Quantum and Circuits statements and proofs · Komorebi

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

Fourier conventions and the finite proof chain

Coprimality becomes an integer signal

Rational character-order projectors

Theorem QA06: Rational kernels and complete divisor decomposition Read proof

Selection, residuals, and information loss

Theorem QA07: Projector geometry and exact residual reconstruction Read proof

Coefficient-residue lanes and polyphase transport

The exact modulus-twelve arithmetic calculation

Theorem QA10: The exact phase and transform tables Read proof

Filtering a signal with something to remove

Theorem QA11: Exact filter recovery and orthogonal energy Read proof

Normalized states and literal unitary transport

Theorem QA12: Exact preparation and recovery Read proof

Relative phase is absent from one histogram and present in another

The selector as a projective measurement

Theorem QA16: Selection probability and conditional state Read proof

Exact arithmetic coordinates and their complex meaning

Theorem QA17: Sound interpretation of the exact operations Read proof

The central finite bridge theorem

Theorem QA18: Arithmetic, rational signal, and unitary realization Read proof

A continuous phase family with a fixed initial histogram

Theorem QA19: Relative-phase response Read proof

Phase replacement and dephasing are different operations

Theorem QA20: The dephased selected probability Read proof

The symmetry-collapse question

Projectors and quantum packets

Exact packet simulation and polynomial collapse

Approximate symmetry and leakage certificates

Moving packets, dynamic frames, and sparse transfer graphs

Fixed charge and exact-number circuits

Corollary 4.1: Fixed-charge polynomial simulation Read proof

Permutation-symmetric circuits and Dicke coordinates

Schur–Weyl coordinates and total-spin packets

Principle 4.4: Multiplicity audit

Permutation-invariant mixed states and commutant compression

Orbit-quotient quantum walks

Theorem 4.9: Orbit-quotient collapse Read proof

Few bosons and few fermions

Corollary 4.10: Few-particle collapse Read proof

Scaling phase diagrams for charge, particles, and local dimension

Measurement, conditioning, and classical feedback

Proposition 5.1: Branch-merging simulation Read proof

Compressed packet simulation

Simulation complexity versus decision complexity

Conjecture 6.3: Universal polynomial packetization

Open problems

Nested gate models and distortion invariants

Catalytic holonomy, excursion geometry, and resource reuse

First arrival, spectral packets, algebraic rarity, and tensor regularization

Matroidal state geometry, quiver factorizations, and root-system catalysis

Circuit model and metric comparison

The universal six-row atom

Catalytic normal forms and the six-role threshold

Exact finite enumeration

Lemma 9.28: Meet-in-the-middle completeness Read proof

Projective, matroidal, and spectral structure

Interval-tree certificates and connected higher-dimensional families

Bounded and weighted catalytic resources

Algebraic rarity of catalytic defects

Recursive, block, and tensor amplification

Discussion and open problems

Search algorithms and correctness

Qudit SUM gates and ring-valued linear logic

Catalytic basis escape in symplectic Clifford synthesis

Temporary nonlinearity: affine logic versus Toffoli-assisted synthesis

Protocol 10.10: First four-bit threshold search

Phase-polynomial and Clifford+T catalysis

Gaussian, matchgate, and fermionic catalytic models

Definition 10.13: Bogoliubov catalytic defect

Symmetry, ancillas, measurement, and architecture

Definition 10.14: Branch-catalytic realization

Packets, types, holonomy, and restricted realization

The exact computational atlas

Algorithm 12.1: Breadth-first defect census

Reusable theorem templates

Research program and paper architecture

Comparison with ordinary circuits

Theorem 13.1: Classical/restored simulation comparison Read proof

The degree bound used throughout

Lemma 13.2: Degree per multiplication Read proof

A degree invariant that retains sharing

Theorem 13.3: Register-product degree invariant Read proof

Exact cut comparison and safe discrepancies

Theorem 14.1: Cut-comparison identity and complete coverage Read proof

Complete safe-packet degree

The restored lower bound

Theorem 15.3: Affine-optimized complete-cover bound Read proof

Exact evaluation-span certificates

Monotone specialization

Calibration families and exact small oracles

All functions on at most three inputs

Theorem 16.5: Exact small restored-oracle calibration Read proof

Homogeneous projection: cost and safety

A compact whole function

Theorem 17.3: Exact nonlinear cost of the whole function Read proof

Extracting the quadratic grade

Faithful linear gluing algebras

Weighted witness fingerprints

Theorem 19.1: Exact witness fingerprint Read proof

Exact top-coefficient extraction

The conditional algorithmic consequence

The overlap obstruction to composition