Paper VIIStatements and proofs
Statements and proofs
Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.
Statements by chapter
The quantum seed realization
Quantum seed rank and cyclicity
Cyclic symmetry packets
Quantum circuits as packet paths
Symmetry-resolved Hamiltonians and leakage
Moving packets, connections, and curvature
Measurements and quantum channels
Kraus transfer
Quantum error correction as obstruction-packet annihilation
Exact effective dynamics
Finite quantum geometry
Spectral dimension and continuum diagnostics
Constraint packets and the quantum-spacetime boundary
II. Effective dynamics and refinement towers
- Definition II.2.1: Finite quantum seed system
- Theorem II.2.2: Orbit dimension of one seed Read proof
- Theorem II.2.3: Quantum seed rank Read proof
- Corollary II.2.4: Local orbit and Schmidt rank Read proof
- Proposition II.2.5: Computational address orbit Read proof
- Remark II.2.6
- Theorem II.3.1: Finite group isotypic projectors Read proof
- Theorem II.3.2: Representation matrix units Read proof
- Corollary II.3.3: Cyclic charge projectors Read proof
- Definition II.3.4: Operator charge
- Theorem II.3.5: Charge selection rule Read proof
- Corollary II.3.6: Charge addition Read proof
- Theorem II.3.7: Tensor charge refinement Read proof
- Corollary II.3.8: Tensor selection rule Read proof
- Remark II.3.9
- Theorem II.4.1: Exact circuit path expansion Read proof
- Theorem II.4.2: Packet unitarity identities Read proof
- Corollary II.4.3: Probability conservation from one packet Read proof
- Theorem II.4.4: Path interference formula Read proof
- Proposition II.4.5: Charge pruning of circuit paths Read proof
- Example II.4.6: Bell circuit
- Theorem II.5.1: Exact invariant packets Read proof
- Lemma II.5.2: Off diagonal norm Read proof
- Theorem II.5.3: Fixed packet leakage Read proof
- Corollary II.5.4: Leakage probability Read proof
- Theorem II.6.1: Canonical packet connection Read proof
- Theorem II.6.2: Exact Kato transport Read proof
- Theorem II.6.3: Moving frame Hamiltonian Read proof
- Theorem II.6.4: Moving packet leakage Read proof
- Theorem II.6.5: Direct packet rotation Read proof
- Theorem II.6.6: Convergence of short packet rotations Read proof
- Theorem II.6.7: Nonabelian packet connection Read proof
- Corollary II.6.8: Bloch sphere curvature Read proof
- Definition II.7.1: Two sided operator packet
- Theorem II.7.2: Complete operator packet system Read proof
- Definition II.7.3: Packet dephasing
- Theorem II.7.4: Dephasing as conditional expectation Read proof
- Corollary II.7.5: Block diagonal measurement invariance Read proof
- Theorem II.7.6: Exact channel block calculus Read proof
- Standard fact II.7.7: Finite Kraus and Choi criteria
- Theorem II.7.8: Coherence order projectors Read proof
- Theorem II.7.9: Covariant channels preserve coherence order Read proof
- Theorem II.7.10: Choi symmetry criterion Read proof
- Corollary II.7.11: Choi packet decomposition Read proof
- Example II.7.12: Amplitude damping
- Standard fact II.7.13: Finite Lindblad generator
- Theorem II.7.14: Charge preservation by a Lindblad generator Read proof
- Corollary II.7.15: Independent coherence evolution Read proof
- Definition II.8.1: Code compression obstruction
- Theorem II.8.2: Knill Laflamme as obstruction vanishing Read proof
- Theorem II.8.3: Orthogonal syndrome construction Read proof
- Theorem II.8.4: Exact recovery on the error span Read proof
- Corollary II.8.5: Syndrome measurement preserves logical information Read proof
- Definition II.8.6: Approximate Knill Laflamme defect
- Proposition II.8.7: Meaning of the approximate defect Read proof
- Example II.8.8: Three qubit repetition code
- Standard fact II.8.9: Correctable observable algebras
- Definition II.9.1: Feshbach map and self energy
- Theorem II.9.2: Exact projected resolvent Read proof
- Theorem II.9.3: Spectral equivalence Read proof
- Theorem II.9.4: Upper half plane sign Read proof
- Proposition II.9.5: Uniform self energy bounds Read proof
- Proposition II.9.6: Low energy expansion Read proof
- Proposition II.9.7: Bare truncation error identity Read proof
- Theorem II.9.8: Exact retained memory equation Read proof
- Theorem II.9.9: Positive definite quantum memory Read proof
- Corollary II.9.10: Finite oscillatory memory expansion Read proof
- Theorem II.9.11: Exact local closure criterion Read proof
- Theorem II.9.12: Stability of the self energy Read proof
- Corollary II.9.13: Uniform Feshbach control Read proof
- Theorem II.10.1: Finite Euclidean transfer Read proof
- Theorem II.10.2: Hamiltonian reconstruction from transfer Read proof
- Theorem II.10.3: Finite reflection positivity Read proof
- Corollary II.10.4: Reflection positive correlators Read proof
- Proposition II.10.5: Ground packet convergence Read proof
- Proposition II.10.6: Symmetry resolved transfer Read proof
- Definition II.11.1: Heat trace and spectral dimension
- Proposition II.11.2: Finite scale limits Read proof
- Proposition II.11.3: Heat packet decomposition Read proof
- Theorem II.11.4: Low mode dispersion in dimension (d ) Read proof
- Theorem II.11.5: Fixed time heat trace convergence Read proof
- Standard fact II.11.6: Theta transformation
- Theorem II.11.7: Continuum torus small time dimension Read proof
- Corollary II.11.8: Rigorous intermediate dimension window Read proof
- Definition II.12.1: Universal closure obstruction
- Theorem II.12.2: Finite constraint closure criterion Read proof
- Proposition II.12.3: Physical compression is not a closure test Read proof
- Theorem II.12.4: Invariance of the physical space Read proof
- Theorem II.12.5: Exact constraint transport Read proof
- Proposition II.12.6: Approximate constraint transport Read proof
- Definition II.13.1: Finite quantum refinement tower
- Theorem II.13.2: Bounded Hamiltonian direct limit Read proof
- Theorem II.13.3: Exact packet direct limit Read proof
- Theorem II.13.4: Norm summable Hamiltonian tower Read proof
- Corollary II.13.5: Limit of packet leakage Read proof
- Theorem II.13.6: Reflection positivity is closed under limits Read proof
- Theorem II.13.7: Constraint closure under norm convergence Read proof
- Theorem II.13.8: Heat trace convergence from trace norm Read proof
- Remark II.13.9
- Definition II.14.1: Anomaly free continuum constraint system
- Definition II.14.2: Reflection positive Euclidean limit
- Definition II.14.3: Einstein infrared criterion
- Definition II.14.4: Massless spin two criterion