C\\log\\rho_C=-\\frac{x_1}{d}.\n\\end{equation}\nTheir sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gauge transformation.","x_1^2+mn>0.\n\\end{equation}\nWriting $\\mu=mn/(m+n)$ and $d=x_1^2+\\mu$, the stationary detail values are\n$y_*=x_1x_2/d$ and $z_*=my_* /(m+n)$. Substitution gives\n\\begin{equation}\n S_C=\\frac{\\mu x_2^2}{2d},\\qquad\n R_C=\\partial_{x_1}+\\frac{x_1x_2}{d}\\partial_{x_2}.\n\\end{equation}\nDifferentiation proves $R_CS_C=0$. Gaussian integration, rather than stationary substitution alone, also gives the density $\\rho_C\\propto D^{-1/2}$, up to a constant. It satisfies\n\\begin{equation}\n \\operatorname{div}R_C=\\frac{x_1}{d},\\qquad\n R_C\\log\\rho_C=-\\frac{x_1}{d}.\n\\end{equation}\nTheir sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gauge transformation.","stationary detail values are\n$y_*=x_1x_2/d$ and $z_*=my_* /(m+n)$. Substitution gives\n\\begin{equation}\n S_C=\\frac{\\mu x_2^2}{2d},\\qquad\n R_C=\\partial_{x_1}+\\frac{x_1x_2}{d}\\partial_{x_2}.\n\\end{equation}\nDifferentiation proves $R_CS_C=0$. Gaussian integration, rather than stationary substitution alone, also gives the density $\\rho_C\\propto D^{-1/2}$, up to a constant. It satisfies\n\\begin{equation}\n \\operatorname{div}R_C=\\frac{x_1}{d},\\qquad\n R_C\\log\\rho_C=-\\frac{x_1}{d}.\n\\end{equation}\nTheir sum vanishes. The effective measure therefore restores the divergence balance of the nonlinear induced gauge transformation.","Eliminating $z$ first gives the effective coefficient $\\mu$ for $y^2$ and the factor $(m+n)^{-1/2}$. Eliminating $y$ next gives $d^{-1/2}$. Their product is $D^{-1/2}$, exactly as in direct elimination. For the Euclidean weight $e^{-S/\\hbar}$, using a flat coarse measure instead records this determinant as\n$S_{C,\\hbar}=S_C+\\tfrac\\hbar2\\log D$, up to an additive constant. A nonlinear symmetry that preserves the classical action can still require a nonconstant induced measure.","\\subsection{The conditional BV pushforward statement}\nSuppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies\n\\begin{equation}\n \\tfrac12(S_F,S_F)-i\\hbar\\Delta_FS_F=0.\n \\label{c:constraints:qme}\n\\end{equation}\nSuppose the variables split into retained and detail sectors with\n$\\Delta_F=\\Delta_C+\\Delta_D$, and choose an admissible detail Lagrangian $\\mathcal L_D$. Define the coarse weight by\n\\begin{equation}\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n\\end{equation}\nWhen this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution,\n\\begin{equation}\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n\\end{equation}\nThis is preservation of the quantum master equation under compatible elimination. It assumes the fine quantum master equation. The classi","\\subsection{The conditional BV pushforward statement}\nSuppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies\n\\begin{equation}\n \\tfrac12(S_F,S_F)-i\\hbar\\Delta_FS_F=0.\n \\label{c:constraints:qme}\n\\end{equation}\nSuppose the variables split into retained and detail sectors with\n$\\Delta_F=\\Delta_C+\\Delta_D$, and choose an admissible detail Lagrangian $\\mathcal L_D$. Define the coarse weight by\n\\begin{equation}\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n\\end{equation}\nWhen this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution,\n\\begin{equation}\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n\\end{equation}\nThis is preservation of the quantum master equation under compatible elimination. It assumes the fine quantum master equation. The classical parent in~\\eqref{c:constraints:bv-action} supplies none of the missing regulator, measure, or anomaly-co","ection{The conditional BV pushforward statement}\nSuppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies\n\\begin{equation}\n \\tfrac12(S_F,S_F)-i\\hbar\\Delta_FS_F=0.\n \\label{c:constraints:qme}\n\\end{equation}\nSuppose the variables split into retained and detail sectors with\n$\\Delta_F=\\Delta_C+\\Delta_D$, and choose an admissible detail Lagrangian $\\mathcal L_D$. Define the coarse weight by\n\\begin{equation}\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n\\end{equation}\nWhen this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution,\n\\begin{equation}\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n\\end{equation}\nThis is preservation of the quantum master equation under compatible elimination. It assumes the fine quantum master equation. The classical parent in~\\eqref{c:constraints:bv-action} supplies none of the missing regulator, measure, or anomaly-control data required by~\\eqref{c:constraints:qme…","regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies\n\\begin{equation}\n \\tfrac12(S_F,S_F)-i\\hbar\\Delta_FS_F=0.\n \\label{c:constraints:qme}\n\\end{equation}\nSuppose the variables split into retained and detail sectors with\n$\\Delta_F=\\Delta_C+\\Delta_D$, and choose an admissible detail Lagrangian $\\mathcal L_D$. Define the coarse weight by\n\\begin{equation}\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n\\end{equation}\nWhen this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution,\n\\begin{equation}\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n\\end{equation}\nThis is preservation of the quantum master equation under compatible elimination. It assumes the fine quantum master equation. The classical parent in~\\eqref{c:constraints:bv-action} supplies none of the missing regulator, measure, or anomaly-control data required by~\\eqref{c:constraints:qme}.","ined and detail sectors with\n$\\Delta_F=\\Delta_C+\\Delta_D$, and choose an admissible detail Lagrangian $\\mathcal L_D$. Define the coarse weight by\n\\begin{equation}\n e^{iS_C/\\hbar}=\\mathcal N^{-1}\\int_{\\mathcal L_D}e^{iS_F/\\hbar}.\n\\end{equation}\nWhen this integral is well defined in the declared analytic or perturbative sense and BV integration by parts has no boundary contribution,\n\\begin{equation}\n \\Delta_C e^{iS_C/\\hbar}\n       =-\\mathcal N^{-1}\\int_{\\mathcal L_D}\\Delta_D e^{iS_F/\\hbar}=0.\n\\end{equation}\nThis is preservation of the quantum master equation under compatible elimination. It assumes the fine quantum master equation. The classical parent in~\\eqref{c:constraints:bv-action} supplies none of the missing regulator, measure, or anomaly-control data required by~\\eqref{c:constraints:qme}.","Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination,\n\\begin{equation}\n L_{\\mathrm{eff}}=L_{CC}-L_{CD}L_{DD}^{-1}L_{DC},\n\\end{equation}\nwith the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local coarse action. Exact detail elimination can create long-range couplings.","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equ","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations wit","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddo","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\var","Take a periodic chain with $N\\geq3$ sites, spacing $h>0$, and indices modulo\n$N$. The continuum metric motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:…","motivating its coefficients is\n\\[\n ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2,\n \\qquad N(t,x)>0,\\quad a(t,x)>0.\n\\]\nThe repeated letter $N$ in this metric is the lapse, whereas the integer\n$N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\n…","N$ counts lattice sites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.","ites. In the finite formulas below only the coefficient\nnames $w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.","$w$, $c$, and $v$ are needed:\n\\[\n w_j=\\frac{a_j}{N_j}>0,\\qquad\n c_{j+1/2}=\\frac{N_{j+1/2}}{a_{j+1/2}}>0,\\qquad\n v_j=N_ja_jm^2\\geq0.\n\\]\nThus $N/a$ is the coordinate characteristic speed of the continuum\nmassless equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.","equation. Assume these finite coefficient arrays are continuously\ndifferentiable in time. For a real field $\\varphi_j(t)$ set\n$\\Delta_j=\\varphi_{j+1}-\\varphi_j$ and define\n\\begin{equation}\n L_h=\\frac h2\\sum_jw_j\\dot\\varphi_j^2\n -\\frac1{2h}\\sum_jc_{j+1/2}\\Delta_j^2\n -\\frac h2\\sum_jv_j\\varphi_j^2.\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.",".\n \\label{c:core:scalar-action}\n\\end{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.","nd{equation}\nVariations with compact time support give\n\\begin{equation}\n w_j\\ddot\\varphi_j+\\dot w_j\\dot\\varphi_j\n =\\frac{c_{j+1/2}\\Delta_j-c_{j-1/2}\\Delta_{j-1}}{h^2}\n -v_j\\varphi_j.\n \\label{c:core:scalar-evolution}\n\\end{equation}\nThe neighboring edge terms in the action produce the two terms in the\nspatial numerator. The time integration by parts differentiates\n$w_j\\dot\\varphi_j$, including the $\\dot w_j\\dot\\varphi_j$ term.","\\Needspace{20\\baselineskip}\n\\begin{proposition}[Local energy, flux, and geometric work]\n\\label{c:core:scalar-work}\nFor every twice differentiable solution of\n\\eqref{c:core:scalar-evolution}, define\n\\begin{align}\n e_j&=\\frac h2w_j\\dot\\varphi_j^2+\\frac h2v_j\\varphi_j^2\n +\\frac{c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{c_{j-1/2}}{4h}\\Delta_{j-1}^2,\\nonumber\\\\\n F_{j+1/2}&=-\\frac{c_{j+1/2}}{2h}\\Delta_j\n             (\\dot\\varphi_j+\\dot\\varphi_{j+1}),\\nonumber\\\\\n s_j&=-\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2\n +\\frac{\\dot c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{\\dot c_{j-1/2}}{4h}\\Delta_{j-1}^2.\n \\label{c:core:scalar-densities}\n\\end{align}\nThen\n\\begin{equation}\n \\dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.\n \\label{c:core:scalar-balance}\n\\end{equation}\nIn particular, $E_h=\\sum_je_j$ satisfies\n$\\dot E_h=\\sum_js_j=-\\partial_tL_h$, where the partial derivative holds\n$\\varphi$ and $\\dot\\varphi$ fixed.\n\\end{proposition}","s_j&=-\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2\n +\\frac{\\dot c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{\\dot c_{j-1/2}}{4h}\\Delta_{j-1}^2.\n \\label{c:core:scalar-densities}\n\\end{align}\nThen\n\\begin{equation}\n \\dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.\n \\label{c:core:scalar-balance}\n\\end{equation}\nIn particular, $E_h=\\sum_je_j$ satisfies\n$\\dot E_h=\\sum_js_j=-\\partial_tL_h$, where the partial derivative holds\n$\\varphi$ and $\\dot\\varphi$ fixed.\n\\end{proposition}","h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2\n +\\frac{\\dot c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{\\dot c_{j-1/2}}{4h}\\Delta_{j-1}^2.\n \\label{c:core:scalar-densities}\n\\end{align}\nThen\n\\begin{equation}\n \\dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.\n \\label{c:core:scalar-balance}\n\\end{equation}\nIn particular, $E_h=\\sum_je_j$ satisfies\n$\\dot E_h=\\sum_js_j=-\\partial_tL_h$, where the partial derivative holds\n$\\varphi$ and $\\dot\\varphi$ fixed.\n\\end{proposition}","t\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2\n +\\frac{\\dot c_{j+1/2}}{4h}\\Delta_j^2\n +\\frac{\\dot c_{j-1/2}}{4h}\\Delta_{j-1}^2.\n \\label{c:core:scalar-densities}\n\\end{align}\nThen\n\\begin{equation}\n \\dot e_j+F_{j+1/2}-F_{j-1/2}=s_j.\n \\label{c:core:scalar-balance}\n\\end{equation}\nIn particular, $E_h=\\sum_je_j$ satisfies\n$\\dot E_h=\\sum_js_j=-\\partial_tL_h$, where the partial derivative holds\n$\\varphi$ and $\\dot\\varphi$ fixed.\n\\end{proposition}","\\begin{proof}\nPut $B_j=c_{j+1/2}\\Delta_j$. Using the equation of motion, the derivative\nof the kinetic and mass terms at site $j$ is\n\\[\n \\frac{\\dot\\varphi_j}{h}(B_j-B_{j-1})\n -\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2.\n\\]\nThe right edge contribution differentiates to\n\\[\n \\frac{\\dot c_{j+1/2}\\Delta_j^2}{4h}\n +\\frac{B_j}{2h}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweigh","\\begin{proof}\nPut $B_j=c_{j+1/2}\\Delta_j$. Using the equation of motion, the derivative\nof the kinetic and mass terms at site $j$ is\n\\[\n \\frac{\\dot\\varphi_j}{h}(B_j-B_{j-1})\n -\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2.\n\\]\nThe right edge contribution differentiates to\n\\[\n \\frac{\\dot c_{j+1/2}\\Delta_j^2}{4h}\n +\\frac{B_j}{2h}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweight $1/(4h)$, giving $-\\partial_tL_h$.\n\\end{proof}","tion of motion, the derivative\nof the kinetic and mass terms at site $j$ is\n\\[\n \\frac{\\dot\\varphi_j}{h}(B_j-B_{j-1})\n -\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2.\n\\]\nThe right edge contribution differentiates to\n\\[\n \\frac{\\dot c_{j+1/2}\\Delta_j^2}{4h}\n +\\frac{B_j}{2h}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweight $1/(4h)$, giving $-\\partial_tL_h$.\n\\end{proof}","B_j-B_{j-1})\n -\\frac h2\\dot w_j\\dot\\varphi_j^2\n +\\frac h2\\dot v_j\\varphi_j^2.\n\\]\nThe right edge contribution differentiates to\n\\[\n \\frac{\\dot c_{j+1/2}\\Delta_j^2}{4h}\n +\\frac{B_j}{2h}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweight $1/(4h)$, giving $-\\partial_tL_h$.\n\\end{proof}","}(\\dot\\varphi_{j+1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweight $1/(4h)$, giving $-\\partial_tL_h$.\n\\end{proof}","1}-\\dot\\varphi_j),\n\\]\nand the left edge has the analogous expression. Adding these terms\nleaves\n$B_j(\\dot\\varphi_j+\\dot\\varphi_{j+1})/(2h)$ on the right and its\nnegative counterpart on the left, plus $s_j$. These are exactly the flux\nterms in \\eqref{c:core:scalar-balance}. Periodic summation cancels every\ninterface flux. Each edge coefficient derivative appears twice with\nweight $1/(4h)$, giving $-\\partial_tL_h$.\n\\end{proof}","Every $e_j$ is nonnegative under the stated coefficient assumptions. An\ninterval of cells $r,\\ldots,s$ obeys the exact coarse balance\n\\[\n \\frac{d}{dt}\\sum_{j=r}^se_j\n +F_{s+1/2}-F_{r-1/2}=\\sum_{j=r}^ss_j.\n\\]\nThus coarse energy is obtained by adding fine cell energies and retaining\nonly the exposed boundary fluxes. This is the same cancellation mechanism\nas projector transfer, applied here to a classical field energy. Its\nphysical units and constitutive coefficients come from the action.","\\begin{example}[Time dependent expansion]\nLet the lapse equal one, let $a(t)>0$ be spatially uniform, and let $m=0$.\nThen $w=a$, $c=1/a$, and\n\\[\n \\dot E_h=-\\frac h2\\dot a\\sum_j\\dot\\varphi_j^2\n -\\frac{\\dot a}{2ha^2}\\sum_j\\Delta_j^2.\n\\]\nAn increasing $a$ transfers energy out of this coordinate-time field\nHamiltonian. The supplied background receives no evolution equation\nfrom this calculation. A self-consistent matter--geometry model must\nalso evolve its gravitational variables.\n\\end{example}","For a smooth field sampled at sites and a smooth coefficient sampled at\nedge midpoints, the spatial expression is second-order consistent with\n$\\partial_x(c\\partial_x\\varphi)$. Indeed, expanding at $x_j$ gives\n\\[\n c(x_j+h/2)[\\varphi(x_j+h)-\\varphi(x_j)]\n -c(x_j-h/2)[\\varphi(x_j)-\\varphi(x_j-h)]\n =h^2(c\\varphi_x)_x(x_j)+O(h^4).\n\\]\nUniform bounds on the requisite fourth derivatives make the remainder\nuniform on a compact spatial region. The exact finite balance holds\nwithout any Taylor approximation.","Consider a twice continuously differentiable real function $\\phi$ on an\noriented $(t,x)$ rectangle, and the coframe\n\\[\n e^0=e^\\phi dt,\\qquad e^1=e^\\phi dx,\n \\qquad ds^2=e^{2\\phi}(-dt^2+dx^2).\n\\]\nWrite\n\\[\n K=\\begin{pmatrix}0&1","Consider a twice continuously differentiable real function $\\phi$ on an\noriented $(t,x)$ rectangle, and the coframe\n\\[\n e^0=e^\\phi dt,\\qquad e^1=e^\\phi dx,\n \\qquad ds^2=e^{2\\phi}(-dt^2+dx^2).\n\\]\nWrite\n\\[\n K=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix},\\qquad\n \\omega=(\\phi_xdt+\\phi_tdx)K.\n\\]\nThe coframe satisfies the torsion-free Cartan equations with this\nconnection. Since every connection value is a multiple of the same\nmatrix, $\\omega\\wedge\\omega=0$. Therefore\n\\begin{equation}\n \\Omega=d\\omega=(\\phi_{tt}-\\phi_{xx})K\\,dt\\wedge dx,\n \\qquad R=2e^{-2\\phi}(\\phi_{tt}-\\phi_{xx}).\n \\label{c:core:conformal-curvature}\n\\end{equation}\nThis fixes the scalar-curvature sign convention used in this monograph.","1&0\\end{pmatrix},\\qquad\n \\omega=(\\phi_xdt+\\phi_tdx)K.\n\\]\nThe coframe satisfies the torsion-free Cartan equations with this\nconnection. Since every connection value is a multiple of the same\nmatrix, $\\omega\\wedge\\omega=0$. Therefore\n\\begin{equation}\n \\Omega=d\\omega=(\\phi_{tt}-\\phi_{xx})K\\,dt\\wedge dx,\n \\qquad R=2e^{-2\\phi}(\\phi_{tt}-\\phi_{xx}).\n \\label{c:core:conformal-curvature}\n\\end{equation}\nThis fixes the scalar-curvature sign convention used in this monograph.","We use the archive's plus-sign edge convention: for each oriented\nboundary edge define $U_e=\\exp(+\\int_e\\omega)$. This specifies the\nsigned transport observable used below. Multiplying in boundary order\nyields an exact identity\n\\begin{equation}\n W_\\square\n =\\exp\\left(K\\oint_{\\partial\\square}(\\phi_xdt+\\phi_tdx)\\right)\n =\\exp\\left(\\frac K2\\int_\\square e^{2\\phi}R\\,dt\\,dx\\right).\n \\label{c:core:conformal-holonomy}\n\\end{equation}\nTo see the second equality directly, integrate $\\phi_{tt}$ along the\ntwo horizontal endpoints and $\\phi_{xx}$ along the two vertical\nendpoints, with the boundary orientation from $dt\\wedge dx$. This is\nthe rectangular fundamental theorem of calculus. No path-order\ncorrection survives because the edge exponents commute.","ge define $U_e=\\exp(+\\int_e\\omega)$. This specifies the\nsigned transport observable used below. Multiplying in boundary order\nyields an exact identity\n\\begin{equation}\n W_\\square\n =\\exp\\left(K\\oint_{\\partial\\square}(\\phi_xdt+\\phi_tdx)\\right)\n =\\exp\\left(\\frac K2\\int_\\square e^{2\\phi}R\\,dt\\,dx\\right).\n \\label{c:core:conformal-holonomy}\n\\end{equation}\nTo see the second equality directly, integrate $\\phi_{tt}$ along the\ntwo horizontal endpoints and $\\phi_{xx}$ along the two vertical\nendpoints, with the boundary orientation from $dt\\wedge dx$. This is\nthe rectangular fundamental theorem of calculus. No path-order\ncorrection survives because the edge exponents commute.","the\nsigned transport observable used below. Multiplying in boundary order\nyields an exact identity\n\\begin{equation}\n W_\\square\n =\\exp\\left(K\\oint_{\\partial\\square}(\\phi_xdt+\\phi_tdx)\\right)\n =\\exp\\left(\\frac K2\\int_\\square e^{2\\phi}R\\,dt\\,dx\\right).\n \\label{c:core:conformal-holonomy}\n\\end{equation}\nTo see the second equality directly, integrate $\\phi_{tt}$ along the\ntwo horizontal endpoints and $\\phi_{xx}$ along the two vertical\nendpoints, with the boundary orientation from $dt\\wedge dx$. This is\nthe rectangular fundamental theorem of calculus. No path-order\ncorrection survives because the edge exponents commute.","an exact identity\n\\begin{equation}\n W_\\square\n =\\exp\\left(K\\oint_{\\partial\\square}(\\phi_xdt+\\phi_tdx)\\right)\n =\\exp\\left(\\frac K2\\int_\\square e^{2\\phi}R\\,dt\\,dx\\right).\n \\label{c:core:conformal-holonomy}\n\\end{equation}\nTo see the second equality directly, integrate $\\phi_{tt}$ along the\ntwo horizontal endpoints and $\\phi_{xx}$ along the two vertical\nendpoints, with the boundary orientation from $dt\\wedge dx$. This is\nthe rectangular fundamental theorem of calculus. No path-order\ncorrection survives because the edge exponents commute.","If the integrated coefficient is $\\chi$, then\n\\[\n e^{\\chi K}=\\cosh\\chi\\,I+\\sinh\\chi\\,K.\n\\]\nThis preserves the Lorentz form $\\operatorname{diag}(-1,1)$ and has\npositive eigenvalues $e^{\\chi}$ and $e^{-\\chi}$. Its real principal\nmatrix logarithm is $\\chi K$. These are Lorentz boosts. The positive\nHilbert norm used for unitary color transport in the gauge monograph\nis a different structure.","\\begin{example}[Constant curvature density]\nFor $\\phi(t,x)=b(t^2-x^2)/2$, one has\n\\[\n \\phi_{tt}-\\phi_{xx}=2b,\n \\qquad R=4b\\,e^{-2\\phi},\n \\qquad W_\\square=e^{2b\\,\\operatorname{area}(\\square)K}.\n\\]\nThe curvature density $e^{2\\phi}R/2$ is constant although $R$ varies\nwith position. Adding oriented subrectangle integrals reconstructs\nthe coarse rapidity exactly.\n\\end{example}","The plus-sign observable is assigned the endpoint action\n\\[\n G(t,x)=e^{\\chi(t,x)K},\\qquad\n U_{a\\to b}'=G(b)U_{a\\to b}G(a)^{-1}.\n\\]\nIt corresponds to the signed connection rule\n$\\omega_+'=\\omega_++d\\chi\\,K$. Intermediate endpoint factors cancel,\nso the loop observable is conjugated at its base point. For the\nCartan convention $D=d+\\omega$, a coframe transformation $e\\mapsto Ge$\ninstead gives $\\omega'=\\omega-d\\chi\\,K$ and parallel transport\n$\\exp(-\\int_e\\omega)$. Its closed-loop rapidity is the negative of the\nrapidity of $W_\\square$ defined here. The fixed-frame curvature identity\n\\eqref{c:core:conformal-holonomy} is unchanged. Declaring these signs\navoids mixing the archive's edge observable with a different Cartan\nparallel-transport convention.","point action\n\\[\n G(t,x)=e^{\\chi(t,x)K},\\qquad\n U_{a\\to b}'=G(b)U_{a\\to b}G(a)^{-1}.\n\\]\nIt corresponds to the signed connection rule\n$\\omega_+'=\\omega_++d\\chi\\,K$. Intermediate endpoint factors cancel,\nso the loop observable is conjugated at its base point. For the\nCartan convention $D=d+\\omega$, a coframe transformation $e\\mapsto Ge$\ninstead gives $\\omega'=\\omega-d\\chi\\,K$ and parallel transport\n$\\exp(-\\int_e\\omega)$. Its closed-loop rapidity is the negative of the\nrapidity of $W_\\square$ defined here. The fixed-frame curvature identity\n\\eqref{c:core:conformal-holonomy} is unchanged. Declaring these signs\navoids mixing the archive's edge observable with a different Cartan\nparallel-transport convention.","\\]\nIt corresponds to the signed connection rule\n$\\omega_+'=\\omega_++d\\chi\\,K$. Intermediate endpoint factors cancel,\nso the loop observable is conjugated at its base point. For the\nCartan convention $D=d+\\omega$, a coframe transformation $e\\mapsto Ge$\ninstead gives $\\omega'=\\omega-d\\chi\\,K$ and parallel transport\n$\\exp(-\\int_e\\omega)$. Its closed-loop rapidity is the negative of the\nrapidity of $W_\\square$ defined here. The fixed-frame curvature identity\n\\eqref{c:core:conformal-holonomy} is unchanged. Declaring these signs\navoids mixing the archive's edge observable with a different Cartan\nparallel-transport convention.","Let $G>0$, let $\\rho(r)$ be a continuous spherical density, and impose\nregularity $\\lim_{r\\downarrow0}r^2\\Phi'(r)=0$. The radial Poisson\nequation gives\n\\[\n (r^2\\Phi')'=4\\pi G r^2\\rho,\n \\qquad M_{\\rm enc}(r)=4\\pi\\int_0^rs^2\\rho(s)\\,ds,\n \\qquad \\Phi'(r)=\\frac{GM_{\\rm enc}(r)}{r^2}.\n\\]\nThe acceleration is $a_r=-\\Phi'$. A positive density therefore gives\nan inward radial acceleration.","Putting $u(r)=r\\Phi(r)$ removes the radial first derivative:\n\\[\n r^2\\Phi'=ru'-u,\\qquad\n (r^2\\Phi')'=ru'',\\qquad u''=4\\pi G r\\rho.\n\\]\nFor fixed endpoint values, this equation is the stationary condition of\n\\[\n S[u]=\\int_0^{R_0}\\left[\\frac{(u')^2}{2G}+4\\pi r\\rho u\\right]dr.\n\\]\nOn $r_i=ih$, $i=0,\\ldots,M$, with $Mh=R_0$, its finite counterpart is\n\\begin{equation}\n S_h[u]=\\sum_{i=0}^{M-1}\\frac{(u_{i+1}-u_i)^2}{2Gh}\n +4\\pi h\\sum_{i=1}^{M-1}r_i\\rho_i u_i.\n \\label{c:core:poisson-action}\n\\end{equation}","\\begin{proposition}[Unique finite radial solution]\n\\label{c:core:poisson-minimum}\nWith $u_0,u_M$ fixed, $S_h$ has a unique minimizer. Its interior values\nsatisfy\n\\[\n \\frac{u_{i+1}-2u_i+u_{i-1}}{h^2}=4\\pi G r_i\\rho_i.\n\\]\n\\end{proposition}","\\begin{proof}\nDifferentiation with respect to $u_i$ gives\n$(2u_i-u_{i-1}-u_{i+1})/(Gh)+4\\pi h r_i\\rho_i$. Setting it to zero\ngives the equation. For a variation $v$ with $v_0=v_M=0$, the Hessian\nquadratic form is\n\\[\n \\frac1{Gh}\\sum_{i=0}^{M-1}(v_{i+1}-v_i)^2.\n\\]\nIts vanishing forces $v$ to be constant and hence zero. The finite\nHessian is positive definite, so the quadratic action has one\nstationary point and that point is its unique minimizer.\n\\end{proof}","The boundary value $u_0=0$ allows a regular finite potential at the\norigin. The quotient $u/r$ is used only for $r>0$, with its continuous\nlimit determined separately at $r=0$.","For the normalized Gaussian density\n\\[\n \\rho(r)=\\frac{M_0}{(2\\pi)^{3/2}\\sigma^3}e^{-r^2/(2\\sigma^2)},\n \\qquad \\sigma>0,\n\\]\ndirect differentiation verifies\n\\begin{align*}\n M_{\\rm enc}(r)&=M_0\\left[\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right)\n -\\sqrt{\\frac2\\pi}\\frac r\\sigma e^{-r^2/(2\\sigma^2)}\\right],","For the normalized Gaussian density\n\\[\n \\rho(r)=\\frac{M_0}{(2\\pi)^{3/2}\\sigma^3}e^{-r^2/(2\\sigma^2)},\n \\qquad \\sigma>0,\n\\]\ndirect differentiation verifies\n\\begin{align*}\n M_{\\rm enc}(r)&=M_0\\left[\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right)\n -\\sqrt{\\frac2\\pi}\\frac r\\sigma e^{-r^2/(2\\sigma^2)}\\right],\\\\\n \\Phi(r)&=-\\frac{GM_0}{r}\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right).\n\\end{align*}\nThe potential tends to zero at infinity and has the finite central\nvalue $-\\sqrt{2/\\pi}\\,GM_0/\\sigma$. This removes the point-source\nsingularity while preserving the attractive sign.","\\Phi(r)&=-\\frac{GM_0}{r}\n \\operatorname{erf}\\left(\\frac r{\\sqrt2\\sigma}\\right).\n\\end{align*}\nThe potential tends to zero at infinity and has the finite central\nvalue $-\\sqrt{2/\\pi}\\,GM_0/\\sigma$. This removes the point-source\nsingularity while preserving the attractive sign.","In the stipulated first-order isotropic weak-field metric,\n\\[\n ds^2=-\\left(1+\\frac{2\\Phi}{c_0^2}\\right)c_0^2dt^2\n       +\\left(1-\\frac{2\\Phi}{c_0^2}\\right)d\\mathbf x^2,\n \\qquad \\left|\\frac{\\Phi}{c_0^2}\\right|\\ll1,\n\\]\na static clock has lapse $\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}$.\nThe static-frequency ratio in this approximation is\n\\[\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n\\]\nFor $\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}$, $b>0$, evaluating the\nfirst-order straight-ray integrals over $[-L,L]$ gives\n\\begin{align*}\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),","+\\left(1-\\frac{2\\Phi}{c_0^2}\\right)d\\mathbf x^2,\n \\qquad \\left|\\frac{\\Phi}{c_0^2}\\right|\\ll1,\n\\]\na static clock has lapse $\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}$.\nThe static-frequency ratio in this approximation is\n\\[\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n\\]\nFor $\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}$, $b>0$, evaluating the\nfirst-order straight-ray integrals over $[-L,L]$ gives\n\\begin{align*}\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),","ht)d\\mathbf x^2,\n \\qquad \\left|\\frac{\\Phi}{c_0^2}\\right|\\ll1,\n\\]\na static clock has lapse $\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}$.\nThe static-frequency ratio in this approximation is\n\\[\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n\\]\nFor $\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}$, $b>0$, evaluating the\nfirst-order straight-ray integrals over $[-L,L]$ gives\n\\begin{align*}\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),","\\]\na static clock has lapse $\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}$.\nThe static-frequency ratio in this approximation is\n\\[\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n\\]\nFor $\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}$, $b>0$, evaluating the\nfirst-order straight-ray integrals over $[-L,L]$ gives\n\\begin{align*}\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),","$\\mathcal N=\\sqrt{1+2\\Phi/c_0^2}$.\nThe static-frequency ratio in this approximation is\n\\[\n \\frac{\\nu_{\\rm obs}}{\\nu_{\\rm emit}}\n =\\frac{\\mathcal N_{\\rm emit}}{\\mathcal N_{\\rm obs}}\n =1+\\frac{\\Phi_{\\rm emit}-\\Phi_{\\rm obs}}{c_0^2}\n +O\\left(\\frac{\\Phi^2}{c_0^4}\\right).\n\\]\nFor $\\Phi(x,b)=-GM_0/\\sqrt{x^2+b^2}$, $b>0$, evaluating the\nfirst-order straight-ray integrals over $[-L,L]$ gives\n\\begin{align*}\n \\Delta t_{\\rm Shapiro}\n &=-\\frac2{c_0^3}\\int_{-L}^L\\Phi(x,b)\\,dx\n =\\frac{4GM_0}{c_0^3}\\operatorname{asinh}\\left(\\frac Lb\\right),\\\\\n \\alpha(L,b)\n &=\\frac2{c_0^2}\\left|\\int_{-L}^L\\partial_b\\Phi(x,b)\\,dx\\right|\n =\\frac{4GM_0L}{c_0^2b\\sqrt{L^2+b^2}}.\n\\end{align*}\nHere $\\alpha$ is the magnitude of the deflection toward the source.\nThe limit $L\\to\\infty$ gives $4GM_0/(c_0^2b)$. The displayed\nequalities evaluate the weak-field integrals exactly. The approximation\nto the physical ray already entered through the metric expansion and\nthe undeviated integration path.","\\alpha(L,b)\n &=\\frac2{c_0^2}\\left|\\int_{-L}^L\\partial_b\\Phi(x,b)\\,dx\\right|\n =\\frac{4GM_0L}{c_0^2b\\sqrt{L^2+b^2}}.\n\\end{align*}\nHere $\\alpha$ is the magnitude of the deflection toward the source.\nThe limit $L\\to\\infty$ gives $4GM_0/(c_0^2b)$. The displayed\nequalities evaluate the weak-field integrals exactly. The approximation\nto the physical ray already entered through the metric expansion and\nthe undeviated integration path.","Let $P,Q$ be real smooth fields on $I\\times\\T$, with\n$\\theta\\in\\R/(2\\pi\\Z)$ and $\\gamma(\\tau)=e^{-2\\tau}$. Define\n\\begin{equation}\n S_G=\\frac12\\int_I\\int_0^{2\\pi}\n \\left[P_\\tau^2-\\gamma P_\\theta^2\n +e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2)\\right]d\\theta\\,d\\tau.\n \\label{c:core:gowdy-action}\n\\end{equation}\nTime variations vanish at the endpoints of $I$ and spatial variations\nare periodic. Integration by parts gives the two Euler--Lagrange\nexpressions\n\\begin{align}\n \\mathcal E_P&=P_{\\tau\\tau}-\\gamma P_{\\theta\\theta}\n -e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2),\\nonumber","\\in\\R/(2\\pi\\Z)$ and $\\gamma(\\tau)=e^{-2\\tau}$. Define\n\\begin{equation}\n S_G=\\frac12\\int_I\\int_0^{2\\pi}\n \\left[P_\\tau^2-\\gamma P_\\theta^2\n +e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2)\\right]d\\theta\\,d\\tau.\n \\label{c:core:gowdy-action}\n\\end{equation}\nTime variations vanish at the endpoints of $I$ and spatial variations\nare periodic. Integration by parts gives the two Euler--Lagrange\nexpressions\n\\begin{align}\n \\mathcal E_P&=P_{\\tau\\tau}-\\gamma P_{\\theta\\theta}\n -e^{2P}(Q_\\tau^2-\\gamma Q_\\theta^2),\\nonumber\\\\\n \\mathcal E_Q&=Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta}\n +2(P_\\tau Q_\\tau-\\gamma P_\\theta Q_\\theta).\n \\label{c:core:gowdy-euler}\n\\end{align}\nIn particular, the $Q$ equation can first be written as\n$\\partial_\\tau(e^{2P}Q_\\tau)-\\partial_\\theta(e^{2P}\\gamma Q_\\theta)=0$.\nThe stationary equations are $\\mathcal E_P=\\mathcal E_Q=0$.","\\mathcal E_Q&=Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta}\n +2(P_\\tau Q_\\tau-\\gamma P_\\theta Q_\\theta).\n \\label{c:core:gowdy-euler}\n\\end{align}\nIn particular, the $Q$ equation can first be written as\n$\\partial_\\tau(e^{2P}Q_\\tau)-\\partial_\\theta(e^{2P}\\gamma Q_\\theta)=0$.\nThe stationary equations are $\\mathcal E_P=\\mathcal E_Q=0$.","Define the densities\n\\begin{align}\n H_G&=P_\\tau^2+\\gamma P_\\theta^2\n +e^{2P}(Q_\\tau^2+\\gamma Q_\\theta^2),\\nonumber\\\\\n M_G&=2(P_\\tau P_\\theta+e^{2P}Q_\\tau Q_\\theta).\n \\label{c:core:gowdy-densities}\n\\end{align}\nThe remaining reconstruction equations are\n$\\lambda_\\tau=H_G$ and $\\lambda_\\theta=M_G$.","M_G&=2(P_\\tau P_\\theta+e^{2P}Q_\\tau Q_\\theta).\n \\label{c:core:gowdy-densities}\n\\end{align}\nThe remaining reconstruction equations are\n$\\lambda_\\tau=H_G$ and $\\lambda_\\theta=M_G$.","\\Needspace{14\\baselineskip}\n\\begin{proposition}[Local integrability and global periodicity]\n\\label{c:core:gowdy-integrability}\nFor twice continuously differentiable $P,Q$,\n\\begin{equation}\n \\partial_\\tau M_G-\\partial_\\theta H_G\n =2P_\\theta\\mathcal E_P+2e^{2P}Q_\\theta\\mathcal E_Q.\n \\label{c:core:gowdy-identity}\n\\end{equation}\nFor a solution on $I\\times\\T$, a single-valued periodic $\\lambda$ with\nthese two derivatives exists if and only if\n$\\int_0^{2\\pi}M_G\\,d\\theta=0$ at one, and hence every, time. It is\nunique up to an additive constant.\n\\end{proposition}","ion}[Local integrability and global periodicity]\n\\label{c:core:gowdy-integrability}\nFor twice continuously differentiable $P,Q$,\n\\begin{equation}\n \\partial_\\tau M_G-\\partial_\\theta H_G\n =2P_\\theta\\mathcal E_P+2e^{2P}Q_\\theta\\mathcal E_Q.\n \\label{c:core:gowdy-identity}\n\\end{equation}\nFor a solution on $I\\times\\T$, a single-valued periodic $\\lambda$ with\nthese two derivatives exists if and only if\n$\\int_0^{2\\pi}M_G\\,d\\theta=0$ at one, and hence every, time. It is\nunique up to an additive constant.\n\\end{proposition}","\\begin{proof}\nDifferentiate the two densities. The terms with $P_\\tau\nP_{\\tau\\theta}$ and $Q_\\tau Q_{\\tau\\theta}$ cancel, leaving\n\\[\n\\begin{aligned}\n \\partial_\\tau M_G-\\partial_\\theta H_G\n ={}&2P_\\theta(P_{\\tau\\tau}-\\gamma P_{\\theta\\theta})","\\begin{proof}\nDifferentiate the two densities. The terms with $P_\\tau\nP_{\\tau\\theta}$ and $Q_\\tau Q_{\\tau\\theta}$ cancel, leaving\n\\[\n\\begin{aligned}\n \\partial_\\tau M_G-\\partial_\\theta H_G\n ={}&2P_\\theta(P_{\\tau\\tau}-\\gamma P_{\\theta\\theta})\\\\\n &+2e^{2P}\\big[\n Q_\\theta(Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta})\n +2P_\\tau Q_\\tau Q_\\theta\n -P_\\theta Q_\\tau^2-\\gamma P_\\theta Q_\\theta^2\\big].\n\\end{aligned}\n\\]\nSubstitute \\eqref{c:core:gowdy-euler} and combine the two occurrences of\n$\\gamma P_\\theta Q_\\theta^2$. This gives\n\\eqref{c:core:gowdy-identity}. On a solution the identity vanishes,\nso periodicity gives $d\\int M_G\\,d\\theta/d\\tau=0$.","&+2e^{2P}\\big[\n Q_\\theta(Q_{\\tau\\tau}-\\gamma Q_{\\theta\\theta})\n +2P_\\tau Q_\\tau Q_\\theta\n -P_\\theta Q_\\tau^2-\\gamma P_\\theta Q_\\theta^2\\big].\n\\end{aligned}\n\\]\nSubstitute \\eqref{c:core:gowdy-euler} and combine the two occurrences of\n$\\gamma P_\\theta Q_\\theta^2$. This gives\n\\eqref{c:core:gowdy-identity}. On a solution the identity vanishes,\nso periodicity gives $d\\int M_G\\,d\\theta/d\\tau=0$.","If a periodic $\\lambda$ exists, integrating $\\lambda_\\theta=M_G$\nforces the zero-period condition. Conversely, fix $\\tau_0$ and define\non the lifted strip\n\\[\n \\lambda(\\tau,\\theta)=\\lambda_0\n +\\int_{\\tau_0}^{\\tau}H_G(s,0)\\,ds\n +\\int_0^{\\theta}M_G(\\tau,\\eta)\\,d\\eta.\n\\]\nIts $\\theta$ derivative is $M_G$. Its $\\tau$ derivative is $H_G$ by\nthe integrability identity and the fundamental theorem of calculus.\nThe zero-period condition makes it periodic. The difference of two\nsuch functions has both derivatives zero, so is constant.\n\\end{proof}","Set $Q=0$ and take an integer $n\\geq1$. Define the entire function\n\\[\n J_0(x)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m(x^2/4)^m}{(m!)^2}.\n\\]\nThe series and its differentiated series converge uniformly on compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","integer $n\\geq1$. Define the entire function\n\\[\n J_0(x)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m(x^2/4)^m}{(m!)^2}.\n\\]\nThe series and its differentiated series converge uniformly on compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","$. Define the entire function\n\\[\n J_0(x)=\\sum_{m=0}^{\\infty}\\frac{(-1)^m(x^2/4)^m}{(m!)^2}.\n\\]\nThe series and its differentiated series converge uniformly on compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","{(-1)^m(x^2/4)^m}{(m!)^2}.\n\\]\nThe series and its differentiated series converge uniformly on compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","he series and its differentiated series converge uniformly on compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","compact\nsets. Coefficient comparison gives\n$x^2J_0''+xJ_0'+x^2J_0=0$. Thus\n\\begin{equation}\n P(\\tau,\\theta)=A J_0(ne^{-\\tau})\\cos(n\\theta)\n \\label{c:core:gowdy-mode}\n\\end{equation}\nis an exact polarized solution. Indeed, for $x=ne^{-\\tau}$,\n$\\partial_\\tau^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x)$,\nwhereas $P_{\\theta\\theta}=-n^2P$. Also\n$M_G=2P_\\tau P_\\theta$ has zero spatial integral, so periodic\n$\\lambda$ can be reconstructed.","The Taylor channels can be separated without changing the physical\nmode. Put $y=x^2/4$, $\\mathcal B(y)=J_0(2\\sqrt y)$, and for an integer\n$r\\geq2$ define\n\\[\n \\mathcal B_{r,j}(y)=\\sum_{q=0}^{\\infty}\n \\frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},\n \\qquad j=0,\\ldots,r-1.\n\\]\nThen $\\mathcal B=\\sum_j\\mathcal B_{r,j}$. With\n$\\vartheta=y\\,d/dy$, the coefficient identity\n\\[\n \\vartheta^2\\mathcal B_{r,j}+y\\mathcal B_{r,j-1}=0,\n \\qquad j\\pmod r,\n\\]\nfollows from $m^2/(m!)^2=1/((m-1)!)^2$ for $m\\geq1$.\nConsequently the separated Taylor channels are coupled. Their sum\nsatisfies the Bessel equation. This decomposition resolves series\norder and does not introduce additional independent gravitational\ndegrees of freedom.","ing the physical\nmode. Put $y=x^2/4$, $\\mathcal B(y)=J_0(2\\sqrt y)$, and for an integer\n$r\\geq2$ define\n\\[\n \\mathcal B_{r,j}(y)=\\sum_{q=0}^{\\infty}\n \\frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},\n \\qquad j=0,\\ldots,r-1.\n\\]\nThen $\\mathcal B=\\sum_j\\mathcal B_{r,j}$. With\n$\\vartheta=y\\,d/dy$, the coefficient identity\n\\[\n \\vartheta^2\\mathcal B_{r,j}+y\\mathcal B_{r,j-1}=0,\n \\qquad j\\pmod r,\n\\]\nfollows from $m^2/(m!)^2=1/((m-1)!)^2$ for $m\\geq1$.\nConsequently the separated Taylor channels are coupled. Their sum\nsatisfies the Bessel equation. This decomposition resolves series\norder and does not introduce additional independent gravitational\ndegrees of freedom.","x^2/4$, $\\mathcal B(y)=J_0(2\\sqrt y)$, and for an integer\n$r\\geq2$ define\n\\[\n \\mathcal B_{r,j}(y)=\\sum_{q=0}^{\\infty}\n \\frac{(-1)^{rq+j}y^{rq+j}}{((rq+j)!)^2},\n \\qquad j=0,\\ldots,r-1.\n\\]\nThen $\\mathcal B=\\sum_j\\mathcal B_{r,j}$. With\n$\\vartheta=y\\,d/dy$, the coefficient identity\n\\[\n \\vartheta^2\\mathcal B_{r,j}+y\\mathcal B_{r,j-1}=0,\n \\qquad j\\pmod r,\n\\]\nfollows from $m^2/(m!)^2=1/((m-1)!)^2$ for $m\\geq1$.\nConsequently the separated Taylor channels are coupled. Their sum\nsatisfies the Bessel equation. This decomposition resolves series\norder and does not introduce additional independent gravitational\ndegrees of freedom.","Let $h=2\\pi/N$ and use the periodic centered derivative\n\\[\n (D_hf)_j=\\frac{f_{j+1}-f_{j-1}}{2h},\\qquad\n (\\Delta_hf)_j=\\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.\n\\]\nHere $D_h^{\\mathsf T}=-D_h$. With $U=\\dot P$, $V=\\dot Q$, the raw\nsemidiscrete vector field $f_h(\\tau,z)$ for $z=(P,Q,U,V)$ is\n\\begin{align}\n \\dot P&=U,&\\dot Q&=V,\\nonumber","Let $h=2\\pi/N$ and use the periodic centered derivative\n\\[\n (D_hf)_j=\\frac{f_{j+1}-f_{j-1}}{2h},\\qquad\n (\\Delta_hf)_j=\\frac{f_{j+1}-2f_j+f_{j-1}}{h^2}.\n\\]\nHere $D_h^{\\mathsf T}=-D_h$. With $U=\\dot P$, $V=\\dot Q$, the raw\nsemidiscrete vector field $f_h(\\tau,z)$ for $z=(P,Q,U,V)$ is\n\\begin{align}\n \\dot P&=U,&\\dot Q&=V,\\nonumber\\\\\n \\dot U&=\\gamma\\Delta_hP+e^{2P}[V^2-\\gamma(D_hQ)^2],\\nonumber\\\\\n \\dot V&=\\gamma\\Delta_hQ-2[UV-\\gamma(D_hP)(D_hQ)].\n \\label{c:core:gowdy-raw}\n\\end{align}\nProducts and exponentials in these formulas are componentwise.\nThe finite global momentum is\n\\begin{equation}\n C_h(z)=2h\\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].\n \\label{c:core:gowdy-momentum}\n\\end{equation}\nIt is linear in the velocity variables when the configuration $P,Q$\nis fixed.","\\dot V&=\\gamma\\Delta_hQ-2[UV-\\gamma(D_hP)(D_hQ)].\n \\label{c:core:gowdy-raw}\n\\end{align}\nProducts and exponentials in these formulas are componentwise.\nThe finite global momentum is\n\\begin{equation}\n C_h(z)=2h\\sum_j[U_j(D_hP)_j+e^{2P_j}V_j(D_hQ)_j].\n \\label{c:core:gowdy-momentum}\n\\end{equation}\nIt is linear in the velocity variables when the configuration $P,Q$\nis fixed.","For the ordinary Euclidean gradients, skew-adjointness of $D_h$ gives\n\\begin{align*}\n \\nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),","For the ordinary Euclidean gradients, skew-adjointness of $D_h$ gives\n\\begin{align*}\n \\nabla_PC_h&=-2hD_hU+4h e^{2P}V(D_hQ),\\\\\n \\nabla_QC_h&=-2hD_h(e^{2P}V),\\\\\n g_U:=\\nabla_UC_h&=2hD_hP,\n &g_V:=\\nabla_VC_h&=2h e^{2P}D_hQ.\n\\end{align*}\nPut $G_h=(0,0,g_U,g_V)$ and $d_h=\\norm{g_U}_2^2+\\norm{g_V}_2^2$.\nOn the open region $d_h>0$, define\n\\[\n \\Lambda_h=\\frac{\\nabla C_h\\cdot f_h}{d_h},\\qquad\n \\widehat f_h=f_h-\\Lambda_hG_h.\n\\]\nThe chain rule proves the exact tangency identity\n\\[\n \\nabla C_h\\cdot\\widehat f_h\n =\\nabla C_h\\cdot f_h-\\Lambda_hd_h=0.\n\\]\nEvery differentiable solution that remains in this region therefore\nkeeps $C_h$ constant.","g_U:=\\nabla_UC_h&=2hD_hP,\n &g_V:=\\nabla_VC_h&=2h e^{2P}D_hQ.\n\\end{align*}\nPut $G_h=(0,0,g_U,g_V)$ and $d_h=\\norm{g_U}_2^2+\\norm{g_V}_2^2$.\nOn the open region $d_h>0$, define\n\\[\n \\Lambda_h=\\frac{\\nabla C_h\\cdot f_h}{d_h},\\qquad\n \\widehat f_h=f_h-\\Lambda_hG_h.\n\\]\nThe chain rule proves the exact tangency identity\n\\[\n \\nabla C_h\\cdot\\widehat f_h\n =\\nabla C_h\\cdot f_h-\\Lambda_hd_h=0.\n\\]\nEvery differentiable solution that remains in this region therefore\nkeeps $C_h$ constant.","\\end{align*}\nPut $G_h=(0,0,g_U,g_V)$ and $d_h=\\norm{g_U}_2^2+\\norm{g_V}_2^2$.\nOn the open region $d_h>0$, define\n\\[\n \\Lambda_h=\\frac{\\nabla C_h\\cdot f_h}{d_h},\\qquad\n \\widehat f_h=f_h-\\Lambda_hG_h.\n\\]\nThe chain rule proves the exact tangency identity\n\\[\n \\nabla C_h\\cdot\\widehat f_h\n =\\nabla C_h\\cdot f_h-\\Lambda_hd_h=0.\n\\]\nEvery differentiable solution that remains in this region therefore\nkeeps $C_h$ constant.","\\begin{proposition}[Exact endpoint momentum]\n\\label{c:core:gowdy-endpoint}\nStart from $z_n$ and let a provisional step produce\n$\\widetilde z=(\\widetilde P,\\widetilde Q,\\widetilde U,\\widetilde V)$.\nForm $\\widetilde g_U,\\widetilde g_V$ at its configuration and assume\n$\\widetilde d_h>0$. The correction\n\\[\n \\mu=\\frac{C_h(\\widetilde z)-C_h(z_n)}{\\widetilde d_h},\\qquad\n z_{n+1}=(\\widetilde P,\\widetilde Q,\n \\widetilde U-\\mu\\widetilde g_U,\n \\widetilde V-\\mu\\widetilde g_V)\n\\]\nsatisfies $C_h(z_{n+1})=C_h(z_n)$ exactly. It is the correction of\nsmallest Euclidean velocity norm at the fixed configuration.\n\\end{proposition}","om $z_n$ and let a provisional step produce\n$\\widetilde z=(\\widetilde P,\\widetilde Q,\\widetilde U,\\widetilde V)$.\nForm $\\widetilde g_U,\\widetilde g_V$ at its configuration and assume\n$\\widetilde d_h>0$. The correction\n\\[\n \\mu=\\frac{C_h(\\widetilde z)-C_h(z_n)}{\\widetilde d_h},\\qquad\n z_{n+1}=(\\widetilde P,\\widetilde Q,\n \\widetilde U-\\mu\\widetilde g_U,\n \\widetilde V-\\mu\\widetilde g_V)\n\\]\nsatisfies $C_h(z_{n+1})=C_h(z_n)$ exactly. It is the correction of\nsmallest Euclidean velocity norm at the fixed configuration.\n\\end{proposition}","\\begin{proof}\nLinearity in the velocities gives\n$C_h(z_{n+1})=C_h(\\widetilde z)-\\mu\\widetilde d_h$.\nFor minimality, write $g=(\\widetilde g_U,\\widetilde g_V)$ and\n$b=C_h(z_n)-C_h(\\widetilde z)$. Every admissible velocity displacement\n$v$ satisfies $g\\cdot v=b$. Cauchy--Schwarz implies\n$\\norm v\\geq|b|/\\norm g$, with equality only at\n$v=bg/\\norm g^2$. This is the displayed correction.\n\\end{proof}","\\Needspace{11\\baselineskip}\n\\begin{proposition}[Fixed-mesh second-order time accuracy]\n\\label{c:core:gowdy-time-order}\nFix $h$. Suppose a smooth exact solution of\n$\\dot z=\\widehat f_h(\\tau,z)$ and its numerical steps remain in a\ncompact regular neighborhood where $d_h\\geq d_0>0$. Assume the mixed time and state\nderivatives of $\\widehat f_h$ through order three and the state\nderivatives of $C_h$ through order four are bounded there. Apply the endpoint correction\nof Proposition~\\ref{c:core:gowdy-endpoint} after explicit midpoint.\nThen its local error is $O(\\Delta\\tau^3)$ and its global error on a\nfixed time interval is $O(\\Delta\\tau^2)$.\n\\end{proposition}","of\n$\\dot z=\\widehat f_h(\\tau,z)$ and its numerical steps remain in a\ncompact regular neighborhood where $d_h\\geq d_0>0$. Assume the mixed time and state\nderivatives of $\\widehat f_h$ through order three and the state\nderivatives of $C_h$ through order four are bounded there. Apply the endpoint correction\nof Proposition~\\ref{c:core:gowdy-endpoint} after explicit midpoint.\nThen its local error is $O(\\Delta\\tau^3)$ and its global error on a\nfixed time interval is $O(\\Delta\\tau^2)$.\n\\end{proposition}","a\ncompact regular neighborhood where $d_h\\geq d_0>0$. Assume the mixed time and state\nderivatives of $\\widehat f_h$ through order three and the state\nderivatives of $C_h$ through order four are bounded there. Apply the endpoint correction\nof Proposition~\\ref{c:core:gowdy-endpoint} after explicit midpoint.\nThen its local error is $O(\\Delta\\tau^3)$ and its global error on a\nfixed time interval is $O(\\Delta\\tau^2)$.\n\\end{proposition}","\\begin{proof}\nThe explicit midpoint expansion agrees with the Taylor expansion of\nthe exact flow through order two. Its provisional endpoint differs\nfrom the exact endpoint by $O(\\Delta\\tau^3)$. Since the exact flow\nkeeps $C_h$ constant and $C_h$ has bounded first derivatives, the\nprovisional constraint defect is also $O(\\Delta\\tau^3)$. The lower\nbound on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","icit midpoint expansion agrees with the Taylor expansion of\nthe exact flow through order two. Its provisional endpoint differs\nfrom the exact endpoint by $O(\\Delta\\tau^3)$. Since the exact flow\nkeeps $C_h$ constant and $C_h$ has bounded first derivatives, the\nprovisional constraint defect is also $O(\\Delta\\tau^3)$. The lower\nbound on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","elta\\tau^3)$. Since the exact flow\nkeeps $C_h$ constant and $C_h$ has bounded first derivatives, the\nprovisional constraint defect is also $O(\\Delta\\tau^3)$. The lower\nbound on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","eeps $C_h$ constant and $C_h$ has bounded first derivatives, the\nprovisional constraint defect is also $O(\\Delta\\tau^3)$. The lower\nbound on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","atives, the\nprovisional constraint defect is also $O(\\Delta\\tau^3)$. The lower\nbound on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","und on the denominator and the upper bound on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","nd on $g$ make the\ncorrection $O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","$O(\\Delta\\tau^3)$. Thus the corrected local error has the\nsame order. Under the stated derivative bounds the corrected step\nmap has Lipschitz constant at most $1+L\\Delta\\tau$ for small steps.\nIf $e_n$ is its error against the exact solution, then\n\\[\n e_{n+1}\\leq(1+L\\Delta\\tau)e_n+M\\Delta\\tau^3.\n\\]\nIterating this inequality for $n\\Delta\\tau\\leq T$ bounds the error by\n$MT e^{LT}\\Delta\\tau^2$ when $e_0=0$.\n\\end{proof}","These are regular-chart results. For a spatially homogeneous\nconfiguration, $g_U=g_V=0$, and the displayed multiplier is\nundefined even though $C_h=0$. Constants in the accuracy theorem\ncan depend on $h$. A claim about simultaneous spatial and temporal\nrefinement additionally needs bounds uniform in $h$.","The local compatibility diagnostic is a separate quantity. Define\n$H_{G,h},M_{G,h}$ by replacing spatial derivatives in\n\\eqref{c:core:gowdy-densities} with $D_h$. If\n$\\mathcal E_{P,h}=\\dot U-\\dot U_{\\rm raw}$ and\n$\\mathcal E_{Q,h}=\\dot V-\\dot V_{\\rm raw}$, then\n\\[\n \\mathcal I_h=\\partial_\\tau M_{G,h}-D_hH_{G,h}\n -2(D_hP)\\mathcal E_{P,h}\n -2e^{2P}(D_hQ)\\mathcal E_{Q,h}\n\\]\nsubtracts the explicit constraint-force contribution. Exact global\nmomentum preservation does not force $\\mathcal I_h$ to vanish at each\nsite. Its refinement behavior tests whether the modified evolution\napproaches the local continuum reconstruction identity. The archived\nsecond-order numerical observations are experimental evidence for\nthat behavior, with their tested range recorded in the companion.","G,h}$ by replacing spatial derivatives in\n\\eqref{c:core:gowdy-densities} with $D_h$. If\n$\\mathcal E_{P,h}=\\dot U-\\dot U_{\\rm raw}$ and\n$\\mathcal E_{Q,h}=\\dot V-\\dot V_{\\rm raw}$, then\n\\[\n \\mathcal I_h=\\partial_\\tau M_{G,h}-D_hH_{G,h}\n -2(D_hP)\\mathcal E_{P,h}\n -2e^{2P}(D_hQ)\\mathcal E_{Q,h}\n\\]\nsubtracts the explicit constraint-force contribution. Exact global\nmomentum preservation does not force $\\mathcal I_h$ to vanish at each\nsite. Its refinement behavior tests whether the modified evolution\napproaches the local continuum reconstruction identity. The archived\nsecond-order numerical observations are experimental evidence for\nthat behavior, with their tested range recorded in the companion.","Let $V_h$ be a finite real inner-product space. Let\n$\\delta_1,\\delta_2,\\delta_3$ be commuting skew-adjoint operators on it:\n\\[\n \\delta_i\\delta_j=\\delta_j\\delta_i,\n \\qquad\\inner{\\delta_iX}{Y}_h=-\\inner{X}{\\delta_iY}_h,\n \\qquad\\delta^2=\\sum_i\\delta_i^2.\n\\]\nThe periodic centered differences in three coordinate directions\nprovide one example. All tensor inner products below sum over both\nindices, using the Frobenius inner product on symmetric tensors.","Let $h_{ij}=h_{ji}$ and $\\pi_{ij}=\\pi_{ji}$ take values in $V_h$.\nRepeated spatial indices are summed. Put $h_{\\rm tr}=h_{kk}$,\n$\\pi_{\\rm tr}=\\pi_{kk}$ and define\n\\begin{align}\n C&=\\delta_i\\delta_jh_{ij}-\\delta^2h_{\\rm tr},\n &C_i&=-2\\delta_j\\pi_{ij},\\nonumber","Let $h_{ij}=h_{ji}$ and $\\pi_{ij}=\\pi_{ji}$ take values in $V_h$.\nRepeated spatial indices are summed. Put $h_{\\rm tr}=h_{kk}$,\n$\\pi_{\\rm tr}=\\pi_{kk}$ and define\n\\begin{align}\n C&=\\delta_i\\delta_jh_{ij}-\\delta^2h_{\\rm tr},\n &C_i&=-2\\delta_j\\pi_{ij},\\nonumber\\\\\n \\mathcal G_{ij}(h)&=\\frac12\\big[\n \\delta_k\\delta_ih_{jk}+\\delta_k\\delta_jh_{ik}\n -\\delta^2h_{ij}-\\delta_i\\delta_jh_{\\rm tr}\n -\\delta_{ij}C\\big].\n \\label{c:core:adm-operators}\n\\end{align}","\\begin{lemma}[Linearized finite Bianchi and adjoint identities]\n\\label{c:core:adm-bianchi}\nThe spatial operator obeys $\\delta_j\\mathcal G_{ij}(h)=0$ and is\nself-adjoint on symmetric tensors.\n\\end{lemma}","\\begin{proof}\nTwice its divergence expands as\n\\begin{align*}\n 2\\delta_j\\mathcal G_{ij}\n ={}&\\delta_i\\delta_j\\delta_kh_{jk}\n +\\delta^2\\delta_kh_{ik}-\\delta^2\\delta_jh_{ij}\n -\\delta_i\\delta^2h_{\\rm tr}\\\\\n &-\\delta_i\\delta_k\\delta_lh_{kl}\n +\\delta_i\\delta^2h_{\\rm tr}.\n\\end{align*}\nThe terms cancel in three pairs. For self-adjointness, write\n$B_i(a)=\\delta_ja_{ij}$ and $A(a)=\\delta_i\\delta_ja_{ij}$. Two\napplications of the adjoint identity give\n\\begin{align*}\n 2\\inner{a}{\\mathcal G(b)}_h\n ={}&-2\\inner{B(a)}{B(b)}_h\n +\\sum_k\\inner{\\delta_ka}{\\delta_kb}_h","&-\\delta_i\\delta_k\\delta_lh_{kl}\n +\\delta_i\\delta^2h_{\\rm tr}.\n\\end{align*}\nThe terms cancel in three pairs. For self-adjointness, write\n$B_i(a)=\\delta_ja_{ij}$ and $A(a)=\\delta_i\\delta_ja_{ij}$. Two\napplications of the adjoint identity give\n\\begin{align*}\n 2\\inner{a}{\\mathcal G(b)}_h\n ={}&-2\\inner{B(a)}{B(b)}_h\n +\\sum_k\\inner{\\delta_ka}{\\delta_kb}_h","&-\\delta_i\\delta_k\\delta_lh_{kl}\n +\\delta_i\\delta^2h_{\\rm tr}.\n\\end{align*}\nThe terms cancel in three pairs. For self-adjointness, write\n$B_i(a)=\\delta_ja_{ij}$ and $A(a)=\\delta_i\\delta_ja_{ij}$. Two\napplications of the adjoint identity give\n\\begin{align*}\n 2\\inner{a}{\\mathcal G(b)}_h\n ={}&-2\\inner{B(a)}{B(b)}_h\n +\\sum_k\\inner{\\delta_ka}{\\delta_kb}_h\\\\\n &-\\inner{A(a)}{b_{\\rm tr}}_h\n -\\inner{a_{\\rm tr}}{A(b)}_h\n -\\sum_k\\inner{\\delta_ka_{\\rm tr}}{\\delta_kb_{\\rm tr}}_h.\n\\end{align*}\nThis expression is symmetric in $a,b$.\n\\end{proof}","&-\\inner{A(a)}{b_{\\rm tr}}_h\n -\\inner{a_{\\rm tr}}{A(b)}_h\n -\\sum_k\\inner{\\delta_ka_{\\rm tr}}{\\delta_kb_{\\rm tr}}_h.\n\\end{align*}\nThis expression is symmetric in $a,b$.\n\\end{proof}","Set\n\\[\n T_h=\\inner{\\pi}{\\pi}_h-\\tfrac12\\inner{\\pi_{\\rm tr}}{\\pi_{\\rm tr}}_h,\n \\qquad U_h=\\tfrac12\\inner{h}{\\mathcal G(h)}_h.\n\\]\nWith lapse perturbation $n$ and shift perturbation $n_i$, the action is\n\\begin{equation}\n S_h^{(2)}=\\int\\left[\n \\inner{\\pi}{\\dot h}_h-T_h-U_h\n +\\inner{n}{C}_h-\\inner{n_i}{C_i}_h\\right]dt.\n \\label{c:core:adm-action}\n\\end{equation}\nVariation of $n,n_i$ imposes $C=C_i=0$. The remaining equations are\n\\begin{align}\n \\dot h_{ij}&=2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}\n             +\\delta_in_j+\\delta_jn_i,\\nonumber","\\inner{\\pi_{\\rm tr}}{\\pi_{\\rm tr}}_h,\n \\qquad U_h=\\tfrac12\\inner{h}{\\mathcal G(h)}_h.\n\\]\nWith lapse perturbation $n$ and shift perturbation $n_i$, the action is\n\\begin{equation}\n S_h^{(2)}=\\int\\left[\n \\inner{\\pi}{\\dot h}_h-T_h-U_h\n +\\inner{n}{C}_h-\\inner{n_i}{C_i}_h\\right]dt.\n \\label{c:core:adm-action}\n\\end{equation}\nVariation of $n,n_i$ imposes $C=C_i=0$. The remaining equations are\n\\begin{align}\n \\dot h_{ij}&=2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}\n             +\\delta_in_j+\\delta_jn_i,\\nonumber\\\\\n \\dot\\pi_{ij}&=-\\mathcal G_{ij}(h)\n              +\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n.\n \\label{c:core:adm-evolution}\n\\end{align}\nThe kinetic gradient is $2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}$.\nLemma~\\ref{c:core:adm-bianchi} makes the potential gradient\n$\\mathcal G(h)$. The lapse gradient follows by two spatial\nintegrations by parts. Integrating the shift term once gives\n$-\\langle\\delta_in_j+\\delta_jn_i,\\pi_{ij}\\rangle_h$ in the\nLagrangian, fixing the sign in the first equation.","\\dot\\pi_{ij}&=-\\mathcal G_{ij}(h)\n              +\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n.\n \\label{c:core:adm-evolution}\n\\end{align}\nThe kinetic gradient is $2\\pi_{ij}-\\delta_{ij}\\pi_{\\rm tr}$.\nLemma~\\ref{c:core:adm-bianchi} makes the potential gradient\n$\\mathcal G(h)$. The lapse gradient follows by two spatial\nintegrations by parts. Integrating the shift term once gives\n$-\\langle\\delta_in_j+\\delta_jn_i,\\pi_{ij}\\rangle_h$ in the\nLagrangian, fixing the sign in the first equation.","\\Needspace{11\\baselineskip}\n\\begin{theorem}[Exact finite linearized closure]\n\\label{c:core:adm-closure}\nFor every solution of \\eqref{c:core:adm-evolution},\n\\[\n \\dot C=-\\delta_iC_i,\\qquad\\dot C_i=0.\n\\]\nThus initially vanishing constraints remain zero. With the canonical\nFrobenius bracket and smearings\n$C[n]=\\langle n,C\\rangle_h$, $D[\\xi]=\\langle\\xi_i,C_i\\rangle_h$,\nall three brackets $\\{C[n],C[m]\\}$, $\\{D[\\xi],D[\\eta]\\}$, and\n$\\{C[n],D[\\xi]\\}$ vanish identically.\n\\end{theorem}","dspace{11\\baselineskip}\n\\begin{theorem}[Exact finite linearized closure]\n\\label{c:core:adm-closure}\nFor every solution of \\eqref{c:core:adm-evolution},\n\\[\n \\dot C=-\\delta_iC_i,\\qquad\\dot C_i=0.\n\\]\nThus initially vanishing constraints remain zero. With the canonical\nFrobenius bracket and smearings\n$C[n]=\\langle n,C\\rangle_h$, $D[\\xi]=\\langle\\xi_i,C_i\\rangle_h$,\nall three brackets $\\{C[n],C[m]\\}$, $\\{D[\\xi],D[\\eta]\\}$, and\n$\\{C[n],D[\\xi]\\}$ vanish identically.\n\\end{theorem}","egin{theorem}[Exact finite linearized closure]\n\\label{c:core:adm-closure}\nFor every solution of \\eqref{c:core:adm-evolution},\n\\[\n \\dot C=-\\delta_iC_i,\\qquad\\dot C_i=0.\n\\]\nThus initially vanishing constraints remain zero. With the canonical\nFrobenius bracket and smearings\n$C[n]=\\langle n,C\\rangle_h$, $D[\\xi]=\\langle\\xi_i,C_i\\rangle_h$,\nall three brackets $\\{C[n],C[m]\\}$, $\\{D[\\xi],D[\\eta]\\}$, and\n$\\{C[n],D[\\xi]\\}$ vanish identically.\n\\end{theorem}","\\begin{proof}\nThe trace of the momentum part of $\\dot h$ is $-\\pi_{\\rm tr}$.\nIts contribution to $\\dot C$ is therefore\n$2\\delta_i\\delta_j\\pi_{ij}=-\\delta_iC_i$. The shift contribution is\n\\[\n \\delta_i\\delta_j(\\delta_in_j+\\delta_jn_i)\n -\\delta^2(2\\delta_kn_k)=0.\n\\]\nFor $\\dot C_i$, the divergence of $\\mathcal G$ vanishes by\nLemma~\\ref{c:core:adm-bianchi}, and\n$\\delta_j(\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n)=0$.\nThis proves the propagation equations and their stated implication.","For the bracket, use\n$\\{F,G\\}=\\langle\\nabla_hF,\\nabla_\\pi G\\rangle_h\n-\\langle\\nabla_\\pi F,\\nabla_hG\\rangle_h$.\nTwo scalar constraints depend only on $h$ and two momentum\nconstraints depend only on $\\pi$, so their brackets vanish.\nThe gradients for the mixed bracket are\n\\[\n (\\nabla_hC[n])_{ij}=\\delta_i\\delta_jn-\\delta_{ij}\\delta^2n,\n \\qquad(\\nabla_\\pi D[\\xi])_{ij}=\\delta_i\\xi_j+\\delta_j\\xi_i.\n\\]\nTheir contraction vanishes after one integration by parts because\nthe first tensor is divergence-free.\n\\end{proof}","The momentum generator changes the metric perturbation by\n$h_{ij}\\mapsto h_{ij}+\\delta_i\\xi_j+\\delta_j\\xi_i$.\nThe corresponding change of the scalar constraint is\n$2\\delta^2\\delta_k\\xi_k-2\\delta^2\\delta_k\\xi_k=0$.\nNo product rule is used in these cancellations.","For a partition of fine spatial sites into coarse regions $A$, the\nnonnegative diagnostic\n\\[\n E_A^{\\rm con}=\\sum_{x\\in A}h^3\n \\left(|C_x|^2+\\sum_i|C_{i,x}|^2\\right)\n\\]\nadds exactly to the full fine diagnostic. This is an exact partition\nidentity. It does not assert that differentiating a coarse field\nreconstructs its fine constraint values.","Choose $n=n_i=0$ and impose the transverse and traceless conditions\n\\[\n \\delta_jh_{ij}=0,\\quad h_{\\rm tr}=0,\n \\qquad\\delta_j\\pi_{ij}=0,\\quad\\pi_{\\rm tr}=0.\n\\]\nThey are preserved by the evolution because\n\\[\n \\mathcal G_{ij}(h)=-\\tfrac12\\delta^2h_{ij},\\qquad\n \\dot h=2\\pi,\\qquad\\dot\\pi=\\tfrac12\\delta^2h,\n \\qquad\\ddot h=\\delta^2h.\n\\]\nOn this reduced sector the energy is\n\\[\n H_{\\rm TT}=\\norm{\\pi}_h^2\n +\\frac14\\sum_k\\norm{\\delta_kh}_h^2\\geq0.\n\\]\nIts derivative is zero: substituting the two first-order equations\nreduces it to\n$\\langle\\pi,\\delta^2h\\rangle_h\n+\\sum_k\\langle\\delta_kh,\\delta_k\\pi\\rangle_h=0$.","i_{ij}=0,\\quad\\pi_{\\rm tr}=0.\n\\]\nThey are preserved by the evolution because\n\\[\n \\mathcal G_{ij}(h)=-\\tfrac12\\delta^2h_{ij},\\qquad\n \\dot h=2\\pi,\\qquad\\dot\\pi=\\tfrac12\\delta^2h,\n \\qquad\\ddot h=\\delta^2h.\n\\]\nOn this reduced sector the energy is\n\\[\n H_{\\rm TT}=\\norm{\\pi}_h^2\n +\\frac14\\sum_k\\norm{\\delta_kh}_h^2\\geq0.\n\\]\nIts derivative is zero: substituting the two first-order equations\nreduces it to\n$\\langle\\pi,\\delta^2h\\rangle_h\n+\\sum_k\\langle\\delta_kh,\\delta_k\\pi\\rangle_h=0$.","For a Fourier mode on the periodic centered lattice, let\n$\\widetilde k_i=\\sin(k_ih)/h$. If $\\widetilde{\\mathbf k}\\neq0$,\ndefine\n\\[\n \\Pi_{ij}=\\delta_{ij}\n -\\frac{\\widetilde k_i\\widetilde k_j}{|\\widetilde{\\mathbf k}|^2},\n \\qquad\n \\Lambda^{\\rm TT}_{ij,kl}\n =\\tfrac12(\\Pi_{ik}\\Pi_{jl}+\\Pi_{il}\\Pi_{jk}-\\Pi_{ij}\\Pi_{kl}).\n\\]\nFor a symmetric tensor $X$, its image is\n$\\Pi X\\Pi-\\Pi\\operatorname{tr}(\\Pi X)/2$. It is transverse and\ntraceless because $\\Pi^2=\\Pi$ and $\\operatorname{tr}\\Pi=2$.\nApplying the map again leaves it unchanged. On the transverse\ntwo-plane, symmetric tensors have dimension three and the trace\ncondition removes one dimension. Thus the projector has rank two.","lattice, let\n$\\widetilde k_i=\\sin(k_ih)/h$. If $\\widetilde{\\mathbf k}\\neq0$,\ndefine\n\\[\n \\Pi_{ij}=\\delta_{ij}\n -\\frac{\\widetilde k_i\\widetilde k_j}{|\\widetilde{\\mathbf k}|^2},\n \\qquad\n \\Lambda^{\\rm TT}_{ij,kl}\n =\\tfrac12(\\Pi_{ik}\\Pi_{jl}+\\Pi_{il}\\Pi_{jk}-\\Pi_{ij}\\Pi_{kl}).\n\\]\nFor a symmetric tensor $X$, its image is\n$\\Pi X\\Pi-\\Pi\\operatorname{tr}(\\Pi X)/2$. It is transverse and\ntraceless because $\\Pi^2=\\Pi$ and $\\operatorname{tr}\\Pi=2$.\nApplying the map again leaves it unchanged. On the transverse\ntwo-plane, symmetric tensors have dimension three and the trace\ncondition removes one dimension. Thus the projector has rank two.","widetilde k_i=\\sin(k_ih)/h$. If $\\widetilde{\\mathbf k}\\neq0$,\ndefine\n\\[\n \\Pi_{ij}=\\delta_{ij}\n -\\frac{\\widetilde k_i\\widetilde k_j}{|\\widetilde{\\mathbf k}|^2},\n \\qquad\n \\Lambda^{\\rm TT}_{ij,kl}\n =\\tfrac12(\\Pi_{ik}\\Pi_{jl}+\\Pi_{il}\\Pi_{jk}-\\Pi_{ij}\\Pi_{kl}).\n\\]\nFor a symmetric tensor $X$, its image is\n$\\Pi X\\Pi-\\Pi\\operatorname{tr}(\\Pi X)/2$. It is transverse and\ntraceless because $\\Pi^2=\\Pi$ and $\\operatorname{tr}\\Pi=2$.\nApplying the map again leaves it unchanged. On the transverse\ntwo-plane, symmetric tensors have dimension three and the trace\ncondition removes one dimension. Thus the projector has rank two.","The resulting frequency obeys\n\\begin{equation}\n \\omega_h^2=\\sum_i\\frac{\\sin^2(k_ih)}{h^2}.\n \\label{c:core:adm-dispersion}\n\\end{equation}\nFor fixed $\\mathbf k\\neq0$ as $h\\to0$,\n\\[\n \\omega_h^2=|\\mathbf k|^2-\\frac{h^2}{3}\\sum_i k_i^4+O(h^4),\n \\qquad\n \\omega_h=|\\mathbf k|\n -\\frac{h^2}{6|\\mathbf k|}\\sum_i k_i^4+O(h^4).\n\\]\nThese follow by expanding the sine and then the positive square root.\nThe regime keeps $\\mathbf k$ fixed. It is not uniform up to the\nNyquist boundary.","ulting frequency obeys\n\\begin{equation}\n \\omega_h^2=\\sum_i\\frac{\\sin^2(k_ih)}{h^2}.\n \\label{c:core:adm-dispersion}\n\\end{equation}\nFor fixed $\\mathbf k\\neq0$ as $h\\to0$,\n\\[\n \\omega_h^2=|\\mathbf k|^2-\\frac{h^2}{3}\\sum_i k_i^4+O(h^4),\n \\qquad\n \\omega_h=|\\mathbf k|\n -\\frac{h^2}{6|\\mathbf k|}\\sum_i k_i^4+O(h^4).\n\\]\nThese follow by expanding the sine and then the positive square root.\nThe regime keeps $\\mathbf k$ fixed. It is not uniform up to the\nNyquist boundary.","Let a periodic real lattice have canonical brackets\n$\\{q_a,p_b\\}_h=\\delta_{ab}/h$ and a skew matrix $D$. Define\n\\[\n \\mathcal D_h[\\xi]=h\\sum_a\\xi_ap_a(Dq)_a,\n \\qquad\n \\mathfrak L_D(f,g)=D(fg)-(Df)g-f(Dg).\n\\]\nAll products on this lattice are pointwise.","\\begin{proposition}[Exact nonlinear defect identity]\n\\label{c:core:nonlinear-defect}\nFor arbitrary finite arrays $q,p,\\xi,\\eta$,\n\\begin{align}\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n ={}&\\mathcal D_h[\\xi D\\eta-\\eta D\\xi]\\nonumber","\\begin{proposition}[Exact nonlinear defect identity]\n\\label{c:core:nonlinear-defect}\nFor arbitrary finite arrays $q,p,\\xi,\\eta$,\n\\begin{align}\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n ={}&\\mathcal D_h[\\xi D\\eta-\\eta D\\xi]\\nonumber\\\\\n &+h\\sum_ap_a\\big[\n \\xi_a\\mathfrak L_D(\\eta,Dq)_a\n -\\eta_a\\mathfrak L_D(\\xi,Dq)_a\\big].\n \\label{c:core:nonlinear-defect-identity}\n\\end{align}\n\\end{proposition}","\\begin{proof}\nCoordinate differentiation gives\n$\\partial_p\\mathcal D_h[\\xi]=h\\xi Dq$ and\n$\\partial_q\\mathcal D_h[\\xi]=-hD(\\xi p)$.\nInsert these in the canonical bracket and use $D^{\\mathsf T}=-D$:\n\\[\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n =h\\sum_ap_a\\left[\\xi_aD(\\eta Dq)_a\n                  -\\eta_aD(\\xi Dq)_a\\right].\n\\]\nExpanding each derivative using the definition of $\\mathfrak L_D$\ncancels the two $\\xi\\eta D^2q$ terms. The remaining product-rule\nterms give the first term of\n\\eqref{c:core:nonlinear-defect-identity}, and the two defect terms\ngive the second.\n\\end{proof}","f}\nCoordinate differentiation gives\n$\\partial_p\\mathcal D_h[\\xi]=h\\xi Dq$ and\n$\\partial_q\\mathcal D_h[\\xi]=-hD(\\xi p)$.\nInsert these in the canonical bracket and use $D^{\\mathsf T}=-D$:\n\\[\n \\{\\mathcal D_h[\\xi],\\mathcal D_h[\\eta]\\}_h\n =h\\sum_ap_a\\left[\\xi_aD(\\eta Dq)_a\n                  -\\eta_aD(\\xi Dq)_a\\right].\n\\]\nExpanding each derivative using the definition of $\\mathfrak L_D$\ncancels the two $\\xi\\eta D^2q$ terms. The remaining product-rule\nterms give the first term of\n\\eqref{c:core:nonlinear-defect-identity}, and the two defect terms\ngive the second.\n\\end{proof}","\\begin{theorem}[Finite pointwise derivation obstruction]\n\\label{c:core:pointwise-no-go}\nA linear derivation on the complete pointwise algebra $\\C^N$ is zero.\nThus a nonzero finite pointwise derivative cannot obey the undeformed\nLeibniz rule on every pair of arrays.\n\\end{theorem}","\\begin{proof}\nLet $e_a$ be the coordinate idempotents. If $D$ is a derivation, then\n$D(e_a)=D(e_a^2)=2e_aD(e_a)$. At coordinate $a$ this forces the\ncomponent to be twice itself, hence zero. At every other coordinate\nthe right side is zero. Therefore $D(e_a)=0$ for every $a$. Linearity\nand the coordinate basis imply $D=0$.\n\\end{proof}","\\begin{example}[An exact bracket failure]\n\\label{c:core:bracket-negative-control}\nUse $N=5$, $h=1$, and $(Df)_a=(f_{a+1}-f_{a-1})/2$.\nTake $q=e_0$, $p=\\xi=e_1$, and $\\eta=e_2$. Then\n$(Dq)_1=-1/2$, $(D\\eta)_1=1/2$, and $\\eta Dq=0$.\nThe direct bracket in the preceding proof is zero, while\n\\[\n \\mathcal D_h[\\xi D\\eta-\\eta D\\xi]=-\\tfrac14.\n\\]\nThe defect is therefore exactly $+1/4$. This finite witness shows\nthat the naive centered scalar constraint does fail to close\nuniversally in the proposed undeformed form.\n\\end{example}","For samples of sufficiently smooth periodic functions, the centered\nderivative obeys\n\\[\n D_hf=f'+\\frac{h^2}{6}f'''+O(h^4).\n\\]\nIf $f,g$ have uniformly bounded derivatives through order five, its\nLeibniz defect consequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","nctions, the centered\nderivative obeys\n\\[\n D_hf=f'+\\frac{h^2}{6}f'''+O(h^4).\n\\]\nIf $f,g$ have uniformly bounded derivatives through order five, its\nLeibniz defect consequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","ntered\nderivative obeys\n\\[\n D_hf=f'+\\frac{h^2}{6}f'''+O(h^4).\n\\]\nIf $f,g$ have uniformly bounded derivatives through order five, its\nLeibniz defect consequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","g$ have uniformly bounded derivatives through order five, its\nLeibniz defect consequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","bniz defect consequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","onsequently satisfies\n\\begin{equation}\n \\mathfrak L_{D_h}(f,g)\n =\\frac{h^2}{2}(f''g'+f'g'')+O(h^4).\n \\label{c:core:smooth-defect}\n\\end{equation}\nFor fixed smooth $p,\\xi,\\eta,q$, with sufficiently many additional\nbounded derivatives of $q$ to control $D_hq$, the contraction in\n\\eqref{c:core:nonlinear-defect-identity} is $O(h^2)$ on a fixed periodic\ninterval. Each summand is uniformly $O(h^2)$ and $hN$ is the interval\nlength. The coordinate-idempotent witness above is outside this\nuniformly smooth regime.","For the full matrix algebra, choose a Hermitian matrix $K$ and set\n$\\delta_KX=i[K,X]$. Direct multiplication gives\n\\begin{equation}\n \\delta_K(XY)=(\\delta_KX)Y+X(\\delta_KY).\n \\label{c:core:inner-derivation}\n\\end{equation}\nIt also obeys $\\delta_K(X^*)=(\\delta_KX)^*$,\n$\\operatorname{Tr}\\delta_KX=0$, and is skew-adjoint for the\nHilbert--Schmidt inner product. The last identity follows by moving\n$K$ through the trace. These properties supply an exact finite\nproduct calculus on a noncommutative algebra.","midrule\\endhead\nCanonical brackets & Each sector declares its variables and sign. In standard coordinates, $\\{F,G\\}=\\sum_a(F_{q_a}G_{p_a}-F_{p_a}G_{q_a})$, so $\\dot F=\\{F,H\\}$.","Spectral resolution & $K$ denotes retained degree and $N$ the sampling resolution in the resolved constructions. Conditions such as $16K<N$ are sufficient degree budgets in a declared representation. They are not invariant under every nonlinear change of variables.","The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies","The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hami","The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolutio","The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local…","It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","cal Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","onspherical gravitational radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","nal radiation. Work in units $G=c=4\\pi\\epsilon_0=1$. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","denotes a radial derivative and a dot denotes a time derivative. On a radial interval where $R>0$ and $\\Lambda>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","da>0$, write\n\\begin{equation}\n ds^2=-N^2dt^2+\\Lambda^2(dr+N^r dt)^2+R^2d\\Omega^2,\n \\qquad \\mathcal A=A_rdr+\\Phi_e dt.\n\\end{equation}\nHere $R$ is the areal radius, $\\Lambda$ determines proper radial distance, $N$ is the lapse, and $N^r$ is the radial shift. The Maxwell multiplier is denoted $\\Phi_e$ to distinguish it from the scalar gradient used later. The bulk canonical action is\n\\begin{equation}\n S=\\int dt\\int dr\\,\n \\bigl(P_\\Lambda\\dot\\Lambda+P_R\\dot R+Q\\dot A_r\n       -N\\mathcal H-N^r\\mathcal D-\\Phi_e\\mathcal G\\bigr)+B_\\partial.\n\\end{equation}\nThe boundary term specifies the mass and electromagnetic boundary ensemble. Its treatment cannot be omitted when claiming a global Hamiltonian evolution, even though it does not change the following local bulk identity.","The gauge-improved radial momentum and electric constraints are\n\\begin{equation}\n \\mathcal D=P_RR'-\\Lambda P_\\Lambda',\\qquad \\mathcal G=Q',\n\\end{equation}\nand the Hamiltonian constraint is\n\\begin{equation}\n\\begin{aligned}\n \\mathcal H={}&-\\frac{P_RP_\\Lambda}{R}\n +\\frac{\\Lambda P_\\Lambda^2}{2R^2}\n +\\frac{RR''}{\\Lambda}-\\frac{RR'\\Lambda'}{\\Lambda^2}\n +\\frac{R'^2}{2\\Lambda}-\\frac\\Lambda2\n +\\frac{\\Lambda Q^2}{2R^2}.\n\\end{aligned}\n\\label{c:strong:electrovac-H}\n\\end{equation}\nThe momenta follow the convention\n\\begin{align}\n P_\\Lambda&=-\\frac RN(\\dot R-N^rR'),","uad \\mathcal G=Q',\n\\end{equation}\nand the Hamiltonian constraint is\n\\begin{equation}\n\\begin{aligned}\n \\mathcal H={}&-\\frac{P_RP_\\Lambda}{R}\n +\\frac{\\Lambda P_\\Lambda^2}{2R^2}\n +\\frac{RR''}{\\Lambda}-\\frac{RR'\\Lambda'}{\\Lambda^2}\n +\\frac{R'^2}{2\\Lambda}-\\frac\\Lambda2\n +\\frac{\\Lambda Q^2}{2R^2}.\n\\end{aligned}\n\\label{c:strong:electrovac-H}\n\\end{equation}\nThe momenta follow the convention\n\\begin{align}\n P_\\Lambda&=-\\frac RN(\\dot R-N^rR'),\\\\\n P_R&=-\\frac1N\\left[\\Lambda(\\dot R-N^rR')\n                  +R\\bigl(\\dot\\Lambda-(N^r\\Lambda)'\\bigr)\\right].\n\\end{align}\nThese signs fix which Painlev\\'e--Gullstrand branch describes inward future characteristics.","\\begin{definition}[Charged mass coordinate]\nThe charged Misner--Sharp coordinate in this canonical chart is\n\\begin{equation}\n \\mathcal M=\\frac R2\\left[1-\\left(\\frac{R'}\\Lambda\\right)^2\n                    +\\left(\\frac{P_\\Lambda}R\\right)^2\\right]\n             +\\frac{Q^2}{2R}.\n\\label{c:strong:mass-coordinate}\n\\end{equation}\nThe charge contribution is included in $\\mathcal M$ so that it is constant throughout an electrovac Reissner--Nordstr\\\"om solution.\n\\end{definition}","\\begin{proposition}[Exact triangular mass identity]\nFor differentiable canonical data in the regular chart,\n\\begin{equation}\n \\mathcal M'=-\\frac{R'}\\Lambda\\mathcal H\n             -\\frac{P_\\Lambda}{R\\Lambda}\\mathcal D\n             +\\frac QR\\mathcal G.\n\\label{c:strong:mass-identity}\n\\end{equation}\nConsequently, on a patch where $R'/\\Lambda\\ne0$, the systems\n$(\\mathcal H,\\mathcal D,\\mathcal G)=0$ and\n$(\\mathcal M',\\mathcal D,\\mathcal G)=0$ have the same zero set.\n\\end{proposition}\n\\begin{proof}\nDifferentiating \\eqref{c:strong:mass-coordinate} gives\n\\begin{equation}\n\\begin{aligned}\n \\mathcal M'={}&\\frac{R'}2-\\frac{R'^3}{2\\Lambda^2}\n -\\frac{RR'R''}{\\Lambda^2}+\\frac{RR'^2\\Lambda'}{\\Lambda^3}\n +\\frac{P_\\Lambda P_\\Lambda'}R-\\frac{P_\\Lambda^2R'}{2R^2}","egin{proposition}[Exact triangular mass identity]\nFor differentiable canonical data in the regular chart,\n\\begin{equation}\n \\mathcal M'=-\\frac{R'}\\Lambda\\mathcal H\n             -\\frac{P_\\Lambda}{R\\Lambda}\\mathcal D\n             +\\frac QR\\mathcal G.\n\\label{c:strong:mass-identity}\n\\end{equation}\nConsequently, on a patch where $R'/\\Lambda\\ne0$, the systems\n$(\\mathcal H,\\mathcal D,\\mathcal G)=0$ and\n$(\\mathcal M',\\mathcal D,\\mathcal G)=0$ have the same zero set.\n\\end{proposition}\n\\begin{proof}\nDifferentiating \\eqref{c:strong:mass-coordinate} gives\n\\begin{equation}\n\\begin{aligned}\n \\mathcal M'={}&\\frac{R'}2-\\frac{R'^3}{2\\Lambda^2}\n -\\frac{RR'R''}{\\Lambda^2}+\\frac{RR'^2\\Lambda'}{\\Lambda^3}\n +\\frac{P_\\Lambda P_\\Lambda'}R-\\frac{P_\\Lambda^2R'}{2R^2}","-\\frac{P_\\Lambda}{R\\Lambda}\\mathcal D\n             +\\frac QR\\mathcal G.\n\\label{c:strong:mass-identity}\n\\end{equation}\nConsequently, on a patch where $R'/\\Lambda\\ne0$, the systems\n$(\\mathcal H,\\mathcal D,\\mathcal G)=0$ and\n$(\\mathcal M',\\mathcal D,\\mathcal G)=0$ have the same zero set.\n\\end{proposition}\n\\begin{proof}\nDifferentiating \\eqref{c:strong:mass-coordinate} gives\n\\begin{equation}\n\\begin{aligned}\n \\mathcal M'={}&\\frac{R'}2-\\frac{R'^3}{2\\Lambda^2}\n -\\frac{RR'R''}{\\Lambda^2}+\\frac{RR'^2\\Lambda'}{\\Lambda^3}\n +\\frac{P_\\Lambda P_\\Lambda'}R-\\frac{P_\\Lambda^2R'}{2R^2}\\\\\n &+\\frac{QQ'}R-\\frac{Q^2R'}{2R^2}.\n\\end{aligned}\n\\end{equation}\nExpanding the right side of \\eqref{c:strong:mass-identity} cancels the two $P_R$ terms and produces precisely this expression. The constraint transformation is upper triangular with diagonal $(-R'/\\Lambda,1,1)$, so it is invertible under the stated condition.\n\\end{proof}","&+\\frac{QQ'}R-\\frac{Q^2R'}{2R^2}.\n\\end{aligned}\n\\end{equation}\nExpanding the right side of \\eqref{c:strong:mass-identity} cancels the two $P_R$ terms and produces precisely this expression. The constraint transformation is upper triangular with diagonal $(-R'/\\Lambda,1,1)$, so it is invertible under the stated condition.\n\\end{proof}","Fix $M>0$, a real charge $Q$, and a branch $\\varepsilon\\in\\{+1,-1\\}$. On $2MR-Q^2>0$, define\n\\begin{equation}\n p(R)=\\varepsilon\\sqrt{2MR-Q^2},\\qquad v(R)=\\frac{p(R)}R.\n\\end{equation}\nThe Painlev\\'e--Gullstrand metric takes the form\n\\begin{equation}\n ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2d\\Omega^2.\n\\label{c:strong:RN-PG}\n\\end{equation}\nNo inverse factor of $1-2M/R+Q^2/R^2$ appears. The metric can therefore cross a horizon while the lapse and radial spatial metric remain regular.","Take a smooth radial map $R=F(t,r)$ with $F_r>0$ and identify $t_P=t$. Expanding $dR=F_tdt+F_rdr$ identifies\n\\begin{equation}\n N=1,\\qquad \\Lambda=F_r,\\qquad N^r=\\frac{F_t+v(F)}{F_r}.\n\\end{equation}\nThe corresponding canonical data are\n\\begin{equation}\n P_\\Lambda=p(F),\\qquad P_R=\\frac{MF_r}{p(F)},\n \\qquad A_r=0,\\quad \\Phi_e=-\\frac QF,\\quad Q'=0.\n\\label{c:strong:RN-data}\n\\end{equation}\nThe sign in $P_R$ is contained in $p(F)$.","ap $R=F(t,r)$ with $F_r>0$ and identify $t_P=t$. Expanding $dR=F_tdt+F_rdr$ identifies\n\\begin{equation}\n N=1,\\qquad \\Lambda=F_r,\\qquad N^r=\\frac{F_t+v(F)}{F_r}.\n\\end{equation}\nThe corresponding canonical data are\n\\begin{equation}\n P_\\Lambda=p(F),\\qquad P_R=\\frac{MF_r}{p(F)},\n \\qquad A_r=0,\\quad \\Phi_e=-\\frac QF,\\quad Q'=0.\n\\label{c:strong:RN-data}\n\\end{equation}\nThe sign in $P_R$ is contained in $p(F)$.","\\begin{proposition}[A time-dependent exact gauge family]\nThe data \\eqref{c:strong:RN-data} satisfy the local electrovac constraints and canonical evolution equations wherever $F_r>0$, $F>0$, and $2MF-Q^2>0$.\n\\end{proposition}\n\\begin{proof}\nFirst, $R'/\\Lambda=1$ and $p(F)^2=2MF-Q^2$ imply $\\mathcal M=M$. Since\n\\begin{equation}\n P_\\Lambda'=\\frac{MF_r}{p(F)}=P_R,\n\\end{equation}\nthe momentum constraint vanishes. The mass identity and $Q'=0$ then give $\\mathcal H=0$.","t exact gauge family]\nThe data \\eqref{c:strong:RN-data} satisfy the local electrovac constraints and canonical evolution equations wherever $F_r>0$, $F>0$, and $2MF-Q^2>0$.\n\\end{proposition}\n\\begin{proof}\nFirst, $R'/\\Lambda=1$ and $p(F)^2=2MF-Q^2$ imply $\\mathcal M=M$. Since\n\\begin{equation}\n P_\\Lambda'=\\frac{MF_r}{p(F)}=P_R,\n\\end{equation}\nthe momentum constraint vanishes. The mass identity and $Q'=0$ then give $\\mathcal H=0$.","ily]\nThe data \\eqref{c:strong:RN-data} satisfy the local electrovac constraints and canonical evolution equations wherever $F_r>0$, $F>0$, and $2MF-Q^2>0$.\n\\end{proposition}\n\\begin{proof}\nFirst, $R'/\\Lambda=1$ and $p(F)^2=2MF-Q^2$ imply $\\mathcal M=M$. Since\n\\begin{equation}\n P_\\Lambda'=\\frac{MF_r}{p(F)}=P_R,\n\\end{equation}\nthe momentum constraint vanishes. The mass identity and $Q'=0$ then give $\\mathcal H=0$.","For evolution, the radius equation gives\n\\begin{equation}\n \\dot R=-\\frac{P_\\Lambda}R+N^rR'\n        =-v(F)+F_t+v(F)=F_t.\n\\end{equation}\nThe radial-metric equation is\n\\begin{equation}\n \\dot\\Lambda=\\frac{\\Lambda P_\\Lambda}{R^2}\n             -\\frac{P_R}R+(N^r\\Lambda)'.\n\\end{equation}\nUse $p_R=M/p$ and $v_R=M/(pR)-p/R^2$. The non-$F_{tr}$ terms cancel, leaving $\\dot\\Lambda=F_{tr}$. The momentum equations in the $N=1$ convention are\n\\begin{align}\n \\dot P_\\Lambda&=N^rP_\\Lambda'\n +\\frac12\\left[1-\\frac{P_\\Lambda^2+Q^2}{R^2}\n                 -\\left(\\frac{R'}\\Lambda\\right)^2\\right],","on, the radius equation gives\n\\begin{equation}\n \\dot R=-\\frac{P_\\Lambda}R+N^rR'\n        =-v(F)+F_t+v(F)=F_t.\n\\end{equation}\nThe radial-metric equation is\n\\begin{equation}\n \\dot\\Lambda=\\frac{\\Lambda P_\\Lambda}{R^2}\n             -\\frac{P_R}R+(N^r\\Lambda)'.\n\\end{equation}\nUse $p_R=M/p$ and $v_R=M/(pR)-p/R^2$. The non-$F_{tr}$ terms cancel, leaving $\\dot\\Lambda=F_{tr}$. The momentum equations in the $N=1$ convention are\n\\begin{align}\n \\dot P_\\Lambda&=N^rP_\\Lambda'\n +\\frac12\\left[1-\\frac{P_\\Lambda^2+Q^2}{R^2}\n                 -\\left(\\frac{R'}\\Lambda\\right)^2\\right],","gin{equation}\n \\dot R=-\\frac{P_\\Lambda}R+N^rR'\n        =-v(F)+F_t+v(F)=F_t.\n\\end{equation}\nThe radial-metric equation is\n\\begin{equation}\n \\dot\\Lambda=\\frac{\\Lambda P_\\Lambda}{R^2}\n             -\\frac{P_R}R+(N^r\\Lambda)'.\n\\end{equation}\nUse $p_R=M/p$ and $v_R=M/(pR)-p/R^2$. The non-$F_{tr}$ terms cancel, leaving $\\dot\\Lambda=F_{tr}$. The momentum equations in the $N=1$ convention are\n\\begin{align}\n \\dot P_\\Lambda&=N^rP_\\Lambda'\n +\\frac12\\left[1-\\frac{P_\\Lambda^2+Q^2}{R^2}\n                 -\\left(\\frac{R'}\\Lambda\\right)^2\\right],\\\\\n \\dot P_R&=\\frac{\\Lambda(P_\\Lambda^2+Q^2)}{R^3}\n -\\frac{P_\\Lambda P_R}{R^2}+(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n\\end{align}\nSubstituting the data and differentiating $p(F)^2=2MF-Q^2$ reduces their right sides to\n\\begin{equation}\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n\\end{equation}\nwhich are the direct time derivatives of the two momenta. Finally,\n$\\Phi_e'=QF_r/F^2$ cancels $N\\Lambda Q/R^2$ in\n$\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'$.\n\\end{proof}","\\dot P_R&=\\frac{\\Lambda(P_\\Lambda^2+Q^2)}{R^3}\n -\\frac{P_\\Lambda P_R}{R^2}+(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n\\end{align}\nSubstituting the data and differentiating $p(F)^2=2MF-Q^2$ reduces their right sides to\n\\begin{equation}\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n\\end{equation}\nwhich are the direct time derivatives of the two momenta. Finally,\n$\\Phi_e'=QF_r/F^2$ cancels $N\\Lambda Q/R^2$ in\n$\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'$.\n\\end{proof}","ambda^2+Q^2)}{R^3}\n -\\frac{P_\\Lambda P_R}{R^2}+(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n\\end{align}\nSubstituting the data and differentiating $p(F)^2=2MF-Q^2$ reduces their right sides to\n\\begin{equation}\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n\\end{equation}\nwhich are the direct time derivatives of the two momenta. Finally,\n$\\Phi_e'=QF_r/F^2$ cancels $N\\Lambda Q/R^2$ in\n$\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'$.\n\\end{proof}","P_\\Lambda P_R}{R^2}+(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n\\end{align}\nSubstituting the data and differentiating $p(F)^2=2MF-Q^2$ reduces their right sides to\n\\begin{equation}\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n\\end{equation}\nwhich are the direct time derivatives of the two momenta. Finally,\n$\\Phi_e'=QF_r/F^2$ cancels $N\\Lambda Q/R^2$ in\n$\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'$.\n\\end{proof}","(N^rP_R)'\n -\\frac{R''}\\Lambda+\\frac{\\Lambda'R'}{\\Lambda^2}.\n\\end{align}\nSubstituting the data and differentiating $p(F)^2=2MF-Q^2$ reduces their right sides to\n\\begin{equation}\n \\frac{MF_t}{p(F)},\\qquad\n M\\left[\\frac{F_{tr}}{p(F)}\n       -\\frac{MF_rF_t}{p(F)^3}\\right],\n\\end{equation}\nwhich are the direct time derivatives of the two momenta. Finally,\n$\\Phi_e'=QF_r/F^2$ cancels $N\\Lambda Q/R^2$ in\n$\\dot A_r=N\\Lambda Q/R^2-\\Phi_e'$.\n\\end{proof}","An arbitrary admissible $F$ therefore creates coordinate motion through one fixed electrovac spacetime. Neither the mass nor the charge changes. A numerical reconstruction of this family is a nonlinear strong-field consistency test, but it is not matter collapse.","\\subsection{Charged horizons and the turning surface}\nWrite\n\\begin{equation}\n f(R)=1-\\frac{2M}R+\\frac{Q^2}{R^2}.\n\\end{equation}\nFor $0<|Q|<M$, its two positive roots are\n\\begin{equation}\n r_\\pm=M\\pm\\sqrt{M^2-Q^2}.\n\\end{equation}\nAt either root $p(R)^2=R^2$, so the canonical quantities in \\eqref{c:strong:RN-data} remain finite if $F_r$ does. For $|Q|=M$, the two roots coincide at $R=M$. For $|Q|>M$ there is no real root. At $Q=0$, the outer root is $2M$ and the formal inner root $0$ is the singular endpoint, not a second regular horizon.","and the turning surface}\nWrite\n\\begin{equation}\n f(R)=1-\\frac{2M}R+\\frac{Q^2}{R^2}.\n\\end{equation}\nFor $0<|Q|<M$, its two positive roots are\n\\begin{equation}\n r_\\pm=M\\pm\\sqrt{M^2-Q^2}.\n\\end{equation}\nAt either root $p(R)^2=R^2$, so the canonical quantities in \\eqref{c:strong:RN-data} remain finite if $F_r$ does. For $|Q|=M$, the two roots coincide at $R=M$. For $|Q|>M$ there is no real root. At $Q=0$, the outer root is $2M$ and the formal inner root $0$ is the singular endpoint, not a second regular horizon.","Write\n\\begin{equation}\n f(R)=1-\\frac{2M}R+\\frac{Q^2}{R^2}.\n\\end{equation}\nFor $0<|Q|<M$, its two positive roots are\n\\begin{equation}\n r_\\pm=M\\pm\\sqrt{M^2-Q^2}.\n\\end{equation}\nAt either root $p(R)^2=R^2$, so the canonical quantities in \\eqref{c:strong:RN-data} remain finite if $F_r$ does. For $|Q|=M$, the two roots coincide at $R=M$. For $|Q|>M$ there is no real root. At $Q=0$, the outer root is $2M$ and the formal inner root $0$ is the singular endpoint, not a second regular horizon.","-\\frac{2M}R+\\frac{Q^2}{R^2}.\n\\end{equation}\nFor $0<|Q|<M$, its two positive roots are\n\\begin{equation}\n r_\\pm=M\\pm\\sqrt{M^2-Q^2}.\n\\end{equation}\nAt either root $p(R)^2=R^2$, so the canonical quantities in \\eqref{c:strong:RN-data} remain finite if $F_r$ does. For $|Q|=M$, the two roots coincide at $R=M$. For $|Q|>M$ there is no real root. At $Q=0$, the outer root is $2M$ and the formal inner root $0$ is the singular endpoint, not a second regular horizon.","The square-root chart instead meets a turning surface at\n\\begin{equation}\n R_{\\rm turn}=\\frac{Q^2}{2M}.\n\\end{equation}\nThere $p=0$, and $P_R=MF_r/p$ becomes unbounded for nonzero $F_r$. This is a limitation of the chosen flat spatial slice and canonical chart. Horizon penetration does not remove the need for a positive distance from this turning surface. It also does not remove the physical curvature singularity at $R=0$.","t instead meets a turning surface at\n\\begin{equation}\n R_{\\rm turn}=\\frac{Q^2}{2M}.\n\\end{equation}\nThere $p=0$, and $P_R=MF_r/p$ becomes unbounded for nonzero $F_r$. This is a limitation of the chosen flat spatial slice and canonical chart. Horizon penetration does not remove the need for a positive distance from this turning surface. It also does not remove the physical curvature singularity at $R=0$.","The invariant curvature diagnostic is\n\\begin{equation}\n \\mathcal K=\\frac{48M^2}{R^6}-\\frac{96MQ^2}{R^7}\n                            +\\frac{56Q^4}{R^8}.\n\\end{equation}\nFor $M=1$, $Q=0.8$, the radii are $r_+=1.6$, $r_-=0.4$, and $R_{\\rm turn}=0.32$. The benchmark interval $R\\in[0.6,6]$ crosses the outer horizon only. At its inner end $\\mathcal K\\simeq199.6646853$, while $\\mathcal K(r_+)\\simeq1.106262207$. High curvature and regular horizon-crossing coordinates coexist on this interval.","\\subsection{Horizon conditioning}\nFor a subextremal outer horizon, put $d=\\sqrt{M^2-Q^2}>0$. Then\n\\begin{equation}\n f'(r_+)=\\frac{2d}{r_+^2}.\n\\end{equation}\nIf a nearby continuous numerical profile has a selected root near this simple root, the local inverse estimate gives\n\\begin{equation}\n |\\delta r_+|\\le\\frac{r_+^2}{2d}\\,\\norm{\\delta f}_\\infty\n                    +O(\\norm{\\delta f}_\\infty^2).\n\\label{c:strong:root-bound}\n\\end{equation}\nThe estimate describes the nearby root and presupposes its localization to a region where the exact derivative is nonzero. It does not make every zero of an arbitrarily oscillatory perturbation physically relevant. The factor is $2.133333333\\ldots$ in the benchmark and diverges as $d\\downarrow0$.","es\n\\begin{equation}\n |\\delta r_+|\\le\\frac{r_+^2}{2d}\\,\\norm{\\delta f}_\\infty\n                    +O(\\norm{\\delta f}_\\infty^2).\n\\label{c:strong:root-bound}\n\\end{equation}\nThe estimate describes the nearby root and presupposes its localization to a region where the exact derivative is nonzero. It does not make every zero of an arbitrarily oscillatory perturbation physically relevant. The factor is $2.133333333\\ldots$ in the benchmark and diverges as $d\\downarrow0$.",",\\norm{\\delta f}_\\infty\n                    +O(\\norm{\\delta f}_\\infty^2).\n\\label{c:strong:root-bound}\n\\end{equation}\nThe estimate describes the nearby root and presupposes its localization to a region where the exact derivative is nonzero. It does not make every zero of an arbitrarily oscillatory perturbation physically relevant. The factor is $2.133333333\\ldots$ in the benchmark and diverges as $d\\downarrow0$.","At extremality $f(R)=(1-M/R)^2$ has a double root. A perturbation of size $\\epsilon$ can move the root by order $\\sqrt\\epsilon$ or remove it entirely. Finite canonical data at a horizon therefore do not imply uniformly well-conditioned horizon detection. The first trapped sphere in the scalar calculation below also occurs at a spatial compactness maximum, and must not be assigned this simple-root estimate without checking its event geometry.","On interior points of a uniform radial grid, use the fourth-order derivative\n\\begin{equation}\n (D_4u)_j=\\frac{-u_{j+2}+8u_{j+1}-8u_{j-1}+u_{j-2}}{12h}.\n\\end{equation}\nIts stencil radius is two. Given $M$, $Q$, and a fixed branch $\\varepsilon$, a radius array satisfying $D_4R_j>0$ and $2MR_j-Q^2>0$ determines\n\\begin{equation}\n \\Pi_h[R]_j=\\left(R_j,D_4R_j,\n                 \\varepsilon\\sqrt{2MR_j-Q^2},\n                 \\frac{M D_4R_j}{\\varepsilon\\sqrt{2MR_j-Q^2}},Q\\right).\n\\label{c:strong:reconstruct}\n\\end{equation}\nThis is a local reconstruction of $(R,\\Lambda,P_\\Lambda,P_R,Q)$, rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining $R$, it is idempotent on its admissible image.","nd{equation}\nIts stencil radius is two. Given $M$, $Q$, and a fixed branch $\\varepsilon$, a radius array satisfying $D_4R_j>0$ and $2MR_j-Q^2>0$ determines\n\\begin{equation}\n \\Pi_h[R]_j=\\left(R_j,D_4R_j,\n                 \\varepsilon\\sqrt{2MR_j-Q^2},\n                 \\frac{M D_4R_j}{\\varepsilon\\sqrt{2MR_j-Q^2}},Q\\right).\n\\label{c:strong:reconstruct}\n\\end{equation}\nThis is a local reconstruction of $(R,\\Lambda,P_\\Lambda,P_R,Q)$, rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining $R$, it is idempotent on its admissible image.","quation}\n \\Pi_h[R]_j=\\left(R_j,D_4R_j,\n                 \\varepsilon\\sqrt{2MR_j-Q^2},\n                 \\frac{M D_4R_j}{\\varepsilon\\sqrt{2MR_j-Q^2}},Q\\right).\n\\label{c:strong:reconstruct}\n\\end{equation}\nThis is a local reconstruction of $(R,\\Lambda,P_\\Lambda,P_R,Q)$, rather than an orthogonal Hilbert-space projector. If it is extended to a full state by discarding the other variables and retaining $R$, it is idempotent on its admissible image.","\\Needspace{18\\baselineskip}\n\\begin{proposition}[Exact finite reduced constraints]\nOn the image of \\eqref{c:strong:reconstruct}, the four reduced constraints\n\\begin{equation}\n\\begin{aligned}\n C_{\\Lambda,j}&=\\Lambda_j-D_4R_j,&\n C_{M,j}&=P_{\\Lambda,j}^2-2MR_j+Q^2,\\\\\n C_{P,j}&=P_{R,j}P_{\\Lambda,j}-M\\Lambda_j,&\n C_{Q,j}&=Q_j-Q\n\\end{aligned}\n\\end{equation}\nvanish in exact arithmetic. The discrete mass coordinate obtained by replacing $R'$ with $D_4R$ in \\eqref{c:strong:mass-coordinate} equals $M$ at every reconstructed node.\n\\end{proposition}\n\\begin{proof}\nEach constraint vanishes by direct substitution. In the discrete mass coordinate $D_4R/\\Lambda=1$ and $P_\\Lambda^2=2MR-Q^2$, leaving\n$\\tfrac{R}{2}(2M/R-Q^2/R^2)+Q^2/(2R)=M$.\n\\end{proof}","C_{P,j}&=P_{R,j}P_{\\Lambda,j}-M\\Lambda_j,&\n C_{Q,j}&=Q_j-Q\n\\end{aligned}\n\\end{equation}\nvanish in exact arithmetic. The discrete mass coordinate obtained by replacing $R'$ with $D_4R$ in \\eqref{c:strong:mass-coordinate} equals $M$ at every reconstructed node.\n\\end{proposition}\n\\begin{proof}\nEach constraint vanishes by direct substitution. In the discrete mass coordinate $D_4R/\\Lambda=1$ and $P_\\Lambda^2=2MR-Q^2$, leaving\n$\\tfrac{R}{2}(2M/R-Q^2/R^2)+Q^2/(2R)=M$.\n\\end{proof}",",j}-M\\Lambda_j,&\n C_{Q,j}&=Q_j-Q\n\\end{aligned}\n\\end{equation}\nvanish in exact arithmetic. The discrete mass coordinate obtained by replacing $R'$ with $D_4R$ in \\eqref{c:strong:mass-coordinate} equals $M$ at every reconstructed node.\n\\end{proposition}\n\\begin{proof}\nEach constraint vanishes by direct substitution. In the discrete mass coordinate $D_4R/\\Lambda=1$ and $P_\\Lambda^2=2MR-Q^2$, leaving\n$\\tfrac{R}{2}(2M/R-Q^2/R^2)+Q^2/(2R)=M$.\n\\end{proof}","These identities are exact algebraic constraints on the reconstruction manifold. The original ADM expressions evaluated with finite differences have a different status: the stencil does not obey a pointwise chain rule. Thus $D_4\\sqrt{2MR-Q^2}$ need not equal $MD_4R/\\sqrt{2MR-Q^2}$ exactly. The original momentum and Hamiltonian constraints are retained as continuum-consistency diagnostics, rather than being relabeled as exact finite invariants.","\\subsection{A level-independent reconstruction estimate}\nAssume a uniform regular set\n\\begin{equation}\n R_j\\ge R_*>0,\\quad 2MR_j-Q^2\\ge p_*^2>0,\n \\quad 0<d_*\\le D_4R_j\\le d^*.\n\\end{equation}\nFor two arrays in this set, with the same $(M,Q,\\varepsilon)$, differentiation along the segment between them gives\n\\begin{align}\n \\norm{\\delta\\Lambda}_\\infty&\\le\\norm{D_4\\delta R}_\\infty,","\\subsection{A level-independent reconstruction estimate}\nAssume a uniform regular set\n\\begin{equation}\n R_j\\ge R_*>0,\\quad 2MR_j-Q^2\\ge p_*^2>0,\n \\quad 0<d_*\\le D_4R_j\\le d^*.\n\\end{equation}\nFor two arrays in this set, with the same $(M,Q,\\varepsilon)$, differentiation along the segment between them gives\n\\begin{align}\n \\norm{\\delta\\Lambda}_\\infty&\\le\\norm{D_4\\delta R}_\\infty,\\\\\n \\norm{\\delta P_\\Lambda}_\\infty&\\le\\frac M{p_*}\\norm{\\delta R}_\\infty,\\\\\n \\norm{\\delta P_R}_\\infty&\\le\\frac M{p_*}\\norm{D_4\\delta R}_\\infty\n                          +\\frac{M^2d^*}{p_*^3}\\norm{\\delta R}_\\infty.\n\\label{c:strong:projection-bound}\n\\end{align}\nFor example, $\\partial_Rp=M/p$ and the two partial derivatives of $M\\Lambda/p$ have magnitudes at most $M/p_*$ and $M^2d^*/p_*^3$. The admissible segment preserves the displayed linear radius and derivative bounds. The constants are independent of $h$.","\\norm{\\delta P_R}_\\infty&\\le\\frac M{p_*}\\norm{D_4\\delta R}_\\infty\n                          +\\frac{M^2d^*}{p_*^3}\\norm{\\delta R}_\\infty.\n\\label{c:strong:projection-bound}\n\\end{align}\nFor example, $\\partial_Rp=M/p$ and the two partial derivatives of $M\\Lambda/p$ have magnitudes at most $M/p_*$ and $M^2d^*/p_*^3$. The admissible segment preserves the displayed linear radius and derivative bounds. The constants are independent of $h$.","c M{p_*}\\norm{D_4\\delta R}_\\infty\n                          +\\frac{M^2d^*}{p_*^3}\\norm{\\delta R}_\\infty.\n\\label{c:strong:projection-bound}\n\\end{align}\nFor example, $\\partial_Rp=M/p$ and the two partial derivatives of $M\\Lambda/p$ have magnitudes at most $M/p_*$ and $M^2d^*/p_*^3$. The admissible segment preserves the displayed linear radius and derivative bounds. The constants are independent of $h$.","The input norm contains a radial derivative. A small pointwise radius error alone therefore does not establish a small error in every reconstructed canonical field. This estimate transfers an already controlled discrete $W^{1,\\infty}$ radius error to the reconstructed state. It does not create that derivative control from an $L^2$ estimate.","\\subsection{Direct and split radius evolution}\nOn the positive branch, restriction of the radius equation to the reconstruction manifold gives\n\\begin{equation}\n \\dot R_j=g(R_j)+b_j(D_4R)_j,\n \\qquad g(R)=-\\frac{\\sqrt{2MR-Q^2}}R,\n\\label{c:strong:direct-radius}\n\\end{equation}\nwhere $b=N^r$ is prescribed in the electrovac benchmark. A split alternative is\n\\begin{equation}\n \\dot R=g(R)+A_h[b]R,\\qquad\n A_h[b]u=\\frac12\\bigl(bD_4u+D_4(bu)-(D_4b)u\\bigr).\n\\label{c:strong:split-radius}\n\\end{equation}\nOn smooth data these formulas differ by half of the discrete product-rule defect, which is fourth order away from boundaries. Their exact finite-grid operators differ.","The archived implementation advances the radius explicitly with RK4. The map $\\Pi_h$ can reconstruct the other fields from each admissible stage radius, and its reduced identities hold on that reconstructed image. At output, however, the code overwrites two canonical boundary nodes on each side with exact manufactured data. Those boundary nodes are not generally the exact image of the finite derivative reconstruction. Its reported constraint norms exclude four nodes from each end. This procedure does not use an implicit discrete-gradient slab solve. Conservation by an implicit invariant-preserving method elsewhere in this monograph remains conditional on actually solving its nonlinear step equation. It cannot be used as evidence that this explicit radius algorithm preserve","\\begin{proposition}[Split-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives","\\begin{proposition}[Split-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the","\\begin{proposition}[Split-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\wid","\\begin{proposition}[Split-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_","\\begin{proposition}[Split-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proo","it-form energy identity]\nLet $D$ be skew adjoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","djoint for the grid inner product $\\inner{u}{w}_h$, and let multiplication by the real array $b$ be self-adjoint. Then\n\\begin{equation}\n \\inner{u}{A_h[b]u}_h\n       =-\\frac12\\inner{u}{(Db)u}_h.\n\\label{c:strong:split-energy}\n\\end{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","nd{equation}\nFor two solutions of the split equation with the same prescribed $b$, remaining in a positive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","sitive-radicand interval where $|g'|\\le L_g$, one obtains\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","ns\n\\begin{equation}\n \\norm{R(t)-\\widetilde R(t)}_h\n \\le e^{(L_g+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","+S_1/2)t}\\norm{R(0)-\\widetilde R(0)}_h,\n \\qquad \\norm{Db}_\\infty\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","y\\le S_1.\n\\label{c:strong:semidiscrete-bound}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nSkew adjointness cancels $\\inner{u}{bDu}_h+\\inner{u}{D(bu)}_h$. The remaining term gives \\eqref{c:strong:split-energy}. With $e=R-\\widetilde R$, the nonlinear difference satisfies\n$|g(R_j)-g(\\widetilde R_j)|\\le L_g|e_j|$, where\n\\begin{equation}\n g'(R)=\\frac{MR-Q^2}{R^2\\sqrt{2MR-Q^2}}.\n\\end{equation}\nThus $\\tfrac12\\partial_t\\norm e_h^2\\le(L_g+S_1/2)\\norm e_h^2$, and integration gives the claim.\n\\end{proof}","The skew-adjoint hypothesis needs care on a physical radial interval. A periodic $D_4$ satisfies it, but a globally periodic radius array cannot also satisfy $D_4R_j>0$ everywhere: summing the derivative over one period gives zero. The periodic identity is an exact algebraic check of the split stencil. It does not by itself give a nonvacuous stability theorem for the globally monotone spherical reconstruction.","For an interval summation-by-parts derivative with a diagonal positive norm matrix $H$ and $HD+D^TH=B$, multiplication by $b$ commutes with $H$. The uncancelled term is instead\n\\begin{equation}\n \\inner{u}{A_h[b]u}_H\n =\\frac12u^TB\\,\\operatorname{diag}(b)u\n  -\\frac12\\inner{u}{(Db)u}_H.\n\\end{equation}\nIts boundary flux must be controlled using an actual boundary closure and appropriate boundary data. Saying that a stencil is centered in the interior does not establish this property for the complete interval operator.","The periodic Fourier symbol is\n\\begin{equation}\n \\widehat D_4(\\theta)=\\frac{i}{h}\\frac{8\\sin\\theta-\\sin2\\theta}{6}.\n\\end{equation}\nIts maximum dimensionless magnitude is $1.3722219798\\ldots$. Applying the RK4 imaginary-axis stability interval to frozen constant advection gives\n\\begin{equation}\n \\frac{|b|\\Delta t}{h}\\le2.0612023174\\ldots.\n\\end{equation}\nThis is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requires a uniform stability argument for the implemented boundary closure and a suitable stronger norm. Fourth-order consistency, the semidiscrete $L^2$ estimate, and this one CFL number do not alone prove fourth-order $W^{1,\\infty}$ convergence.","inary-axis stability interval to frozen constant advection gives\n\\begin{equation}\n \\frac{|b|\\Delta t}{h}\\le2.0612023174\\ldots.\n\\end{equation}\nThis is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requires a uniform stability argument for the implemented boundary closure and a suitable stronger norm. Fourth-order consistency, the semidiscrete $L^2$ estimate, and this one CFL number do not alone prove fourth-order $W^{1,\\infty}$ convergence.","equation}\n \\frac{|b|\\Delta t}{h}\\le2.0612023174\\ldots.\n\\end{equation}\nThis is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requires a uniform stability argument for the implemented boundary closure and a suitable stronger norm. Fourth-order consistency, the semidiscrete $L^2$ estimate, and this one CFL number do not alone prove fourth-order $W^{1,\\infty}$ convergence.","The retained run uses\n\\begin{equation}\n F(t,r)=r+0.08\\sin(0.7t)\n                \\sin^2\\!\\left(\\pi\\frac{r-0.6}{5.4}\\right),\n \\quad 0\\le t\\le0.4,\n\\end{equation}\nwith $M=1$, $Q=0.8$, and $\\Delta t/h$ close to $0.1$. The sampled minimum radial Jacobian is $0.9871380683$ and the minimum momentum radicand is $0.56$. The exact shift, ghost values, and two boundary nodes on either side are supplied from the manufactured family. Errors are measured on interior nodes after excluding four points from each end.","The recorded state errors are\n\\begin{center}\\small\n\\begin{tabular}{rrrrr}\n\\toprule\n$N$&$\\norm{\\delta R}_\\infty$&$\\norm{\\delta\\Lambda}_\\infty$&$\\norm{\\delta P_\\Lambda}_\\infty$&$\\norm{\\delta P_R}_\\infty$","\\midrule\n64&$8.3247\\,10^{-9}$&$4.1362\\,10^{-8}$&$5.7367\\,10^{-9}$&$2.3732\\,10^{-8}$","128&$5.0495\\,10^{-10}$&$2.5079\\,10^{-9}$&$3.3079\\,10^{-10}$&$1.3795\\,10^{-9}$","256&$3.1077\\,10^{-11}$&$1.5438\\,10^{-10}$&$2.0248\\,10^{-11}$&$8.4854\\,10^{-11}$","512&$1.9296\\,10^{-12}$&$1.0485\\,10^{-11}$&$1.2559\\,10^{-12}$&$5.3844\\,10^{-12}$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe least-squares orders are respectively $3.99856$, $3.96010$, $4.02393$, and $4.00788$. The ordinary ADM Hamiltonian on exact projected data has fitted order $3.70410$. These are results for a smooth, forced trajectory on four grids. The tiny final errors are close enough to roundoff that they should not be interpreted as arbitrarily extendible asymptotic scaling laws.","On the tested interior of the $N=512$ reconstructed grid, the retained mass residual is $4.44\\,10^{-16}$, the flat-slice residual is zero, the algebraic mass constraint is $2.22\\,10^{-15}$, and the reduced momentum residual is $1.11\\,10^{-16}$. These floating-point numbers are consistent with the exact algebraic reconstruction theorem. They are not an exact-zero assertion about floating arithmetic or the overwritten boundary nodes.","The negative control evolves all four canonical fields without the reconstruction. At $N=128$, it reports mass drift $2.25125\\,10^{-4}$, radial momentum residual $1.27688\\,10^{-3}$, and Hamiltonian residual $1.25735\\,10^{-2}$. This gives the reduced reconstruction a meaningful comparator. It does not show that every other constrained algorithm fails or that the direct ADM system is ill posed.","A smooth perturbation of amplitude $10^{-8}$ evolved with the split form has recorded $L^2$ amplification between $1.03345462$ and $1.03348106$ on $N=128,256,512,1024$. The spread is $2.64438\\,10^{-5}$. This supports the bounded experiment's level-insensitive behavior. It leaves the interval boundary and stronger-norm theorem obligations visible.","The outer horizon remains at the fixed areal radius $1.6$. Its computational coordinate moves because $F$ changes. The retained final coordinate comparison at $N=256$ gives $|r_{+,h}-r_+^{\\rm exact}|=1.89262\\,10^{-11}$. This measures tracking of a known horizon under coordinate deformation. It does not measure the formation or growth of a black hole.","To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use $G=c=1$ and the rescaled field $\\varphi=\\sqrt{4\\pi}\\,\\phi_{\\rm phys}$. The scalar canonical pair is $(\\varphi,P_\\varphi)$, and the matter additions to the Hamiltonian and momentum constraints are\n\\begin{equation}\n \\mathcal H_{\\rm matt}=\\frac{P_\\varphi^2}{2\\Lambda R^2}\n                         +\\frac{R^2\\varphi'^2}{2\\Lambda},\n \\qquad \\mathcal D_{\\rm matt}=P_\\varphi\\varphi'.\n\\end{equation}\nThe gravitational part is \\eqref{c:strong:electrovac-H} with $Q=0$.","\\phi_{\\rm phys}$. The scalar canonical pair is $(\\varphi,P_\\varphi)$, and the matter additions to the Hamiltonian and momentum constraints are\n\\begin{equation}\n \\mathcal H_{\\rm matt}=\\frac{P_\\varphi^2}{2\\Lambda R^2}\n                         +\\frac{R^2\\varphi'^2}{2\\Lambda},\n \\qquad \\mathcal D_{\\rm matt}=P_\\varphi\\varphi'.\n\\end{equation}\nThe gravitational part is \\eqref{c:strong:electrovac-H} with $Q=0$.","Impose $R=r$ and $\\Lambda=1$, and set\n\\begin{equation}\n N=\\sigma>0,\\quad N^r=\\sigma v,\\quad\n P_\\Lambda=rv,\\quad P_\\varphi=r^2\\Pi,\\quad \\Phi=\\varphi'.\n\\end{equation}\nHere $\\Pi$ is the scalar momentum per areal area, not the reconstruction map $\\Pi_h$. Define\n\\begin{equation}\n m=\\frac r2v^2,\\qquad v=\\sqrt{\\frac{2m}{r}}.\n\\end{equation}\nThe inward branch gives\n\\begin{equation}\n ds^2=-\\sigma^2dt^2+(dr+\\sigma vdt)^2+r^2d\\Omega^2.\n\\label{c:strong:scalar-metric}\n\\end{equation}\nThe following derivation applies on smooth patches with $r>0$, $\\sigma>0$, and $v>0$. These hypotheses keep every division legitimate.","uad \\Phi=\\varphi'.\n\\end{equation}\nHere $\\Pi$ is the scalar momentum per areal area, not the reconstruction map $\\Pi_h$. Define\n\\begin{equation}\n m=\\frac r2v^2,\\qquad v=\\sqrt{\\frac{2m}{r}}.\n\\end{equation}\nThe inward branch gives\n\\begin{equation}\n ds^2=-\\sigma^2dt^2+(dr+\\sigma vdt)^2+r^2d\\Omega^2.\n\\label{c:strong:scalar-metric}\n\\end{equation}\nThe following derivation applies on smooth patches with $r>0$, $\\sigma>0$, and $v>0$. These hypotheses keep every division legitimate.","i=\\varphi'.\n\\end{equation}\nHere $\\Pi$ is the scalar momentum per areal area, not the reconstruction map $\\Pi_h$. Define\n\\begin{equation}\n m=\\frac r2v^2,\\qquad v=\\sqrt{\\frac{2m}{r}}.\n\\end{equation}\nThe inward branch gives\n\\begin{equation}\n ds^2=-\\sigma^2dt^2+(dr+\\sigma vdt)^2+r^2d\\Omega^2.\n\\label{c:strong:scalar-metric}\n\\end{equation}\nThe following derivation applies on smooth patches with $r>0$, $\\sigma>0$, and $v>0$. These hypotheses keep every division legitimate.","{equation}\nHere $\\Pi$ is the scalar momentum per areal area, not the reconstruction map $\\Pi_h$. Define\n\\begin{equation}\n m=\\frac r2v^2,\\qquad v=\\sqrt{\\frac{2m}{r}}.\n\\end{equation}\nThe inward branch gives\n\\begin{equation}\n ds^2=-\\sigma^2dt^2+(dr+\\sigma vdt)^2+r^2d\\Omega^2.\n\\label{c:strong:scalar-metric}\n\\end{equation}\nThe following derivation applies on smooth patches with $r>0$, $\\sigma>0$, and $v>0$. These hypotheses keep every division legitimate.","The momentum constraint removes $P_R$ algebraically:\n\\begin{equation}\n P_R=P_\\Lambda'-P_\\varphi\\Phi.\n\\end{equation}\nSince $m'=P_\\Lambda P_\\Lambda'/r-P_\\Lambda^2/(2r^2)$, inserting this relation into the Hamiltonian constraint gives\n\\begin{equation}\n m'=r^2\\mathcal E,\\qquad\n \\mathcal E=\\frac12(\\Pi^2+\\Phi^2)+v\\Pi\\Phi.\n\\label{c:strong:scalar-mass}\n\\end{equation}\nPreservation of $\\Lambda=1$ gives\n\\begin{equation}\n 0=\\sigma\\left(\\frac vr-\\frac{P_R}r\\right)+(\\sigma v)'\n   =\\sigma r\\Pi\\Phi+\\sigma'v,\n\\end{equation}\nhence\n\\begin{equation}\n (\\log\\sigma)'=-\\frac{r\\Pi\\Phi}{v}.\n\\label{c:strong:scalar-lapse}\n\\end{equation}\nThe scalar evolution equations are\n\\begin{align}\n \\dot\\varphi&=\\sigma(\\Pi+v\\Phi),","constraint removes $P_R$ algebraically:\n\\begin{equation}\n P_R=P_\\Lambda'-P_\\varphi\\Phi.\n\\end{equation}\nSince $m'=P_\\Lambda P_\\Lambda'/r-P_\\Lambda^2/(2r^2)$, inserting this relation into the Hamiltonian constraint gives\n\\begin{equation}\n m'=r^2\\mathcal E,\\qquad\n \\mathcal E=\\frac12(\\Pi^2+\\Phi^2)+v\\Pi\\Phi.\n\\label{c:strong:scalar-mass}\n\\end{equation}\nPreservation of $\\Lambda=1$ gives\n\\begin{equation}\n 0=\\sigma\\left(\\frac vr-\\frac{P_R}r\\right)+(\\sigma v)'\n   =\\sigma r\\Pi\\Phi+\\sigma'v,\n\\end{equation}\nhence\n\\begin{equation}\n (\\log\\sigma)'=-\\frac{r\\Pi\\Phi}{v}.\n\\label{c:strong:scalar-lapse}\n\\end{equation}\nThe scalar evolution equations are\n\\begin{align}\n \\dot\\varphi&=\\sigma(\\Pi+v\\Phi),","P_\\Lambda'/r-P_\\Lambda^2/(2r^2)$, inserting this relation into the Hamiltonian constraint gives\n\\begin{equation}\n m'=r^2\\mathcal E,\\qquad\n \\mathcal E=\\frac12(\\Pi^2+\\Phi^2)+v\\Pi\\Phi.\n\\label{c:strong:scalar-mass}\n\\end{equation}\nPreservation of $\\Lambda=1$ gives\n\\begin{equation}\n 0=\\sigma\\left(\\frac vr-\\frac{P_R}r\\right)+(\\sigma v)'\n   =\\sigma r\\Pi\\Phi+\\sigma'v,\n\\end{equation}\nhence\n\\begin{equation}\n (\\log\\sigma)'=-\\frac{r\\Pi\\Phi}{v}.\n\\label{c:strong:scalar-lapse}\n\\end{equation}\nThe scalar evolution equations are\n\\begin{align}\n \\dot\\varphi&=\\sigma(\\Pi+v\\Phi),","E,\\qquad\n \\mathcal E=\\frac12(\\Pi^2+\\Phi^2)+v\\Pi\\Phi.\n\\label{c:strong:scalar-mass}\n\\end{equation}\nPreservation of $\\Lambda=1$ gives\n\\begin{equation}\n 0=\\sigma\\left(\\frac vr-\\frac{P_R}r\\right)+(\\sigma v)'\n   =\\sigma r\\Pi\\Phi+\\sigma'v,\n\\end{equation}\nhence\n\\begin{equation}\n (\\log\\sigma)'=-\\frac{r\\Pi\\Phi}{v}.\n\\label{c:strong:scalar-lapse}\n\\end{equation}\nThe scalar evolution equations are\n\\begin{align}\n \\dot\\varphi&=\\sigma(\\Pi+v\\Phi),\\\\\n \\dot\\Phi&=\\partial_r[\\sigma(\\Pi+v\\Phi)],\\\\\n \\dot\\Pi&=r^{-2}\\partial_r[r^2\\sigma(\\Phi+v\\Pi)].\n\\label{c:strong:scalar-evolution}\n\\end{align}\nIf $\\Phi=\\varphi'$ initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","\\dot\\Pi&=r^{-2}\\partial_r[r^2\\sigma(\\Phi+v\\Pi)].\n\\label{c:strong:scalar-evolution}\n\\end{align}\nIf $\\Phi=\\varphi'$ initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","At $v=0$, the lapse equation requires a separately specified extension. Smoothness of $\\Pi$ and $\\Phi$ alone does not establish that $r\\Pi\\Phi/v$ is removable. A vacuum interval with vanishing scalar data can provide an appropriate extension, but that must be part of the argument. The Gaussian benchmark has very small tails rather than exact compact support, and its numerical treatment of small $v$ is described below. This calculation does not prove a regular-center theorem.","\\subsection{Exact mass flux and compatibility}\nPut $A=\\Pi+v\\Phi$ and $B=\\Phi+v\\Pi$. The remaining Einstein momentum evolution gives\n\\begin{equation}\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\P","\\subsection{Exact mass flux and compatibility}\nPut $A=\\Pi+v\\Phi$ and $B=\\Phi+v\\Pi$. The remaining Einstein momentum evolution gives\n\\begin{equation}\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\P","\\subsection{Exact mass flux and compatibility}\nPut $A=\\Pi+v\\Phi$ and $B=\\Phi+v\\Pi$. The remaining Einstein momentum evolution gives\n\\begin{equation}\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]$, which is \\eqref{c:strong:scalar-flux}.","i$. The remaining Einstein momentum evolution gives\n\\begin{equation}\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]$, which is \\eqref{c:strong:scalar-flux}.","um evolution gives\n\\begin{equation}\n \\dot m=r^2\\mathcal J,\\qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]$, which is \\eqref{c:strong:scalar-flux}.","qquad \\mathcal J=\\sigma AB.\n\\label{c:strong:scalar-flux}\n\\end{equation}\nOne can verify its origin directly from the $P_\\Lambda$ equation. In the flat-slice gauge, the gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]$, which is \\eqref{c:strong:scalar-flux}.","he gravitational terms give $\\sigma vP_\\Lambda'-\\sigma v^2/2-r\\sigma'$. The lapse-gradient term is essential. The matter variation contributes $\\sigma r^2(\\Pi^2+\\Phi^2)/2$. As $\\dot m=v\\dot P_\\Lambda$, their sum becomes\n\\begin{equation}\n \\dot m=\\sigma v\\left[m'+\\frac{r^2}{2}(\\Pi^2+\\Phi^2)\\right]-rv\\sigma'.\n\\end{equation}\nUsing \\eqref{c:strong:scalar-mass} and \\eqref{c:strong:scalar-lapse} yields\n$\\dot m=r^2\\sigma[v(\\Pi^2+\\Phi^2)+(1+v^2)\\Pi\\Phi]$, which is \\eqref{c:strong:scalar-flux}.","\\begin{proposition}[Continuum mass-constraint propagation]\nOn the regular patch, the scalar evolution, lapse relation, and mass-flux evolution satisfy\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n\\end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'","\\begin{proposition}[Continuum mass-constraint propagation]\nOn the regular patch, the scalar evolution, lapse relation, and mass-flux evolution satisfy\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n\\end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}","relation, and mass-flux evolution satisfy\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n\\end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","olution satisfy\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n\\end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","mathcal E)=\\partial_r(r^2\\mathcal J).\n\\label{c:strong:scalar-compatibility}\n\\end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","end{equation}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","tion}\nThus $\\partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","partial_t(m'-r^2\\mathcal E)=0$. A radial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","dial mass constraint imposed initially propagates under this continuum system.\n\\end{proposition}\n\\Needspace{20\\baselineskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","eskip}\n\\begin{proof}\nExpanding \\eqref{c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","c:strong:scalar-evolution} gives\n$\\dot\\Phi=(\\sigma A)'$ and $\\dot\\Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","Pi=(\\sigma B)'+2\\sigma B/r$.\nDifferentiate $\\mathcal E$ and subtract the radial derivative of the flux. Terms involving $\\Pi'$, $\\Phi'$, and $v'$ cancel, leaving\n\\begin{equation}\n \\partial_t(r^2\\mathcal E)-\\partial_r(r^2\\mathcal J)\n =r^2\\bigl[\\sigma'AB+\\Pi\\Phi\\dot v\\bigr].\n\\end{equation}\nSince $m=rv^2/2$, the flux equation implies $\\dot v=r\\sigma AB/v$. The remaining expression is\n\\begin{equation}\n r^2AB\\left[\\sigma'+\\frac{\\sigma r\\Pi\\Phi}{v}\\right]=0\n\\end{equation}\nby \\eqref{c:strong:scalar-lapse}. This establishes the identity without assuming the radial constraint in its last cancellation step.\n\\end{proof}","On a fixed radial interval $[r_i,r_o]$, the exact balance is\n\\begin{equation}\n \\frac{d}{dt}\\bigl(m(t,r_o)-m(t,r_i)\\bigr)\n =r_o^2\\mathcal J(t,r_o)-r_i^2\\mathcal J(t,r_i).\n\\end{equation}\nBoundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid and time step.","\\subsection{Characteristic channels and the loss of a coordinate energy bound}\nDefine $w_+=\\Pi+\\Phi$ and $w_-=\\Pi-\\Phi$. Their equations are\n\\begin{align}\n \\dot w_+={}&\\sigma(v+1)w_+'\n +[\\sigma v'+\\sigma'(v+1)]w_+\n +\\frac\\sigma r[(v+1)w_++(v-1)w_-],","\\subsection{Characteristic channels and the loss of a coordinate energy bound}\nDefine $w_+=\\Pi+\\Phi$ and $w_-=\\Pi-\\Phi$. Their equations are\n\\begin{align}\n \\dot w_+={}&\\sigma(v+1)w_+'\n +[\\sigma v'+\\sigma'(v+1)]w_+\n +\\frac\\sigma r[(v+1)w_++(v-1)w_-],\\\\\n \\dot w_-={}&\\sigma(v-1)w_-'\n +[\\sigma v'+\\sigma'(v-1)]w_-\n +\\frac\\sigma r[(v-1)w_-+(v+1)w_+].\n\\end{align}\nWritten as transport equations $\\dot w+c\\,w'=\\text{source}$, their coordinate speeds are\n\\begin{equation}\n c_+=-\\sigma(v+1),\\qquad c_-=\\sigma(1-v).\n\\label{c:strong:characteristics}\n\\end{equation}\nThe plus channel is inward on the positive branch. The minus channel is outward when $v<1$, stationary at $v=1$, and inward when $v>1$.","\\dot w_-={}&\\sigma(v-1)w_-'\n +[\\sigma v'+\\sigma'(v-1)]w_-\n +\\frac\\sigma r[(v-1)w_-+(v+1)w_+].\n\\end{align}\nWritten as transport equations $\\dot w+c\\,w'=\\text{source}$, their coordinate speeds are\n\\begin{equation}\n c_+=-\\sigma(v+1),\\qquad c_-=\\sigma(1-v).\n\\label{c:strong:characteristics}\n\\end{equation}\nThe plus channel is inward on the positive branch. The minus channel is outward when $v<1$, stationary at $v=1$, and inward when $v>1$.","{}&\\sigma(v-1)w_-'\n +[\\sigma v'+\\sigma'(v-1)]w_-\n +\\frac\\sigma r[(v-1)w_-+(v+1)w_+].\n\\end{align}\nWritten as transport equations $\\dot w+c\\,w'=\\text{source}$, their coordinate speeds are\n\\begin{equation}\n c_+=-\\sigma(v+1),\\qquad c_-=\\sigma(1-v).\n\\label{c:strong:characteristics}\n\\end{equation}\nThe plus channel is inward on the positive branch. The minus channel is outward when $v<1$, stationary at $v=1$, and inward when $v>1$.","sigma v'+\\sigma'(v-1)]w_-\n +\\frac\\sigma r[(v-1)w_-+(v+1)w_+].\n\\end{align}\nWritten as transport equations $\\dot w+c\\,w'=\\text{source}$, their coordinate speeds are\n\\begin{equation}\n c_+=-\\sigma(v+1),\\qquad c_-=\\sigma(1-v).\n\\label{c:strong:characteristics}\n\\end{equation}\nThe plus channel is inward on the positive branch. The minus channel is outward when $v<1$, stationary at $v=1$, and inward when $v>1$.","The same channels give the exact algebraic identities\n\\begin{align}\n \\mathcal E&=\\frac14[(1+v)w_+^2+(1-v)w_-^2],\\\\\n \\mathcal J&=\\frac\\sigma4[(1+v)^2w_+^2-(1-v)^2w_-^2].\n\\label{c:strong:null-energy}\n\\end{align}\nFor $0\\le v<1$, $\\mathcal E\\ge(1-v)(w_+^2+w_-^2)/4$. This lower bound degenerates at a horizon. For $v>1$, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density $(\\Pi^2+\\Phi^2)/2$ remains nonnegative. The loss of coercivity in $\\mathcal E$ therefore diagnoses this slicing's mass-coordinate energy estimate, rather than negative physical scalar energy.","\\mathcal J&=\\frac\\sigma4[(1+v)^2w_+^2-(1-v)^2w_-^2].\n\\label{c:strong:null-energy}\n\\end{align}\nFor $0\\le v<1$, $\\mathcal E\\ge(1-v)(w_+^2+w_-^2)/4$. This lower bound degenerates at a horizon. For $v>1$, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density $(\\Pi^2+\\Phi^2)/2$ remains nonnegative. The loss of coercivity in $\\mathcal E$ therefore diagnoses this slicing's mass-coordinate energy estimate, rather than negative physical scalar energy.","\\gravityfigure{11_scalar_flux}{The mass density and mass flux resolve into the two scalar characteristic channels. The outgoing characteristic and its contribution to the coordinate mass density degenerate at $v=1$. These are exact algebraic profiles, not a newly evolved collapse solution.}","For \\eqref{c:strong:scalar-metric}, choose future radial null vectors\n\\begin{equation}\n \\ell_\\pm=\\sigma^{-1}\\partial_t+(-v\\pm1)\\partial_r.\n\\end{equation}\nBecause the sphere area is $4\\pi r^2$, its expansion along $\\ell$ is $2\\ell(r)/r$. Consequently\n\\begin{equation}\n \\theta_+=\\frac2r(1-v),\\qquad\n \\theta_-=-\\frac2r(1+v).\n\\label{c:strong:expansions}\n\\end{equation}\nOn the positive branch, $v=1$ is a marginally outer trapped sphere, and $v>1$ gives both expansions negative. Equivalently, the compactness $2m/r$ equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric diagnostic. It is not by itself identification of a global event horizon.","oose future radial null vectors\n\\begin{equation}\n \\ell_\\pm=\\sigma^{-1}\\partial_t+(-v\\pm1)\\partial_r.\n\\end{equation}\nBecause the sphere area is $4\\pi r^2$, its expansion along $\\ell$ is $2\\ell(r)/r$. Consequently\n\\begin{equation}\n \\theta_+=\\frac2r(1-v),\\qquad\n \\theta_-=-\\frac2r(1+v).\n\\label{c:strong:expansions}\n\\end{equation}\nOn the positive branch, $v=1$ is a marginally outer trapped sphere, and $v>1$ gives both expansions negative. Equivalently, the compactness $2m/r$ equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric diagnostic. It is not by itself identification of a global event horizon.","1}\\partial_t+(-v\\pm1)\\partial_r.\n\\end{equation}\nBecause the sphere area is $4\\pi r^2$, its expansion along $\\ell$ is $2\\ell(r)/r$. Consequently\n\\begin{equation}\n \\theta_+=\\frac2r(1-v),\\qquad\n \\theta_-=-\\frac2r(1+v).\n\\label{c:strong:expansions}\n\\end{equation}\nOn the positive branch, $v=1$ is a marginally outer trapped sphere, and $v>1$ gives both expansions negative. Equivalently, the compactness $2m/r$ equals or exceeds one. The normalization of the null vectors affects expansion magnitude but not these signs. A marginally trapped sphere is a local geometric diagnostic. It is not by itself identification of a global event horizon.","\\subsection{What is local in the algorithm}\nAt each RK4 stage the matter channels determine the geometry through two radial solves. The mass constraint is integrated outward with $m(r_{\\rm exc})=0$. The lapse derivative is integrated and the result normalized by $\\sigma(r_{\\max})=1$. These are triangular prefix operations. A matter value near the inner end can affect the reconstructed mass and lapse far along the interval in the same stage. The full stage map is therefore globally coupled.","The characteristic evolution uses the second-order one-sided derivatives\n\\begin{equation}\n (D_+u)_j=\\frac{-3u_j+4u_{j+1}-u_{j+2}}{2h},\\qquad\n (D_-u)_j=\\frac{3u_j-4u_{j-1}+u_{j-2}}{2h}.\n\\end{equation}\nThe inward channel uses $D_+$ and the outgoing channel uses $D_-$ outside a horizon, switching to $D_+$ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interior derivative for $v'$ and separate one-sided boundary formulas. Finite stencil radius for these derivatives must not be promoted to a finite dependency cone for the geometry-coupled step.","s the second-order one-sided derivatives\n\\begin{equation}\n (D_+u)_j=\\frac{-3u_j+4u_{j+1}-u_{j+2}}{2h},\\qquad\n (D_-u)_j=\\frac{3u_j-4u_{j-1}+u_{j-2}}{2h}.\n\\end{equation}\nThe inward channel uses $D_+$ and the outgoing channel uses $D_-$ outside a horizon, switching to $D_+$ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interior derivative for $v'$ and separate one-sided boundary formulas. Finite stencil radius for these derivatives must not be promoted to a finite dependency cone for the geometry-coupled step.","Several implementation conventions are mathematically material. The mass right side and the reconstructed $v$ use $\\sqrt{\\max(2m/r,0)}$. The lapse derivative is set to zero where $v\\le10^{-13}$. Incoming characteristic values are explicitly set after a completed RK4 step, with separate boundary formulas during stages. These choices guard a finite calculation near vacuum and boundaries. An exact continuum proof on $v>0$ does not establish their uniform consistency as the threshold is varied or $h\\downarrow0$.","tion conventions are mathematically material. The mass right side and the reconstructed $v$ use $\\sqrt{\\max(2m/r,0)}$. The lapse derivative is set to zero where $v\\le10^{-13}$. Incoming characteristic values are explicitly set after a completed RK4 step, with separate boundary formulas during stages. These choices guard a finite calculation near vacuum and boundaries. An exact continuum proof on $v>0$ does not establish their uniform consistency as the threshold is varied or $h\\downarrow0$.","ructed $v$ use $\\sqrt{\\max(2m/r,0)}$. The lapse derivative is set to zero where $v\\le10^{-13}$. Incoming characteristic values are explicitly set after a completed RK4 step, with separate boundary formulas during stages. These choices guard a finite calculation near vacuum and boundaries. An exact continuum proof on $v>0$ does not establish their uniform consistency as the threshold is varied or $h\\downarrow0$.","\\subsection{Initial data and boundary scope}\nThe numerical domain is $[r_{\\rm exc},r_{\\max}]=[0.05,30]$. It replaces a regular center by a small excised core. The characteristic data are $w_-(r_{\\rm exc})=0$ and $w_+(r_{\\max})=0$. These prohibit prescribed incoming radiation from the inner and outer boundaries in the respective channels. The radial mass boundary condition additionally fixes the excised mass to zero at every geometry solve.","The initial scalar packet is\n\\begin{equation}\n \\varphi(0,r)=A e^{-(r-10)^2},\\qquad\n \\Phi(0,r)=\\partial_r\\varphi(0,r),\\qquad\n \\Pi(0,r)=\\Phi(0,r)+\\frac{\\varphi(0,r)}r.\n\\end{equation}\nThe last equation is the flat-space ingoing condition for $\\chi=r\\varphi$, used to initialize the scalar. The nonlinear gravitational constraints are then solved. It is therefore an ingoing packet prescription, rather than exactly freezing the geometry to Minkowski space.","The strong amplitude is $A=0.10$ and the control amplitude is $A=0.05$. The strong event occurs near $r=3.1$, well separated from both boundaries in coordinate distance. That separation is useful but is not a substitute for studying boundary influence: the geometry solve is global, and finite-domain mass drift can contain both boundary and discretization effects.","\\subsection{Recorded first marginally trapped sphere}\nThe event finder takes a quadratic interpolation of the spatial maximum of $2m/r$ and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is\n\\begin{center}\\small\n\\begin{tabular}{rrrrr}\n\\toprule\n$N_r$&$h$&$t_H$&$r_H$&$\\Delta M/M_0$","\\midrule\n600&0.0500000&4.7133579&3.0518959&$-1.1205\\,10^{-2}$","800&0.0374844&4.7162491&3.0506543&$-1.6427\\,10^{-2}$","1000&0.0299800&4.7059219&3.0766099&$-1.0696\\,10^{-2}$","1200&0.0249791&4.7004729&3.0905271&$-7.6261\\,10^{-3}$","1600&0.0187305&4.6952990&3.1040860&$-4.6604\\,10^{-3}$","2000&0.0149825&4.6930194&3.1101678&$-3.3322\\,10^{-3}$","2400&0.0124844&4.6918209&3.1134115&$-2.6241\\,10^{-3}$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe coarsest two event coordinates are not monotone with refinement. On the six finest resolutions the retained nonlinear fits are\n\\begin{align}\n t_H(h)&\\simeq4.6893522370+33.88577338\\,h^{2.173853282},\\\\\n r_H(h)&\\simeq3.1204965744-65.59310603\\,h^{2.084394965}.\n\\end{align}\nThese fit exponents are consistent with second-order event convergence on this bounded family. They are not a convergence proof, a reliable error bar for every observable, or a critical-collapse exponent.","For $N_r=2400$, the interpolated event has\n\\begin{equation}\n t_H=4.69182094580,\\quad r_H=3.11341154797,\n \\quad m_H=r_H/2=1.55670577399.\n\\end{equation}\nThe relation $m_H=r_H/2$ is imposed by the interpolated marginal-sphere criterion. It is not an independent numerical conservation test.","The stored event fields come from the next completed step, at $t=4.69206969694$, rather than exactly the interpolated $t_H$. Their recorded outer mass is $1.72538868709$, compared with initial outer mass $1.72992821318$, a fractional change of $-2.6241124\\,10^{-3}$. The same stored fields have minimum lapse $0.999829382946$, lapse at the sampled peak $1.02582462096$, and maximum absolute scalar Ricci curvature $7.35797274672$. This timing distinction is small but matters when those quantities are compared with an independently interpolated event.","t fields come from the next completed step, at $t=4.69206969694$, rather than exactly the interpolated $t_H$. Their recorded outer mass is $1.72538868709$, compared with initial outer mass $1.72992821318$, a fractional change of $-2.6241124\\,10^{-3}$. The same stored fields have minimum lapse $0.999829382946$, lapse at the sampled peak $1.02582462096$, and maximum absolute scalar Ricci curvature $7.35797274672$. This timing distinction is small but matters when those quantities are compared with an independently interpolated event.","The scalar curvature diagnostic follows the rescaled convention\n\\begin{equation}\n \\mathcal R=2(\\Phi^2-\\Pi^2).\n\\end{equation}\nFinite lapse and curvature values show that this calculation can reach the marginal-sphere event without a coordinate blow-up. They do not provide continuation to a singularity or control of subsequent trapped-region evolution.","\\gravityfigure{12_trapped_refinement}{Archived event coordinates under radial refinement, together with the exact expansion and characteristic sign change at $v=1$. The event table is replotted from retained data. No new scalar-collapse evolution was performed for this figure.}","\\subsection{Control and independent diagnostics}\nThe weak pulse on $N_r=1200$ remains untrapped through $t=5$, with maximum compactness $0.146722872367$ and fractional outer-mass change $-4.5106042\\,10^{-3}$. It is a finite-time control in an excised domain. It does not establish global dispersion, passage through a regular center, or a critical amplitude separating two complete physical outcomes.","At the smooth pre-horizon time $t=3$, the $N_r=1200$ retained relative residuals are\n\\begin{center}\n\\begin{tabular}{lr}\n\\toprule\nDiagnostic&Relative residual","\\midrule\nRadial mass constraint&$2.27319745\\,10^{-6}$","Lapse constraint&$2.80774843\\,10^{-6}$","Local mass flux against a time difference&$1.26971871\\,10^{-3}$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe flux diagnostic is independent of simply inserting the geometry solve's own right side into its output. Its finite time difference is still an approximation. The separately reported absolute outer-mass flux-balance defect is $0.00230311127$. A small total outer-mass change cannot be attributed solely to physical boundary flux unless that flux is integrated and its numerical balance is controlled.","nvacuous interval energy theorem must control boundary flux and then establish the stronger norm needed by the reconstruction estimate. A full RK4 theorem needs a variable-coefficient argument, rather than the frozen advection symbol alone. For scalar matter, the mass and lapse prefix solves need stability and consistency estimates coupled to the characteristic evolution, including behavior near $v=0$ and the excision boundary. Event convergence needs quantitative control of the compactness maximum and its temporal crossing. A tangential spatial root cannot inherit a simple-root horizon bound automatically.","Beyond these radial sectors, finite-range nonlinear gravitational closure remains a separate problem. Spherical symmetry removes the radiative gravitational degrees of freedom. A general $3+1$ Einstein--Maxwell--Dirac construction must restore them, choose boundary and gauge conditions, establish the relevant constraint algebra and chart regularity, and prove consistency and stability of its actual evolution. Numerical marginal-sphere formation here establishes no global cosmic-censorship result, singularity theorem, nonspherical stability theorem, or unique event-horizon reconstruction."]}