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It cannot invent a new direction in the row space.","source_text":"$A$ acts on the left. It cannot invent a new direction in the row space.","position":[7.0,0.6],"line":23}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12],"reading_aid":{"alt":"Product and Bell seed matrices have different row ranks.","description":"The left coefficient matrix has one nonzero row direction and the right matrix has two. The adjacent annotations give the corresponding local orbit dimensions. 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The equations on the right pair each column with its shift eigenvalue and rank-one projector. The note below states the common entry magnitude.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column",21,22,23,24],"rows":[[17,1,2,3,4],[18,5,6,7,8],[19,9,10,11,12],[20,13,14,15,16]]},{"kind":"text","nodes":[25,26,27,28,29,30,31,32,33]}]}},"parity":{"name":"parity","plate":7,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/parity.tex","source_sha256":"de743c1cdde4582c932cb9ecc4a1aead7578f3a7ad9af0c188c2f8fa48b5a2ab","image":"assets/atlas/paper-vii/parity.png","image_sha256":"85c055857e9b4edef8fde2f88412e7485d31ec4dc7698f37ddaf43a403ccd590","pdf_page":124,"crop_pdf_points":[78,218,531,398],"nodes":[{"text":"even","source_text":"even","position":[0.5,3.1],"line":3},{"text":"$\\ket{000}$","source_text":"$\\ket{000}$","position":[3.0,3.1],"line":5},{"text":"$\\ket{011}$","source_text":"$\\ket{011}$","position":[6.0,3.1],"line":7},{"text":"$\\ket{101}$","source_text":"$\\ket{101}$","position":[9.0,3.1],"line":9},{"text":"$\\ket{110}$","source_text":"$\\ket{110}$","position":[12.0,3.1],"line":11},{"text":"odd","source_text":"odd","position":[0.5,1.05],"line":12},{"text":"$\\ket{001}$","source_text":"$\\ket{001}$","position":[3.0,1.05],"line":14},{"text":"$\\ket{010}$","source_text":"$\\ket{010}$","position":[6.0,1.05],"line":16},{"text":"$\\ket{100}$","source_text":"$\\ket{100}$","position":[9.0,1.05],"line":18},{"text":"$\\ket{111}$","source_text":"$\\ket{111}$","position":[12.0,1.05],"line":20},{"text":"Each coarse projector has rank four. All eight fine projectors have rank one.","source_text":"Each coarse projector has rank four. All eight fine projectors have rank one.","position":[7.0,4.55],"line":21}],"node_order":[0,1,2,3,4,5,6,7,8,9,10],"reading_aid":{"alt":"Eight fine addresses are grouped into even and odd parity rows.","description":"The upper row contains the four even addresses and the lower row the four odd addresses. 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The notes below identify which parity packets are occupied at each stage, including the coherence in the final even packet.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]}]}},"interference":{"name":"interference","plate":9,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/interference.tex","source_sha256":"4403c1208466eca234a66e5c6f1f00a8822658ec1c0728c6a947688a2e45b224","image":"assets/atlas/paper-vii/interference.png","image_sha256":"b62ab68d1c5e090999a7cecae0bb76b3728574cff23c732441a9ded3af57034c","pdf_page":126,"crop_pdf_points":[78,205,531,385],"nodes":[{"text":"$0$","source_text":"$0$","position":[1.4,2.3],"line":4},{"text":"$0$","source_text":"$0$","position":[6.6,3.6],"line":6},{"text":"$1$","source_text":"$1$","position":[6.6,1.0],"line":8},{"text":"$1$","source_text":"$1$","position":[12.3,2.3],"line":10},{"text":"$1/\\sqrt2$","source_text":"$1/\\sqrt2$","position":[3.975,3.3],"line":12},{"text":"$1/\\sqrt2$","source_text":"$1/\\sqrt2$","position":[3.975,1.3],"line":14},{"text":"$1/\\sqrt2$","source_text":"$1/\\sqrt2$","position":[9.475,3.3],"line":16},{"text":"$-1/\\sqrt2$","source_text":"$-1/\\sqrt2$","position":[9.475,1.3],"line":18},{"text":"Two actual Hadamard paths, not classical alternatives","source_text":"Two actual Hadamard paths, not classical alternatives","position":[6.7,4.6],"line":19},{"text":"$\\text{amplitude}=1/2-1/2=0\\quad\\ne\\quad\\text{sum of path probabilities}=1/2$","source_text":"$\\text{amplitude}=1/2-1/2=0\\quad\\ne\\quad\\text{sum of path probabilities}=1/2$","position":[6.8,-0.18],"line":20}],"node_order":[0,1,2,3,4,5,6,7,8,9],"reading_aid":{"alt":"Two signed Hadamard paths cancel at one output.","description":"The diagram branches through the two intermediate addresses and recombines at the selected output. One second-stage amplitude has a minus sign. The comparison below separates the sum of amplitudes from a sum of path probabilities.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9]}]}},"hamiltonian":{"name":"hamiltonian","plate":10,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/hamiltonian.tex","source_sha256":"d9ce6370138fcb0721eca291a636204402eed781abb8be9b4933fc26a57aae75","image":"assets/atlas/paper-vii/hamiltonian.png","image_sha256":"d3db0c93ac97f959e7df7ee0f416af8889cc78f353d1832b43c1a3b7ae8647c5","pdf_page":127,"crop_pdf_points":[78,203,531,383],"nodes":[{"text":"Computational order","source_text":"Computational order","position":[2.34,4.7],"line":3},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[1.11,3.59],"line":5},{"text":"$0$","source_text":"$0$","position":[1.93,3.59],"line":7},{"text":"$0$","source_text":"$0$","position":[2.75,3.59],"line":9},{"text":"$1$","source_text":"$1$","position":[3.57,3.59],"line":11},{"text":"$0$","source_text":"$0$","position":[1.11,2.77],"line":13},{"text":"$0$","source_text":"$0$","position":[1.93,2.77],"line":15},{"text":"$1$","source_text":"$1$","position":[2.75,2.77],"line":17},{"text":"$0$","source_text":"$0$","position":[3.57,2.77],"line":19},{"text":"$0$","source_text":"$0$","position":[1.11,1.95],"line":21},{"text":"$1$","source_text":"$1$","position":[1.93,1.95],"line":23},{"text":"$0$","source_text":"$0$","position":[2.75,1.95],"line":25},{"text":"$0$","source_text":"$0$","position":[3.57,1.95],"line":27},{"text":"$1$","source_text":"$1$","position":[1.11,1.13],"line":29},{"text":"$0$","source_text":"$0$","position":[1.93,1.13],"line":31},{"text":"$0$","source_text":"$0$","position":[2.75,1.13],"line":33},{"text":"$- \\frac{3}{4}$","source_text":"$- \\frac{3}{4}$","position":[3.57,1.13],"line":35},{"text":"00","source_text":"00","position":[0.55,3.59],"line":36},{"text":"01","source_text":"01","position":[0.55,2.77],"line":37},{"text":"10","source_text":"10","position":[0.55,1.95],"line":38},{"text":"11","source_text":"11","position":[0.55,1.13],"line":39},{"text":"00","source_text":"00","position":[1.11,4.23],"line":40},{"text":"01","source_text":"01","position":[1.93,4.23],"line":41},{"text":"10","source_text":"10","position":[2.75,4.23],"line":42},{"text":"11","source_text":"11","position":[3.57,4.23],"line":43},{"text":"Parity order","source_text":"Parity order","position":[10.34,4.7],"line":44},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[9.11,3.59],"line":46},{"text":"$1$","source_text":"$1$","position":[9.93,3.59],"line":48},{"text":"$0$","source_text":"$0$","position":[10.75,3.59],"line":50},{"text":"$0$","source_text":"$0$","position":[11.57,3.59],"line":52},{"text":"$1$","source_text":"$1$","position":[9.11,2.77],"line":54},{"text":"$- \\frac{3}{4}$","source_text":"$- \\frac{3}{4}$","position":[9.93,2.77],"line":56},{"text":"$0$","source_text":"$0$","position":[10.75,2.77],"line":58},{"text":"$0$","source_text":"$0$","position":[11.57,2.77],"line":60},{"text":"$0$","source_text":"$0$","position":[9.11,1.95],"line":62},{"text":"$0$","source_text":"$0$","position":[9.93,1.95],"line":64},{"text":"$0$","source_text":"$0$","position":[10.75,1.95],"line":66},{"text":"$1$","source_text":"$1$","position":[11.57,1.95],"line":68},{"text":"$0$","source_text":"$0$","position":[9.11,1.13],"line":70},{"text":"$0$","source_text":"$0$","position":[9.93,1.13],"line":72},{"text":"$1$","source_text":"$1$","position":[10.75,1.13],"line":74},{"text":"$0$","source_text":"$0$","position":[11.57,1.13],"line":76},{"text":"00","source_text":"00","position":[8.55,3.59],"line":77},{"text":"11","source_text":"11","position":[8.55,2.77],"line":78},{"text":"01","source_text":"01","position":[8.55,1.95],"line":79},{"text":"10","source_text":"10","position":[8.55,1.13],"line":80},{"text":"00","source_text":"00","position":[9.11,4.23],"line":81},{"text":"11","source_text":"11","position":[9.93,4.23],"line":82},{"text":"01","source_text":"01","position":[10.75,4.23],"line":83},{"text":"10","source_text":"10","position":[11.57,4.23],"line":84},{"text":"same operator\\newline different order","source_text":"same operator\\\\different order","position":[6.8,2.7],"line":85},{"text":"Even eigenvalues: $\\pm5/4$. Odd eigenvalues: $\\pm1$.","source_text":"Even eigenvalues: $\\pm5/4$. Odd eigenvalues: $\\pm1$.","position":[6.9,0.2],"line":86}],"node_order":[0,21,22,23,24,17,1,2,3,4,18,5,6,7,8,19,9,10,11,12,20,13,14,15,16,25,46,47,48,49,42,26,27,28,29,43,30,31,32,33,44,34,35,36,37,45,38,39,40,41,50,51],"reading_aid":{"alt":"The same Hamiltonian is shown in computational and parity orders.","description":"The address labels show the row and column permutation. The reordered matrix displays two blocks, while the middle note identifies the two matrices as the same operator. The bottom annotation lists the two sector spectra.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column",21,22,23,24],"rows":[[17,1,2,3,4],[18,5,6,7,8],[19,9,10,11,12],[20,13,14,15,16]]},{"kind":"text","nodes":[25]},{"kind":"table","caption":"Drawing grid 2 · rows top to bottom, columns left to right","headers":["Row / column",46,47,48,49],"rows":[[42,26,27,28,29],[43,30,31,32,33],[44,34,35,36,37],[45,38,39,40,41]]},{"kind":"text","nodes":[50,51]}]}},"leakage":{"name":"leakage","plate":11,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/leakage.tex","source_sha256":"2b4ccc5ca1558ac7002e1dd141644fd3a15cefdf75e4753850013f688c0b9ee5","image":"assets/atlas/paper-vii/leakage.png","image_sha256":"34723ec6d80f7c3bd3409843b32b8d94360a145cacc42f6bc6da82de22c8a8df","pdf_page":128,"crop_pdf_points":[78,197,531,377],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.44],"line":5},{"text":"0.5","source_text":"0.5","position":[2.5,0.44],"line":8},{"text":"1","source_text":"1","position":[4.2,0.44],"line":11},{"text":"1.5","source_text":"1.5","position":[5.9,0.44],"line":14},{"text":"2","source_text":"2","position":[7.6,0.44],"line":17},{"text":"2.5","source_text":"2.5","position":[9.3,0.44],"line":20},{"text":"0","source_text":"0","position":[0.64,0.7],"line":22},{"text":"0.5","source_text":"0.5","position":[0.64,2.113043],"line":24},{"text":"1","source_text":"1","position":[0.64,3.526087],"line":26},{"text":"$u=|\\varepsilon t|$","source_text":"$u=|\\varepsilon t|$","position":[5.05,0.15],"line":29},{"text":"leakage amplitude","source_text":"leakage amplitude","position":[0.8,4.35],"line":30},{"text":"$H=\\varepsilon X$","source_text":"$H=\\varepsilon X$","position":[11.7,3.95],"line":33},{"text":"solid: $|\\sin u|$\\newline dashed: $\\min(1,u)$","source_text":"solid: $|\\sin u|$\\\\dashed: $\\min(1,u)$","position":[11.7,2.95],"line":34},{"text":"Probability is $\\sin^2u$,\\newline not $|\\sin u|$.","source_text":"Probability is $\\sin^2u$,\\\\not $|\\sin u|$.","position":[11.7,1.5],"line":35}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],"reading_aid":{"alt":"A leakage-amplitude curve is compared with its bound.","description":"The horizontal axis is scaled elapsed time. The solid curve is the absolute sine amplitude and the dashed curve is the stated upper bound. The note on the right distinguishes this amplitude from the squared transition probability.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13]}]}},"moving_frame":{"name":"moving_frame","plate":12,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/moving_frame.tex","source_sha256":"8499161f9da816183ef41bf96928c67cd006b05d5ec46bb871d3fe0a7d8c7741","image":"assets/atlas/paper-vii/moving_frame.png","image_sha256":"4bac60e1ed555119092a4bbc9410c8f17c8fcc3e6a42736cf69742b7b4b63f0f","pdf_page":129,"crop_pdf_points":[78,197,531,377],"nodes":[{"text":"$\\theta=0$","source_text":"$\\theta=0$","position":[2.0,4.2],"line":8},{"text":"$(1,0)$","source_text":"$(1,0)$","position":[2.0,0.25],"line":9},{"text":"$\\theta=\\pi/6$","source_text":"$\\theta=\\pi/6$","position":[6.65,4.2],"line":15},{"text":"$(\\sqrt3/2,-1/2)$","source_text":"$(\\sqrt3/2,-1/2)$","position":[6.65,0.25],"line":16},{"text":"$\\theta=\\pi/3$","source_text":"$\\theta=\\pi/3$","position":[11.3,4.2],"line":22},{"text":"$(1/2,-\\sqrt3/2)$","source_text":"$(1/2,-\\sqrt3/2)$","position":[11.3,0.25],"line":23},{"text":"Solid vector: fixed in the laboratory. Dashed axes: rotating frame.","source_text":"Solid vector: fixed in the laboratory. Dashed axes: rotating frame.","position":[7.0,4.75],"line":24}],"node_order":[0,1,2,3,4,5,6],"reading_aid":{"alt":"A fixed laboratory vector is viewed in three rotating frames.","description":"The solid arrow stays fixed while the dashed coordinate axes rotate from left to right. The coordinate pairs below the panels change even though the laboratory vector does not.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6]}]}},"connection":{"name":"connection","plate":13,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/connection.tex","source_sha256":"02be1c712c5da7c1c1f9095883c8a43ceadcb5a9762f4d586d169d6a781c60fc","image":"assets/atlas/paper-vii/connection.png","image_sha256":"52406ed0c77aeaa5482b8ec238709c698ae372e468eaaf835a3a94e57c808aee","pdf_page":130,"crop_pdf_points":[78,197,531,377],"nodes":[{"text":"$P(0)$","source_text":"$P(0)$","position":[1.72,4.05],"line":3},{"text":"$1$","source_text":"$1$","position":[1.21,2.84],"line":5},{"text":"$0$","source_text":"$0$","position":[2.23,2.84],"line":7},{"text":"$0$","source_text":"$0$","position":[1.21,1.82],"line":9},{"text":"$0$","source_text":"$0$","position":[2.23,1.82],"line":11},{"text":"$\\dot P(0)$","source_text":"$\\dot P(0)$","position":[6.27,4.05],"line":12},{"text":"$0$","source_text":"$0$","position":[5.76,2.84],"line":14},{"text":"$1$","source_text":"$1$","position":[6.78,2.84],"line":16},{"text":"$1$","source_text":"$1$","position":[5.76,1.82],"line":18},{"text":"$0$","source_text":"$0$","position":[6.78,1.82],"line":20},{"text":"$C$","source_text":"$C$","position":[10.82,4.05],"line":21},{"text":"$0$","source_text":"$0$","position":[10.31,2.84],"line":23},{"text":"$-1$","source_text":"$-1$","position":[11.33,2.84],"line":25},{"text":"$1$","source_text":"$1$","position":[10.31,1.82],"line":27},{"text":"$0$","source_text":"$0$","position":[11.33,1.82],"line":29},{"text":"$P\\dot P P=0,\\qquad C^*=-C,\\qquad[C,P]=\\dot P$.","source_text":"$P\\dot P P=0,\\qquad C^*=-C,\\qquad[C,P]=\\dot P$.","position":[7.0,0.3],"line":30}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"Three matrices give a projector, its derivative, and its connection.","description":"Read the matrices from left to right at the same parameter value. The identities below state the tangent condition, skew-adjointness, and the commutator relation retained in the source drawing.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",1,2],["Row 2",3,4]]},{"kind":"text","nodes":[5]},{"kind":"table","caption":"Drawing grid 2 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",6,7],["Row 2",8,9]]},{"kind":"text","nodes":[10]},{"kind":"table","caption":"Drawing grid 3 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",11,12],["Row 2",13,14]]},{"kind":"text","nodes":[15]}]}},"range_atlas":{"name":"range_atlas","plate":14,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/range_atlas.tex","source_sha256":"538674d9307b4e1d3773119d69b601d385eb3f3074e5c6962f178c6551471ab1","image":"assets/atlas/paper-vii/range_atlas.png","image_sha256":"3ebf1e07b0c36dad014c919df8a3865a3ba37019626ce49a34ef6fddff28cb4c","pdf_page":131,"crop_pdf_points":[78,208,531,390],"nodes":[{"text":"$0$","source_text":"$0$","position":[1.6,4.2],"line":6},{"text":"$\\det T=1$","source_text":"$\\det T=1$","position":[1.6,0.45],"line":7},{"text":"$\\pi/6$","source_text":"$\\pi/6$","position":[5.15,4.2],"line":11},{"text":"$\\det T=3/4$","source_text":"$\\det T=3/4$","position":[5.15,0.45],"line":12},{"text":"$\\pi/3$","source_text":"$\\pi/3$","position":[8.7,4.2],"line":16},{"text":"$\\det T=1/4$","source_text":"$\\det T=1/4$","position":[8.7,0.45],"line":17},{"text":"$\\pi/2$","source_text":"$\\pi/2$","position":[12.25,4.2],"line":21},{"text":"$\\det T=0$","source_text":"$\\det T=0$","position":[12.25,0.45],"line":22},{"text":"The ranges of $P(\\theta)$ are shown with a horizontal anchor.","source_text":"The ranges of $P(\\theta)$ are shown with a horizontal anchor.","position":[7.0,4.8],"line":23},{"text":"At the last position, use a new anchor. The fiber still has dimension one.","source_text":"At the last position, use a new anchor. The fiber still has dimension one.","position":[7.0,-0.15],"line":24}],"node_order":[0,1,2,3,4,5,6,7,8,9],"reading_aid":{"alt":"Four rotating one-dimensional ranges are tested against one anchor.","description":"Each panel shows a range direction against the same horizontal anchor. The determinant values decrease to zero at the final panel. The note calls for a new chart there while retaining the same fiber dimension.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9]}]}},"bloch_curvature":{"name":"bloch_curvature","plate":15,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/bloch_curvature.tex","source_sha256":"b0cee1f1fed23902436cf5e20e94397d31ff9fd6e977d4fb479ed04834ec997c","image":"assets/atlas/paper-vii/bloch_curvature.png","image_sha256":"615f36e3a1a0db479c623981f85a6daeb906b37deff616396ffd22664c9581c5","pdf_page":132,"crop_pdf_points":[78,239,531,419],"nodes":[{"text":"Cap flux at $\\theta=\\pi/3$ is $\\pi/2$.","source_text":"Cap flux at $\\theta=\\pi/3$ is $\\pi/2$.","position":[2.8,0.2],"line":8},{"text":"0","source_text":"0","position":[6.4,0.59],"line":11},{"text":"0","source_text":"0","position":[6.24,0.85],"line":13},{"text":"0.25","source_text":"0.25","position":[6.24,2.016667],"line":15},{"text":"0.5","source_text":"0.5","position":[6.24,3.183333],"line":17},{"text":"polar angle $\\theta$","source_text":"polar angle $\\theta$","position":[9.75,0.3],"line":20},{"text":"$F_{\\theta\\phi}=\\frac12\\sin\\theta$","source_text":"$F_{\\theta\\phi}=\\frac12\\sin\\theta$","position":[6.4,4.05],"line":21},{"text":"$\\pi$","source_text":"$\\pi$","position":[13.1,0.58],"line":23},{"text":"The density vanishes at coordinate poles.","source_text":"The density vanishes at coordinate poles.","position":[9.7,4.45],"line":24}],"node_order":[0,1,2,3,4,5,6,7,8],"reading_aid":{"alt":"A spherical cap is paired with a curvature-density curve.","description":"The sphere at left marks a latitude and cap. The curve at right plots the displayed density against polar angle and vanishes at both coordinate poles. 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The right matrix removes it under computational dephasing. 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The middle annotations identify the damping parameter and coherence factor. 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The bottom note distinguishes zero trace from the zero operator.","blocks":[{"kind":"text","nodes":[0]},{"kind":"table","caption":"Drawing grid 1 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",1,2],["Row 2",3,4]]},{"kind":"text","nodes":[5,6,7,8]},{"kind":"table","caption":"Drawing grid 2 · rows top to bottom, columns left to right","headers":["Row / column","Column 1","Column 2"],"rows":[["Row 1",9,10],["Row 2",11,12]]},{"kind":"text","nodes":[13]}]}},"feshbach":{"name":"feshbach","plate":21,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/feshbach.tex","source_sha256":"5d78b4ed4d5fb49e4f0bf8c6c5063939565984b0e1872a1f8881f72259aa80ef","image":"assets/atlas/paper-vii/feshbach.png","image_sha256":"7677ccb04a77c704a4e66a9c04b514ae3aaef5c199d8b9a595671bd9431c64a8","pdf_page":138,"crop_pdf_points":[78,218,531,397],"nodes":[{"text":"$1-\\sqrt2$","source_text":"$1-\\sqrt2$","position":[2.610913,3.4],"line":6},{"text":"$1+\\sqrt2$","source_text":"$1+\\sqrt2$","position":[10.389087,3.4],"line":9},{"text":"$2$","source_text":"$2$","position":[9.25,1.3],"line":12},{"text":"$3$","source_text":"$3$","position":[12.0,1.3],"line":15},{"text":"Filled: full resolvent poles. Open at 2: only the eliminated inverse fails.","source_text":"Filled: full resolvent poles. Open at 2: only the eliminated inverse fails.","position":[6.9,4.55],"line":16},{"text":"At $z=3$: exact retained resolvent $1/2$, bare truncation $1/3$.","source_text":"At $z=3$: exact retained resolvent $1/2$, bare truncation $1/3$.","position":[6.9,0.12],"line":17}],"node_order":[0,1,2,3,4,5],"reading_aid":{"alt":"Filled and open points distinguish two kinds of spectral failure.","description":"The filled points mark poles of the full resolvent. Open points mark the displayed hidden-sector inverses that fail. The bottom comparison evaluates the exact retained resolvent and the bare truncation at the same regular spectral point.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"memory":{"name":"memory","plate":22,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/memory.tex","source_sha256":"48df39ed051c11ec65e5539893bb86187082ab2fb11ddd49322fd9c0da29fde3","image":"assets/atlas/paper-vii/memory.png","image_sha256":"510cd567f738124423d099a17850499cb052d49abf95d1fb90de0fe041cfb42c","pdf_page":139,"crop_pdf_points":[78,204,531,384],"nodes":[{"text":"0","source_text":"0","position":[0.8,0.49],"line":5},{"text":"0","source_text":"0","position":[0.64,0.75],"line":7},{"text":"0.5","source_text":"0.5","position":[0.64,2.159091],"line":9},{"text":"1","source_text":"1","position":[0.64,3.568182],"line":11},{"text":"time $t$","source_text":"time $t$","position":[5.05,0.2],"line":14},{"text":"probability","source_text":"probability","position":[0.8,4.25],"line":15},{"text":"solid: retained\\newline dashed: hidden","source_text":"solid: retained\\\\dashed: hidden","position":[11.7,3.8],"line":18},{"text":"$K(t)=1$","source_text":"$K(t)=1$","position":[11.7,2.4],"line":19},{"text":"The two probabilities\\newline sum to one.","source_text":"The two probabilities\\\\sum to one.","position":[11.7,1.2],"line":20},{"text":"$\\pi$","source_text":"$\\pi$","position":[9.25,0.47],"line":21}],"node_order":[0,1,2,3,4,5,6,7,8,9],"reading_aid":{"alt":"Retained and hidden probabilities exchange over time.","description":"The solid retained curve and dashed hidden curve start at opposite endpoints, exchange, and return. Their sum and the memory-kernel value are annotated on the right. The horizontal axis is real time.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9]}]}},"euclidean":{"name":"euclidean","plate":23,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/euclidean.tex","source_sha256":"b1d3647b4cff8e281e9408148fec0c2507353db17819ddfb85a67b21ba174d34","image":"assets/atlas/paper-vii/euclidean.png","image_sha256":"2a8deb8c6110cab5369e9c1ce7f7aa92398fbc124dbae6ca6115e7ca16371165","pdf_page":140,"crop_pdf_points":[78,210,531,390],"nodes":[{"text":"$T$ at $e^{-2\\tau}=1/2$","source_text":"$T$ at $e^{-2\\tau}=1/2$","position":[2.25,4.2],"line":3},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[1.675,2.925],"line":5},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[2.825,2.925],"line":7},{"text":"$\\frac{1}{4}$","source_text":"$\\frac{1}{4}$","position":[1.675,1.775],"line":9},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[2.825,1.775],"line":11},{"text":"$ZTZ$ in another basis","source_text":"$ZTZ$ in another basis","position":[10.35,4.2],"line":12},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[9.775,2.925],"line":14},{"text":"$- \\frac{1}{4}$","source_text":"$- \\frac{1}{4}$","position":[10.925,2.925],"line":16},{"text":"$- \\frac{1}{4}$","source_text":"$- \\frac{1}{4}$","position":[9.775,1.775],"line":18},{"text":"$\\frac{3}{4}$","source_text":"$\\frac{3}{4}$","position":[10.925,1.775],"line":20},{"text":"Both have\\newline spectrum $\\{1,1/2\\}$.","source_text":"Both have\\\\spectrum $\\{1,1/2\\}$.","position":[6.5,2.4],"line":21},{"text":"Positive-definite operator: $v^*Tv>0$ for $v\\ne0$. Not an entrywise test.","source_text":"Positive-definite operator: $v^*Tv>0$ for $v\\ne0$. Not an entrywise test.","position":[7.0,0.4],"line":22}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11],"reading_aid":{"alt":"Two basis-dependent matrices represent the same positive transfer.","description":"The left matrix is the transfer in one basis and the right matrix is its conjugate in another. Their common spectrum is written between them. 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The solid curve represents the four-cycle and the dashed curve its two-factor product. The notes identify the curves as finite diagnostics and state the limitation on a dimension plateau.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12]}]}},"dispersion":{"name":"dispersion","plate":25,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/dispersion.tex","source_sha256":"d90f3620053a43c81c311c63f453dbdb2bfb17fbab4c3af39202b2075815cc31","image":"assets/atlas/paper-vii/dispersion.png","image_sha256":"4afc4ef611477f4f16c3caec9579fcf13019a90927eba97955aaa4bbc4779cdf","pdf_page":142,"crop_pdf_points":[78,208,531,388],"nodes":[{"text":"$h=\\frac{1}{4}$","source_text":"$h=\\frac{1}{4}$","position":[2.2,4.3],"line":3},{"text":"error (approx.)","source_text":"error (approx.)","position":[2.2,3.3],"line":4},{"text":"$\\simeq 0.005197495$","source_text":"$\\simeq 0.005197495$","position":[2.2,2.65],"line":5},{"text":"bound $h^2/12$","source_text":"bound $h^2/12$","position":[2.2,1.65],"line":6},{"text":"$\\frac{1}{192}$","source_text":"$\\frac{1}{192}$","position":[2.2,1.0],"line":7},{"text":"$h=\\frac{1}{8}$","source_text":"$h=\\frac{1}{8}$","position":[7.0,4.3],"line":8},{"text":"error (approx.)","source_text":"error (approx.)","position":[7.0,3.3],"line":9},{"text":"$\\simeq 0.001301405$","source_text":"$\\simeq 0.001301405$","position":[7.0,2.65],"line":10},{"text":"bound $h^2/12$","source_text":"bound $h^2/12$","position":[7.0,1.65],"line":11},{"text":"$\\frac{1}{768}$","source_text":"$\\frac{1}{768}$","position":[7.0,1.0],"line":12},{"text":"$h=\\frac{1}{16}$","source_text":"$h=\\frac{1}{16}$","position":[11.8,4.3],"line":13},{"text":"error (approx.)","source_text":"error (approx.)","position":[11.8,3.3],"line":14},{"text":"$\\simeq 0.000325478$","source_text":"$\\simeq 0.000325478$","position":[11.8,2.65],"line":15},{"text":"bound $h^2/12$","source_text":"bound $h^2/12$","position":[11.8,1.65],"line":16},{"text":"$\\frac{1}{3072}$","source_text":"$\\frac{1}{3072}$","position":[11.8,1.0],"line":17},{"text":"Successive error ratios lie strictly between $3.99$ and $4$.","source_text":"Successive error ratios lie strictly between $3.99$ and $4$.","position":[7.0,0.05],"line":18}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"reading_aid":{"alt":"Three grid spacings compare dispersion errors with bounds.","description":"Each column gives a spacing, its displayed approximate error, and an exact bound. The bottom note compares successive error ratios. The retained plate explains the exact enclosure behind the displayed approximations.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}]}},"constraints":{"name":"constraints","plate":26,"source":"papers/paper-vii/current/editable/traditional/visual_graphics/constraints.tex","source_sha256":"8ecbddb673c23bd71427f97179aa7eee6eaaf9fa4353812b8c4c57e38426c4e0","image":"assets/atlas/paper-vii/constraints.png","image_sha256":"6f31bb1ae6697e153586c979a11c2355ff44d1b0538d6eed7e0fbcc55cbc5fb0","pdf_page":143,"crop_pdf_points":[78,224,531,402],"nodes":[{"text":"$[C_1,C_2]$","source_text":"$[C_1,C_2]$","position":[2.05,4.45],"line":3},{"text":"$0$","source_text":"$0$","position":[1.15,3.3],"line":5},{"text":"$0$","source_text":"$0$","position":[2.05,3.3],"line":7},{"text":"$0$","source_text":"$0$","position":[2.95,3.3],"line":9},{"text":"$0$","source_text":"$0$","position":[1.15,2.4],"line":11},{"text":"$0$","source_text":"$0$","position":[2.05,2.4],"line":13},{"text":"$2$","source_text":"$2$","position":[2.95,2.4],"line":15},{"text":"$0$","source_text":"$0$","position":[1.15,1.5],"line":17},{"text":"$-2$","source_text":"$-2$","position":[2.05,1.5],"line":19},{"text":"$0$","source_text":"$0$","position":[2.95,1.5],"line":21},{"text":"$P[C_1,C_2]P$","source_text":"$P[C_1,C_2]P$","position":[11.15,4.45],"line":22},{"text":"$0$","source_text":"$0$","position":[10.25,3.3],"line":24},{"text":"$0$","source_text":"$0$","position":[11.15,3.3],"line":26},{"text":"$0$","source_text":"$0$","position":[12.05,3.3],"line":28},{"text":"$0$","source_text":"$0$","position":[10.25,2.4],"line":30},{"text":"$0$","source_text":"$0$","position":[11.15,2.4],"line":32},{"text":"$0$","source_text":"$0$","position":[12.05,2.4],"line":34},{"text":"$0$","source_text":"$0$","position":[10.25,1.5],"line":36},{"text":"$0$","source_text":"$0$","position":[11.15,1.5],"line":38},{"text":"$0$","source_text":"$0$","position":[12.05,1.5],"line":40},{"text":"$P=\\diag(1,0,0)$","source_text":"$P=\\diag(1,0,0)$","position":[6.9,2.7],"line":41},{"text":"$C_1P=C_2P=0$","source_text":"$C_1P=C_2P=0$","position":[6.9,1.4],"line":42},{"text":"The right matrix is zero by the common-kernel choice, not by operator closure.","source_text":"The right matrix is zero by the common-kernel choice, not by operator closure.","position":[7.0,0.3],"line":43}],"node_order":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"reading_aid":{"alt":"A nonzero commutator is compared with its zero compression.","description":"The left matrix is the full constraint commutator. 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The larger standard isotypic box records both its irreducible dimension and multiplicity. 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The plate discusses where this local chart can fail.","blocks":[{"kind":"text","nodes":[0,1,2]}]}},"choi":{"name":"choi","plate":31,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/choi.tex","source_sha256":"3d9c2e948b1df391ab7be03554d4233cb4df2453ffd9ad9538325acde25dced9","image":"assets/atlas/paper-vii/choi.png","image_sha256":"00ee7cb5d79a8eb521b949df7c8ec5f22d24009eef82e55e2edc09be5bce8260","pdf_page":148,"crop_pdf_points":[92,202,517,325],"nodes":[{"text":"even Choi addresses $00,11$\\newline $\\begin{pmatrix}1&4/5\\\\4/5&16/25\\end{pmatrix}$","source_text":"even Choi addresses $00,11$\\\\$\\begin{pmatrix}1&4/5\\\\4/5&16/25\\end{pmatrix}$","position":[3.2,1.8],"line":4},{"text":"odd Choi addresses $01,10$\\newline $\\operatorname{diag}(9/25,0)$","source_text":"odd Choi addresses $01,10$\\\\$\\operatorname{diag}(9/25,0)$","position":[8.5,1.8],"line":5},{"text":"$\\Phi(\\ket0\\bra1)=(4/5)\\ket0\\bra1$","source_text":"$\\Phi(\\ket0\\bra1)=(4/5)\\ket0\\bra1$","position":[6.0,0.5],"line":6}],"node_order":[0,1,2],"reading_aid":{"alt":"Even and odd Choi address packets retain different blocks.","description":"The left box contains the block on addresses 00 and 11; the right box contains the block on addresses 01 and 10. 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The note below distinguishes the defect for one pair from the defect for the stated full operator family.","blocks":[{"kind":"text","nodes":[0,1,2,3,4,5]}]}},"stability":{"name":"stability","plate":34,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/stability.tex","source_sha256":"9bcf9cbfac63aa806fac8eb4006ba153e8e92a42a7a6b8105d2c709349eefeba","image":"assets/atlas/paper-vii/stability.png","image_sha256":"e7fda9684fd6abdb972a36b2ac4063f0099f7df8e4e116d073639641c2707bbf","pdf_page":151,"crop_pdf_points":[92,230,517,352],"nodes":[{"text":"$B=1$, $D=2$\\newline $\\Sigma(3)=1$","source_text":"$B=1$, $D=2$\\\\$\\Sigma(3)=1$","position":[2.6,2.0],"line":4},{"text":"$\\widetilde B=9/8$, $\\widetilde D=17/8$\\newline $\\widetilde\\Sigma(3)=81/56$","source_text":"$\\widetilde B=9/8$, $\\widetilde D=17/8$\\\\$\\widetilde\\Sigma(3)=81/56$","position":[8.9,2.0],"line":5},{"text":"$25/56\\leq21/32$","source_text":"$25/56\\leq21/32$","position":[5.6,1.6],"line":6},{"text":"common hypotheses: $g=9/8$, $\\delta=3/4$","source_text":"common hypotheses: $g=9/8$, $\\delta=3/4$","position":[6.0,0.5],"line":7}],"node_order":[0,1,2,3],"reading_aid":{"alt":"Two hidden-sector models are compared at one spectral point.","description":"The boxes list the original and perturbed coupling data and self-energies. 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The plate uses the corresponding Gram factorization to explain positivity.","blocks":[{"kind":"text","nodes":[0,1,2]}]}},"transfer_ground":{"name":"transfer_ground","plate":36,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/transfer_ground.tex","source_sha256":"4de1a487a03a3050a0c22d02563109b0764c3b162408a7b8f82a3e5297e5bb63","image":"assets/atlas/paper-vii/transfer_ground.png","image_sha256":"b470b5bda7aafb8d6351d4a743f95f8e5d48e07c28cb6d15fce22af69483ab13","pdf_page":153,"crop_pdf_points":[92,235,517,358],"nodes":[{"text":"$1$","source_text":"$1$","position":[2.0,2.9],"line":4},{"text":"$1/2$","source_text":"$1/2$","position":[6.0,1.7],"line":4},{"text":"$1/4$","source_text":"$1/4$","position":[10.0,1.1],"line":4},{"text":"$T^n:\\quad1,\\ 2^{-n},\\ 4^{-n}\\qquad\\|T^n-P_0\\|=2^{-n}$","source_text":"$T^n:\\quad1,\\ 2^{-n},\\ 4^{-n}\\qquad\\|T^n-P_0\\|=2^{-n}$","position":[6.0,0.1],"line":5}],"node_order":[0,1,2,3],"reading_aid":{"alt":"Three positive transfer eigenvalues have different decay rates.","description":"The bars show the three diagonal transfer values at unit time. 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The bottom route first refines the grid at fixed positive time and then takes the small-time limit. The arrows retain the distinct outcomes stated in the source.","blocks":[{"kind":"text","nodes":[0,1,2,3]}]}},"constraint_image":{"name":"constraint_image","plate":38,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/constraint_image.tex","source_sha256":"7014f1c5129efc4211c24c793d9ef76b06a3b6f6316348cf69566bebcc2b4135","image":"assets/atlas/paper-vii/constraint_image.png","image_sha256":"7fa43a62a7b437895d9dc23c61f5c55e642dd1dfbdc441f635a43723ebfc9424","pdf_page":155,"crop_pdf_points":[92,197,517,320],"nodes":[{"text":"coarse $\\CC^3$\\newline $0\\oplus\\sigma_a$","source_text":"coarse $\\CC^3$\\\\$0\\oplus\\sigma_a$","position":[3.0,1.7],"line":4},{"text":"$R$","source_text":"$R$","position":[6.0,1.7],"line":6},{"text":"embedded image\\newline same constraints","source_text":"embedded image\\\\same constraints","position":[8.9,1.7],"line":7},{"text":"new coordinate","source_text":"new coordinate","position":[11.4,1.7],"line":7},{"text":"closure on $\\operatorname{im}R$ transfers, and the new complement needs its own test","source_text":"closure on $\\operatorname{im}R$ transfers, and the new complement needs its own test","position":[6.0,0.15],"line":8}],"node_order":[0,1,2,3,4],"reading_aid":{"alt":"A coarse constraint system embeds into a larger space with a new complement.","description":"The arrow maps the coarse box into the shaded image inside the larger box. A separately marked new coordinate lies outside that image. The note distinguishes transferred closure on the image from the test still needed on the complement.","blocks":[{"kind":"text","nodes":[0,1,2,3,4]}]}},"tower_tail":{"name":"tower_tail","plate":39,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/tower_tail.tex","source_sha256":"e2b7267f8c4760e504a289f33ffe9df01f8016b903ee9ba8dedf83719ac9ba9f","image":"assets/atlas/paper-vii/tower_tail.png","image_sha256":"61097ca7db59152d02c8d130b5ef7c5c24f7a60a9effcfa90d17d73cefef4eec","pdf_page":156,"crop_pdf_points":[92,210,517,333],"nodes":[{"text":"$\\CC^n$\\newline $H_n=\\operatorname{diag}(0,1,1/2,\\ldots)$","source_text":"$\\CC^n$\\\\$H_n=\\operatorname{diag}(0,1,1/2,\\ldots)$","position":[2.0,2.2],"line":4},{"text":"$\\CC^{n+1}$\\newline old energies retained","source_text":"$\\CC^{n+1}$\\\\old energies retained","position":[9.5,2.2],"line":5},{"text":"$R_n$","source_text":"$R_n$","position":null,"line":5},{"text":"fixed $\\CC^2$: $H_n=\\operatorname{diag}(0,1+2^{-n})$\\newline tail $\\|H_n-H\\|=2^{-n}$","source_text":"fixed $\\CC^2$: $H_n=\\operatorname{diag}(0,1+2^{-n})$\\\\tail $\\|H_n-H\\|=2^{-n}$","position":[6.0,0.6],"line":6}],"node_order":[0,1,2,3],"reading_aid":{"alt":"Compatible embedded Hamiltonians are compared with a fixed-space tail estimate.","description":"The upper arrow embeds one coordinate space into the next while retaining the old energies. The lower box gives a different construction on one fixed two-dimensional space and its displayed norm error. The plate keeps the hypotheses of these routes separate.","blocks":[{"kind":"text","nodes":[0,1,2,3]}]}},"trace_topology":{"name":"trace_topology","plate":40,"source":"papers/paper-vii/current/editable/traditional/visual_graphics_extended/trace_topology.tex","source_sha256":"b3d0d084672acd495968d616cb6a35ba66f54372a75f6fff5319f0172fd5224b","image":"assets/atlas/paper-vii/trace_topology.png","image_sha256":"e4bd45371bda30135372fd1d46cea99ec73412cb7f8b74717f46562aa021b82d","pdf_page":157,"crop_pdf_points":[92,220,517,343],"nodes":[{"text":"$A_n=P_n/n$\\newline $n$ equal eigenvalues $1/n$","source_text":"$A_n=P_n/n$\\\\$n$ equal eigenvalues $1/n$","position":[2.8,1.9],"line":4},{"text":"$\\|A_n\\|=1/n\\to0$\\newline $\\|A_n\\|_1=\\operatorname{Tr}A_n=1$","source_text":"$\\|A_n\\|=1/n\\to0$\\\\$\\|A_n\\|_1=\\operatorname{Tr}A_n=1$","position":[8.8,1.9],"line":5},{"text":"positive trace-class control: norm convergence alone does not preserve a trace","source_text":"positive trace-class control: norm convergence alone does not preserve a trace","position":[6.0,0.45],"line":6}],"node_order":[0,1,2],"reading_aid":{"alt":"A growing-rank sequence has vanishing norm but constant trace.","description":"The left box defines the finite-rank operators and their repeated eigenvalues. The right box contrasts their operator norm with their trace norm and trace. The bottom note records the limitation of norm convergence for this quantity.","blocks":[{"kind":"text","nodes":[0,1,2]}]}}},"manuscript_references":{"1":[["Section II.1","read/paper-vii/effective-dynamics-and-towers.html#L1"],["Section II.7","read/paper-vii/effective-dynamics-and-towers.html#L936"],["Section II.9","read/paper-vii/effective-dynamics-and-towers.html#L1488"]],"2":[["Definition II.2.1","read/paper-vii/effective-dynamics-and-towers.html#L87"],["Definition II.7.1","read/paper-vii/effective-dynamics-and-towers.html#L943"]],"3":[["Proposition II.2.5","read/paper-vii/effective-dynamics-and-towers.html#L174"]],"4":[["Theorem II.2.2","read/paper-vii/effective-dynamics-and-towers.html#L119"],["Corollary II.2.4","read/paper-vii/effective-dynamics-and-towers.html#L153"]],"5":[["Theorem II.2.3","read/paper-vii/effective-dynamics-and-towers.html#L133"]],"6":[["Corollary II.3.3","read/paper-vii/effective-dynamics-and-towers.html#L272"]],"7":[["Theorem 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