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Contents Preface and volume guide (1) 1 Preface and volume guide Chapter 1. Quantum–Arithmetic Bridge (17) 2 Overview 3 The mathematical object being transported 4 Fourier conventions and the finite proof chain 5 Coprimality becomes an integer signal 6 Rational character-order projectors 7 Selection, residuals, and information loss 8 Coefficient-residue lanes and polyphase transport 9 The exact modulus-twelve arithmetic calculation 10 Filtering a signal with something to remove 11 Normalized states and literal unitary transport 12 Relative phase is absent from one histogram and present in another 13 The selector as a projective measurement 14 Exact arithmetic coordinates and their complex meaning 15 The central finite bridge theorem 16 A continuous phase family with a fixed initial histogram 17 Phase replacement and dephasing are different operations 18 What is complete and what remains a separate question Chapter 2. Arithmetic in Fourier space (25) 19 One arithmetic selector, three realizations 20 Coprimality selects primitive roots 21 The selected phases add and cancel 22 Interference yields an integer signal 23 Why the phase sum is an integer 24 The spectrum recovers the selector 25 A diagonal selector becomes a rational filter 26 Give the projector something to remove 27 Account for the entire input 28 Projection and unitary change are different operations 29 All twelve modes belong to six order classes 30 Six exact filters in one ambient space 31 A packet's capacity is not its occupied energy 32 The arithmetic signal is a coherent amplitude vector 33 Amplitudes and probabilities tell different stories 34 Identical histograms can hide orthogonal states 35 A basis change makes the hidden distinction observable 36 Move continuously between the two states 37 A real three-dimensional view of one amplitude 38 Rearrange twelve samples into three exact lanes 39 Phase factors connect the lane spectra 40 The construction extends beyond this picture 41 State exactly what each passage preserves 42 Exact data, floating rendering, and formal coverage 43 What the construction establishes Chapter 3. Foundations of packet simulation (6) 44 The symmetry-collapse question 45 Projectors and quantum packets 46 Uniform circuits and the classical cost model 47 Exact packet simulation and polynomial collapse 48 Approximate symmetry and leakage certificates 49 Moving packets, dynamic frames, and sparse transfer graphs Chapter 4. Exact symmetry families (11) 50 Fixed charge and exact-number circuits 51 The exact weight-two benchmark 52 Coarse packets and the honest limitation 53 Permutation-symmetric circuits and Dicke coordinates 54 Schur–Weyl coordinates and total-spin packets 55 Permutation-invariant mixed states and commutant compression 56 Orbit-quotient quantum walks 57 Few bosons and few fermions 58 Scaling phase diagrams for charge, particles, and local dimension 59 Stabilizer syndromes and Clifford packets 60 Gaussian, fermionic, and matchgate compression Chapter 5. States, measurements, and open dynamics (4) 61 Density operators: populations and coherences 62 Channels, Kraus maps, and Lindblad generators 63 Measurement, conditioning, and classical feedback 64 Block encodings and polynomial operator transforms Chapter 6. The complexity boundary (4) 65 Compressed packet simulation 66 Global interference and the failure of local packet collapse 67 Simulation complexity versus decision complexity 68 Existence defects and quantum combinatorial realizations Chapter 7. Algorithms and open directions (5) 69 Architecture of a symmetry-collapse compiler 70 Benchmark and validation protocol 71 Atlas of candidate symmetry collapses 72 Open problems 73 Conclusion Chapter 8. General theory of temporary escape (4) 74 Nested gate models and distortion invariants 75 Catalytic holonomy, excursion geometry, and resource reuse 76 First arrival, spectral packets, algebraic rarity, and tensor regularization 77 Matroidal state geometry, quiver factorizations, and root-system catalysis Chapter 9. The complete linear core (14) 78 Introduction 79 Circuit model and metric comparison 80 The universal six-row atom 81 Catalytic normal forms and the six-role threshold 82 Exact finite enumeration 83 Projective, matroidal, and spectral structure 84 Interval-tree certificates and connected higher-dimensional families 85 Bounded and weighted catalytic resources 86 Algebraic rarity of catalytic defects 87 Recursive, block, and tensor amplification 88 Discussion and open problems 89 The twenty binary dimension-six defects 90 Search algorithms and correctness 91 Reproducibility ledger Chapter 10. Quantum and reversible gate families (6) 92 Qudit SUM gates and ring-valued linear logic 93 Catalytic basis escape in symplectic Clifford synthesis 94 Temporary nonlinearity: affine logic versus Toffoli-assisted synthesis 95 Phase-polynomial and Clifford+T catalysis 96 Gaussian, matchgate, and fermionic catalytic models 97 Symmetry, ancillas, measurement, and architecture Chapter 11. Packets, types, holonomy, and restricted realization (1) 98 Packets, types, holonomy, and restricted realization Chapter 12. The computational atlas and open directions (3) 99 The exact computational atlas 100 Reusable theorem templates 101 Research program and paper architecture Chapter 13. Scope, notation, and computational model (5) 102 Scope, notation, and computational model 103 The restored classical oracle 104 Comparison with ordinary circuits 105 The degree bound used throughout 106 A degree invariant that retains sharing Chapter 14. Exact cut comparison and safe discrepancies (2) 107 Exact cut comparison and safe discrepancies 108 What the comparison does and does not require Chapter 15. Complete safe-packet degree (4) 109 Complete safe-packet degree 110 The restored lower bound 111 Exact evaluation-span certificates 112 Monotone specialization Chapter 16. Calibration families and exact small oracles (3) 113 Calibration families and exact small oracles 114 An explicit moment certificate 115 All functions on at most three inputs Chapter 17. Homogeneous projection: cost and safety (3) 116 Homogeneous projection: cost and safety 117 A compact whole function 118 Extracting the quadratic grade Chapter 18. Faithful linear gluing algebras (2) 119 Faithful linear gluing algebras 120 The quantified model boundary Chapter 19. Weighted witness fingerprints (3) 121 Weighted witness fingerprints 122 Exact top-coefficient extraction 123 The conditional algorithmic consequence Chapter 20. The overlap obstruction to composition (1) 124 The overlap obstruction to composition Chapter 21. Circuit examples in view (25) 125 The model: boundary restoration and Boolean degree 126 An arbitrary cut compares two reversible computations 127 Why completeness is pointwise, and why no decoder is needed 128 A dirty scratch wire can be fully restored 129 The output direction can move away from the target 130 Degree bounds must retain shared register products 131 The invariant handles duplicates, NOT, and CNOT 132 One safe packet need not cover every accepting point 133 Balanced cuts turn coverage into a Toffoli bound 134 An affine shift reveals a harder accepting point 135 A degree-one dual certificate for two-input AND 136 The finite evaluation-span dual in full 137 Every function through three inputs has a matching oracle 138 Projection is defined in the original coordinates 139 Exact nonlinear projection costs versus actual input length 140 At m=10, the total-gate accounting points the other way 141 Quadratic projection can destroy safety 142 Witness indicators are orthogonal idempotents 143 Faithful linear multiplicative summaries need 2^m dimensions 144 Distinct witness monomials prevent formal cancellation 145 A short expression inside a trace is not a computed trace 146 Disjoint-variable factorization uses each predicate's own block 147 Overlap breaks the attempted context-free product rule 148 Randomized detection is conditional on efficient evaluation 149 Reproduce the figures and inspect their exact evidence Appendix A. The formal arithmetic bridge (19) 150 What this guide certifies 151 The typed mathematical target 152 Revision reconciliation 153 Architecture of the proof 154 The rational signal and its filter 155 The exact phase arithmetic 156 Finite indices and normalized Fourier certificates 157 Why the coordinate certificates mean complex identities 158 Matrix transport and actual unitarity 159 The exponential sign and square-root normalization 160 Hermitian projection and phase-sensitive states 161 Two exact information-loss statements 162 The publication theorem in full 163 Manuscript-to-Lean concordance 164 Trust boundary and current verification 165 Controls, failed attempts, and reproducibility 166 Complete active theorem inventory 167 What completion means here 168 The original finite bridge and its extension Appendix B. Symmetry: supplementary proofs and computation (4) 169 Proof details and auxiliary derivations 170 Python API and reproduction commands 171 Benchmark data ledger 172 Theorem and evidence-status ledger Appendix C. Gate matrices, census data, and open obligations (4) 173 Gate matrices and symbolic conventions 174 Finite census tables and witnesses 175 Python and C++ interface 176 Conjecture and obligation ledger Appendix D. The formal circuit model (22) 177 The purpose and evidence contract 178 The formal endpoint 179 Model and definitions 180 Why sharing does not break the degree bound 181 From the cut to a numerical lower bound 182 Affine output adjustment is an actual circuit 183 Verification and proof boundaries 184 Reading the formal model literally 185 Certificates, semantic minima and natural arithmetic 186 The proof dependency route 187 Formal and computational evidence 188 Negative controls and the trust boundary 189 Reproduce and inspect the authoritative files 190 Complete declaration register 191 Gates: 16 declarations 192 CutComparison: 13 declarations 193 Examples: 11 declarations 194 Degree: 19 declarations 195 MultiplicativeDegree: 12 declarations 196 SafeDegree: 22 declarations 197 AffineOutput: 14 declarations 198 DegreeChecks: 8 declarations Appendix E. Claim coverage and theorem index (5) 199 Claim coverage and theorem index 200 Dependency structure 201 Complete numbered-result index 202 Baseline domains reproduced by the full run 203 Excluded inferences Appendix F. Verification and proof boundaries (1) 204 Verification and proof boundaries Shared glossary (1) 205 Shared glossary Sources and references (1) 206 Source lineage References (1) 207 References PDF and companion guides This HTML edition preserves the authored wording, mathematics, captions, and recorded limits of the 344-page volume. Verification statements describe the retained edition and its records. This presentation adds no proof review or Lean execution.
