This dashboard provides a complete inventory and formalization status map for Book IV: Categorical Microcosm. It covers particle physics, quantum mechanics, the force architecture, and the complete microcosm. Each registry object carries a scope label (established, tau-effective, or conjectural) and a formalization status. Use the statistics below to assess coverage, the dependency summary to trace proof chains, and the object list to navigate to individual entries in the registry.
Book IV Registry Dashboard
Generated: 2026-03-10
Statistics
- Total objects: 1292
-
By type: corollary: 1 definition: 392 lemma: 12 proposition: 231 remark: 443 theorem: 213 -
Formalization: formalized: 973 not_applicable: 278 skeleton: 41 -
Scope: conjectural: 42 established: 27 metaphorical: 4 tau-effective: 1219
Dependency Summary
- Max chain depth: 88 (IV.P79)
- Most depended-on: IV.T152 (19 uses), III.D13 (15 uses), IV.D07 (15 uses)
- No dependencies (foundational): IV.D10, IV.D12, IV.D16, IV.D22, IV.D26, IV.D38, IV.D44, IV.R12, IV.D49, IV.D55 …
Part Coverage
| Part | Objects | Formalized | Planned | N/A | |——|———|————|———|—–| | 0 | 3 | 0 | 0 | 3 | | 1 | 133 | 93 | 0 | 40 | | 2 | 165 | 146 | 0 | 15 | | 3 | 203 | 156 | 0 | 41 | | 4 | 238 | 202 | 0 | 31 | | 5 | 294 | 190 | 0 | 82 | | 6 | 92 | 74 | 0 | 15 | | 7 | 100 | 69 | 0 | 31 | | 8 | 41 | 21 | 0 | 19 | | part01 | 5 | 5 | 0 | 0 | | part03 | 5 | 5 | 0 | 0 | | part04 | 10 | 10 | 0 | 0 | | part05 | 3 | 2 | 0 | 1 |
Object List
| ID | Type | Name | Scope | Lean |
|—-|——|——|——-|——|
| IV.D01 | definition | Sector Physics Template | tau-e | form |
| IV.D02 | definition | EM Sector at E₁ | tau-e | form |
| IV.D03 | definition | Strong Sector at E₁ | tau-e | form |
| IV.D04 | definition | Higgs Sector at E₁ | tau-e | form |
| IV.D05 | definition | Gravity Sector at E₁ | tau-e | form |
| IV.D06 | definition | Weak Sector at E₁ | tau-e | form |
| IV.D07 | definition | Coupling Formula Map | tau-e | form |
| IV.T01 | theorem | Temporal Complement | tau-e | form |
| IV.T02 | theorem | Temporal Multiplicative Closure | tau-e | form |
| IV.P01 | proposition | All Couplings Positive | tau-e | form |
| IV.D08 | definition | Spectral Fine Structure | tau-e | form |
| IV.P02 | proposition | α Numerical Range | tau-e | form |
| IV.R01 | remark | Holonomy vs Spectral | tau-e | form |
| IV.R02 | remark | Wrong Formula Correction | tau-e | form |
| IV.P03 | proposition | Power Hierarchy | tau-e | form |
| IV.D09 | definition | Primary Invariant | tau-e | form |
| IV.D10 | definition | Carrier Type | tau-e | form |
| IV.D11 | definition | Physical Quantity Template | tau-e | form |
| IV.D12 | definition | Particle Kind | tau-e | form |
| IV.D13 | definition | Planck Character | tau-e | form |
| IV.D14 | definition | Uncertainty Product | tau-e | form |
| IV.D15 | definition | Sector Lift Functor | tau-e | form |
| IV.R03 | remark | Physical Constants Core | tau-e | form |
| IV.T03 | theorem | Sector Lifts σ-Equivariant | tau-e | form |
| IV.D16 | definition | Defect Component | tau-e | form |
| IV.D17 | definition | Defect Tuple | tau-e | form |
| IV.D18 | definition | Fluid Regime | tau-e | form |
| IV.D19 | definition | Regime Signature | tau-e | form |
| IV.D20 | definition | Mass Index | tau-e | form |
| IV.D21 | definition | Energy Index | tau-e | form |
| IV.D23 | definition | Mass-Energy Relation | tau-e | form |
| IV.D22 | definition | Speed Constant | tau-e | form |
| IV.R04 | remark | Neutron First Ontic Particle | tau-e | form |
| IV.D24 | definition | Entropy Splitting | tau-e | form |
| IV.D25 | definition | Defect Budget | tau-e | form |
| IV.P04 | proposition | No-Running Principle | tau-e | form |
| IV.T04 | theorem | Euler Budget Conservation | tau-e | form |
| IV.D26 | definition | SI Reference Table | estab | form |
| IV.D27 | definition | SI Measured Constants | estab | form |
| IV.D28 | definition | Weinberg Near-Match | estab | form |
| IV.D29 | definition | Strong Near-Match | estab | form |
| IV.P05 | proposition | All Near-Matches in Range | tau-e | form |
| IV.D30 | definition | Calibration Anchor | tau-e | form |
| IV.D31 | definition | τ-to-SI Conversion | tau-e | form |
| IV.T05 | theorem | Parameter Count | tau-e | form |
| IV.T06 | theorem | τ-Collapse (5→1) | tau-e | form |
| IV.R07 | remark | Ontological Priority | tau-e | form |
| IV.D32 | definition | Tau Physical Scale | tau-e | form |
| IV.D33 | definition | Speed of Light | tau-e | form |
| IV.D34 | definition | Planck Constant | tau-e | form |
| IV.D35 | definition | Coulomb Constant | tau-e | form |
| IV.D36 | definition | Vacuum Permittivity | tau-e | form |
| IV.D37 | definition | Vacuum Permeability | tau-e | form |
| IV.T07 | theorem | Maxwell Relation | estab | form |
| IV.T08 | theorem | Coulomb-Permittivity | estab | form |
| IV.R08 | remark | G Frontier | conje | form |
| IV.D38 | definition | Ledger Entry | tau-e | form |
| IV.D39 | definition | Complete Constants Ledger | tau-e | form |
| IV.T09 | theorem | Ledger Count | estab | form |
| IV.R09 | remark | Self-Assessment | tau-e | form |
| IV.D40 | definition | Epstein Zeta Structure | tau-e | form |
| IV.D41 | definition | Chowla-Selberg Decomposition | tau-e | form |
| IV.T10 | theorem | Leading Exponent | tau-e | form |
| IV.R10 | remark | Normalization | tau-e | form |
| IV.D42 | definition | Lemniscate Three-Fold | tau-e | form |
| IV.D43 | definition | Spectral Distance √3 | tau-e | form |
| IV.T11 | theorem | Three-Fold Distance Squared | tau-e | form |
| IV.P06 | proposition | √3 Approximation Quality | estab | form |
| IV.R11 | remark | √3 Triad | tau-e | form |
| IV.D44 | definition | Triple Holonomy | tau-e | form |
| IV.D45 | definition | Holonomy Correction | tau-e | form |
| IV.T12 | theorem | Correction Smallness | tau-e | form |
| IV.R12 | remark | Charge Conjugation | tau-e | form |
| IV.D46 | definition | Mass Ratio Bulk Term | tau-e | form |
| IV.D47 | definition | Level 0 Formula | tau-e | form |
| IV.D48 | definition | Level 1+ Formula | tau-e | form |
| IV.T13 | theorem | Bulk Overshoots | tau-e | form |
| IV.T14 | theorem | Level 0 Range | tau-e | form |
| IV.T15 | theorem | Derivation Chain Complete | tau-e | form |
| IV.P07 | proposition | All Links Tau-Effective | tau-e | form |
| IV.D49 | definition | CR-Manifold | tau-e | form |
| IV.P08 | proposition | Integrability Criterion | tau-e | form |
| IV.D50 | definition | CR-Structure on τ³ | tau-e | form |
| IV.P09 | proposition | Integrability of τ³ CR-Structure | tau-e | form |
| IV.D51 | definition | Character Modes | tau-e | form |
| IV.D52 | definition | CR-Address | tau-e | form |
| IV.D53 | definition | Address Precision (ch16) | tau-e | form |
| IV.L01 | lemma | Wedge Holonomy | tau-e | form |
| IV.T16 | theorem | CR Parity Constraint | tau-e | form |
| IV.D54 | definition | CR-Admissible Sublattice | tau-e | form |
| IV.P10 | proposition | Density Halving | tau-e | form |
| IV.R13 | remark | Physical Factor of Two | tau-e | — |
| IV.T17 | theorem | Emergence of Spin-1/2 | tau-e | form |
| IV.D55 | definition | Character on a Space | tau-e | form |
| IV.P11 | proposition | Characters on T² | tau-e | form |
| IV.D56 | definition | Character Variety of T² | tau-e | form |
| IV.P12 | proposition | Automatic Quantization | tau-e | form |
| IV.D57 | definition | Address Precision (ch17) | tau-e | form |
| IV.D58 | definition | Geometric Charge | tau-e | form |
| IV.P13 | proposition | Charge Quantization from Winding | tau-e | form |
| IV.R14 | remark | Fractional Charges and Confinement | tau-e | form |
| IV.P14 | proposition | Energy Duality | tau-e | form |
| IV.D59 | definition | Sharp and Spread States | tau-e | form |
| IV.P15 | proposition | Conjugate Precision Trade-off | tau-e | form |
| IV.R15 | remark | Quasi-Ergodic Coverage | tau-e | form |
| IV.D60 | definition | Space of CR-Functions | tau-e | form |
| IV.P16 | proposition | Algebraic Properties of CR(τ³) | tau-e | form |
| IV.D61 | definition | Canonical Inner Product | tau-e | form |
| IV.P17 | proposition | Inner Product Properties | tau-e | form |
| IV.P18 | proposition | Inner Product Uniqueness | tau-e | form |
| IV.D62 | definition | Holomorphic Hilbert Space | tau-e | form |
| IV.T18 | theorem | Hilbert Space Properties | tau-e | form |
| IV.P19 | proposition | Central Theorem Implies Boundary Determination | tau-e | form |
| IV.T19 | theorem | Orthonormal Basis | tau-e | form |
| IV.P20 | proposition | Spectral Completeness | tau-e | form |
| IV.D63 | definition | Physical State Space | tau-e | form |
| IV.D64 | definition | Entanglement | tau-e | form |
| IV.P21 | proposition | Generic Entanglement | tau-e | form |
| IV.P22 | proposition | Superposition from Linearity | tau-e | form |
| IV.D65 | definition | Holomorphic Vector Field | tau-e | form |
| IV.D66 | definition | Quantum Operator | tau-e | form |
| IV.P23 | proposition | Commutator Equals Lifted Lie Bracket | tau-e | form |
| IV.T20 | theorem | Topological Quantization | tau-e | form |
| IV.T21 | theorem | Canonical Commutation Relation | tau-e | form |
| IV.D67 | definition | Observable | tau-e | form |
| IV.P24 | proposition | X and P Are Self-Adjoint | tau-e | form |
| IV.D68 | definition | Address Uncertainty | tau-e | form |
| IV.D69 | definition | τ-Normal Form for Joint Address | tau-e | form |
| IV.D70 | definition | Phase Transport Witness | tau-e | form |
| IV.D71 | definition | Clopen Position Localization | tau-e | form |
| IV.T22 | theorem | Phase Transport Monotonicity | tau-e | form |
| IV.T23 | theorem | No-Joint-Minimum Theorem | tau-e | form |
| IV.D72 | definition | σ-Equivariant Crossing-Point Mediator | tau-e | form |
| IV.T24 | theorem | Planck Character Uniqueness | tau-e | form |
| IV.D73 | definition | Canonical Saturation State | tau-e | form |
| IV.P25 | proposition | Saturation Equality | tau-e | form |
| IV.T25 | theorem | Heisenberg Uncertainty (Position-Momentum) | tau-e | form |
| IV.T26 | theorem | Heisenberg Uncertainty (Time-Energy) | tau-e | form |
| IV.R16 | remark | Existence Not Measurement | tau-e | — |
| IV.R17 | remark | Role of ℏ_τ in Obstruction | tau-e | — |
| IV.R18 | remark | Determinism and (×,∧)-Tension | tau-e | — |
| IV.D74 | definition | Address Resolution | tau-e | form |
| IV.T27 | theorem | Born Rule | tau-e | form |
| IV.P26 | proposition | Measurement Repeatability | tau-e | form |
| IV.D75 | definition | Decoherence | tau-e | form |
| IV.P27 | proposition | Classical Limit | tau-e | form |
| IV.T28 | theorem | Schrödinger Equation | tau-e | form |
| IV.P28 | proposition | Determinism-Probability Reconciliation | tau-e | form |
| IV.R19 | remark | Measurement Problem Dissolves | tau-e | — |
| IV.R20 | remark | Zero Postulates Significance | tau-e | — |
| IV.D76 | definition | Energy as CR-Tension | tau-e | form |
| IV.D77 | definition | Graph Energy Density | tau-e | form |
| IV.P29 | proposition | Energy-Localization Bound | tau-e | form |
| IV.D78 | definition | Mass as Fiber Stiffness | tau-e | form |
| IV.D79 | definition | Frequency as Base Circulation | tau-e | form |
| IV.T29 | theorem | Energy Duality | tau-e | form |
| IV.T30 | theorem | Energy Conservation | tau-e | form |
| IV.D80 | definition | Holomorphic Entropy | tau-e | form |
| IV.P30 | proposition | Entropy-Mode-Count Bound | tau-e | form |
| IV.D81 | definition | Temporal Direction | tau-e | form |
| IV.T31 | theorem | Second Law of Thermodynamics | tau-e | form |
| IV.T32 | theorem | Structural Arrow of Time | tau-e | form |
| IV.P31 | proposition | Reversibility-Irreversibility Resolution | tau-e | form |
| IV.R21 | remark | Energy-Entropy Complementarity | tau-e | form |
| IV.R22 | remark | No Past Hypothesis Needed | tau-e | form |
| IV.D82 | definition | Photon Mode | tau-e | form |
| IV.T33 | theorem | Photon Masslessness | tau-e | form |
| IV.T34 | theorem | Photon Propagation Speed | tau-e | form |
| IV.P32 | proposition | No Rest Frame | tau-e | form |
| IV.P33 | proposition | Photon Spin and Polarization | tau-e | form |
| IV.T35 | theorem | Dissolution of Wave-Particle Duality | tau-e | form |
| IV.D83 | definition | U(1) Holonomy on T^2 | tau-e | form |
| IV.D84 | definition | Electric Charge | tau-e | form |
| IV.P34 | proposition | Charge of Fundamental Modes | tau-e | form |
| IV.T36 | theorem | Charge Conservation | tau-e | form |
| IV.P35 | proposition | Photon as Boundary Character | tau-e | form |
| IV.P36 | proposition | Photon Coupling Strength | tau-e | form |
| IV.D85 | definition | EM Principal Bundle | tau-e | form |
| IV.D86 | definition | Local Trivialization | tau-e | form |
| IV.D87 | definition | Transition Function | tau-e | form |
| IV.P37 | proposition | EM Bundle Topology | tau-e | form |
| IV.D88 | definition | Section of the EM Bundle | tau-e | form |
| IV.D89 | definition | Gauge Connection | tau-e | form |
| IV.D90 | definition | Covariant Derivative | tau-e | form |
| IV.T37 | theorem | Gauge Invariance Kernel Theorem | tau-e | form |
| IV.D91 | definition | Parallel Transport | tau-e | form |
| IV.P38 | proposition | Parallel Transport is Gauge-Covariant | tau-e | form |
| IV.D92 | definition | Electromagnetic Field Strength | tau-e | form |
| IV.P39 | proposition | Explicit Form of F_mu_nu | tau-e | form |
| IV.T38 | theorem | Gauge Invariance of F_mu_nu | tau-e | form |
| IV.D93 | definition | Aharonov-Bohm Phase | tau-e | form |
| IV.D94 | definition | EM Loop Space | tau-e | form |
| IV.D95 | definition | Sigma-Equivariance | tau-e | form |
| IV.T39 | theorem | tau-AB Kernel Theorem | tau-e | form |
| IV.T40 | theorem | AB Phase Quantization | tau-e | form |
| IV.R23 | remark | Flux Quantization on tau^1 | tau-e | — |
| IV.D96 | definition | Wilson Loop | tau-e | form |
| IV.P40 | proposition | Observable Hierarchy | tau-e | form |
| IV.T41 | theorem | Ambrose-Singer Reconstruction | tau-e | form |
| IV.D97 | definition | Non-Abelian Gauge Field | tau-e | form |
| IV.P41 | proposition | Self-Interaction from Non-Commutativity | tau-e | form |
| IV.P42 | proposition | Non-Abelian Holonomy | tau-e | form |
| IV.P43 | proposition | Categorical Derivation of Gauge Structure | tau-e | form |
| IV.D98 | definition | EM Gauge Bundle | tau-e | form |
| IV.D99 | definition | EM Connection 1-Form | tau-e | form |
| IV.D100 | definition | Electromagnetic Field Tensor | tau-e | form |
| IV.D101 | definition | Electric and Magnetic Field Extraction | tau-e | form |
| IV.T42 | theorem | Homogeneous Maxwell Equations | tau-e | form |
| IV.D102 | definition | Hodge Dual of F | tau-e | form |
| IV.T43 | theorem | Inhomogeneous Maxwell Equations | tau-e | form |
| IV.T44 | theorem | Complete tau-Maxwell System | tau-e | form |
| IV.P44 | proposition | Current Conservation from Gauge Invariance | tau-e | form |
| IV.D103 | definition | Electromagnetic Current 4-Vector | tau-e | form |
| IV.P45 | proposition | Charge Density as Winding-Number Density | tau-e | form |
| IV.P46 | proposition | Current as Defect Transport | tau-e | form |
| IV.T45 | theorem | Source-Defect Correspondence | tau-e | form |
| IV.T46 | theorem | Coulomb’s Law from tau-Maxwell | tau-e | form |
| IV.T47 | theorem | EM Wave Equation | tau-e | form |
| IV.P47 | proposition | Photon-Wave Identity | tau-e | form |
| IV.P48 | proposition | Magnetic Force Does No Work | tau-e | form |
| IV.P49 | proposition | Perturbative Expansion in alpha | tau-e | form |
| IV.D104 | definition | tau-Native Fine-Structure Constant | tau-e | form |
| IV.L02 | lemma | Holonomy Normalization | tau-e | form |
| IV.L03 | lemma | Photon Phase Quantum | tau-e | form |
| IV.L04 | lemma | Mediator Ratio | tau-e | form |
| IV.D105 | definition | Null Transport Mode | tau-e | form |
| IV.P50 | proposition | Photon Uniqueness | tau-e | form |
| IV.T48 | theorem | Spectral Formula for alpha_em | tau-e | form |
| IV.T49 | theorem | Holonomy Formula for alpha_em | tau-e | form |
| IV.D106 | definition | Holonomy Correction Factor | tau-e | form |
| IV.T50 | theorem | No-Running Principle for alpha_em | tau-e | form |
| IV.P51 | proposition | alpha_em versus iota_tau | tau-e | form |
| IV.R24 | remark | Charge Balance |Q_p| = |Q_e| | tau-e | — |
| IV.R25 | remark | Gauge Invariance Is Geometric Not Dynamic | tau-e | form |
| IV.R26 | remark | Lorentz Force as Geometric Deviation | tau-e | — |
| IV.R27 | remark | sqrt(3) Triad | tau-e | form |
| IV.D107 | definition | Sector Polarity | tau-e | form |
| IV.D108 | definition | Polarity-Preserving Interaction | tau-e | form |
| IV.D109 | definition | Channel-Switching Defect Bundle | tau-e | form |
| IV.D110 | definition | W-pm Defect Bundles | tau-e | form |
| IV.D111 | definition | Z0 Defect Bundle | tau-e | form |
| IV.D112 | definition | Chirality | tau-e | form |
| IV.D113 | definition | A-Sector Sigma-Admissibility | tau-e | form |
| IV.D114 | definition | Parity Operator | tau-e | form |
| IV.P52 | proposition | Polarity Assignments | tau-e | form |
| IV.P53 | proposition | Mass of Channel-Switching Defect Bundles | tau-e | form |
| IV.L05 | lemma | Sigma_A-Admissibility Selects Chirality | tau-e | form |
| IV.T51 | theorem | Parity Violation in the A-Sector | tau-e | form |
| IV.P54 | proposition | Parity in Non-Weak Sectors | tau-e | form |
| IV.P55 | proposition | SU(2)L from Lemniscate Automorphisms | tau-e | form |
| IV.R28 | remark | CP Violation and the Crossing Phase | tau-e | form |
| IV.D115 | definition | Crossing-Point Action Space | tau-e | form |
| IV.D116 | definition | Weak Isospin Generators | tau-e | form |
| IV.T52 | theorem | Weak Gauge Group | tau-e | form |
| IV.D117 | definition | Left-Handed Doublets | tau-e | form |
| IV.D118 | definition | Weak Gauge Bosons Before Mixing | tau-e | form |
| IV.P56 | proposition | Charged and Neutral Combinations | tau-e | form |
| IV.D119 | definition | Weak Holonomy Loop | tau-e | form |
| IV.T53 | theorem | Weak Coupling from Sector Parameters | tau-e | form |
| IV.P57 | proposition | Fermi Constant from Weak Coupling | tau-e | form |
| IV.P58 | proposition | EM-Weak Coupling Hierarchy | tau-e | form |
| IV.T54 | theorem | W Mass from Coherence Fixing | conje | form |
| IV.D120 | definition | W Boson Parameters | tau-e | form |
| IV.P59 | proposition | W as Crossing Process | tau-e | form |
| IV.D121 | definition | Weinberg Angle from Tau-Couplings | tau-e | form |
| IV.T55 | theorem | Z Mass from Coherence Fixing | conje | form |
| IV.D122 | definition | Z Boson Parameters | tau-e | form |
| IV.P60 | proposition | Z Field Composition | tau-e | form |
| IV.D123 | definition | The Rho Parameter | tau-e | form |
| IV.T56 | theorem | Tree-Level Rho = 1 | tau-e | form |
| IV.P61 | proposition | Measured Rho | tau-e | form |
| IV.P62 | proposition | Z Heavier Than W | tau-e | form |
| IV.T57 | theorem | Low-Energy Limit of W Exchange (Fermi Limit) | tau-e | form |
| IV.P63 | proposition | Beta-Decay Rate from Tau-Units | tau-e | form |
| IV.P64 | proposition | Invisible Width and N_nu | tau-e | form |
| IV.P65 | proposition | Electroweak Coupling Summary | tau-e | form |
| IV.R29 | remark | Hierarchy Problem in Category Tau | tau-e | form |
| IV.D124 | definition | Neutrino as Time-Eigenmode | tau-e | form |
| IV.T58 | theorem | Spectral Gap Exponent | tau-e | form |
| IV.D125 | definition | Neutrino Flavor Eigenstates | tau-e | form |
| IV.P66 | proposition | Mass Eigenvalue Ratios | conje | form |
| IV.D126 | definition | PMNS Matrix | tau-e | form |
| IV.T59 | theorem | Neutrino Interaction Channels | tau-e | form |
| IV.P67 | proposition | CP Violation from Sigma-Polarity | conje | form |
| IV.R30 | remark | Oscillation as Tau^1 Geometry | tau-e | — |
| IV.D127 | definition | τ-Hypercharge | tau-e | form |
| IV.D128 | definition | Pre-Mixing Electroweak Gauge Group | tau-e | form |
| IV.D129 | definition | Charged W Bosons | tau-e | form |
| IV.D130 | definition | Weinberg Angle | tau-e | form |
| IV.T60 | theorem | Neutral Boson Mixing | tau-e | form |
| IV.P68 | proposition | Coupling Relations | tau-e | form |
| IV.T61 | theorem | Weinberg Angle Prediction | tau-e | form |
| IV.P69 | proposition | Weinberg Angle Residual Analysis | tau-e | form |
| IV.R31 | remark | Weinberg Angle Gap Assessment | tau-e | form |
| IV.D131 | definition | Mixing Compatibility | tau-e | form |
| IV.T62 | theorem | Mixing Uniqueness Theorem | tau-e | form |
| IV.P70 | proposition | No Grand Unification | tau-e | form |
| IV.R32 | remark | Structural Not Energy-Scale Obstruction | tau-e | — |
| IV.D132 | definition | Electroweak Coherence State | tau-e | form |
| IV.D133 | definition | Coherence Fixing (Ch33) | tau-e | form |
| IV.P71 | proposition | Parity Bridge and Mixing | tau-e | form |
| IV.R33 | remark | One Property Two Consequences | tau-e | — |
| IV.D134 | definition | Coherence Fixing (Formal) | tau-e | form |
| IV.R34 | remark | Coherence Fixing vs Mass Generation | tau-e | form |
| IV.P72 | proposition | Crossing-Point Degeneracy | tau-e | form |
| IV.D135 | definition | Finite-Stage Coherence Functional | tau-e | form |
| IV.P73 | proposition | Finite-Stage Vacuum Existence and Uniqueness | tau-e | form |
| IV.D136 | definition | Physical Vacuum Ω*[ω] | tau-e | form |
| IV.T63 | theorem | Vacuum Construction | tau-e | form |
| IV.D137 | definition | Minimality Condition | tau-e | form |
| IV.D138 | definition | EM-Nullity Condition | tau-e | form |
| IV.D139 | definition | σ-Polarity Condition | tau-e | form |
| IV.L06 | lemma | Crossing-Point Mode Classification | tau-e | form |
| IV.T64 | theorem | Higgs Scalarity Theorem | tau-e | form |
| IV.D140 | definition | Vacuum Hessian at Crossing Point | tau-e | form |
| IV.P74 | proposition | Finite-Stage Hessian Properties | tau-e | form |
| IV.D141 | definition | τ-Higgs Mass | tau-e | form |
| IV.P75 | proposition | Higgs Mass Readout | conje | form |
| IV.P76 | proposition | Goldstone Modes at Crossing Point | tau-e | form |
| IV.T65 | theorem | No Hierarchy Problem in Category τ | tau-e | form |
| IV.R35 | remark | Structural Resolution of Hierarchy | tau-e | form |
| IV.D142 | definition | τ-Yukawa Coupling (Ch34) | tau-e | form |
| IV.P77 | proposition | Decay Mode Consistency | tau-e | form |
| IV.R36 | remark | Where τ and SM Could Diverge | tau-e | form |
| IV.D143 | definition | τ-Yukawa Coupling (Overlap Integral) | tau-e | form |
| IV.P78 | proposition | τ-Yukawa Hierarchy | tau-e | form |
| IV.R37 | remark | Explicit Overlap Integrals | conje | form |
| IV.T66 | theorem | Electroweak Prediction Table | tau-e | form |
| IV.R38 | remark | Precision Gap Lesson | tau-e | form |
| IV.R39 | remark | √3 as Category τ Signature | tau-e | — |
| IV.T67 | theorem | Zero Free Parameters in Electroweak Sector | tau-e | form |
| IV.P79 | proposition | Framework Comparison | tau-e | form |
| IV.R40 | remark | Honesty About Precision | tau-e | form |
| IV.L07 | lemma | Discrete Hessian Spectrum | tau-e | form |
| IV.R41 | remark | Ontological Inventory | tau-e | — |
| IV.R42 | remark | Five Conceptual Shifts | tau-e | — |
| IV.R43 | remark | Donut Persistence | tau-e | — |
| IV.R44 | remark | Alpha Formula Correction | tau-e | — |
| IV.R45 | remark | Part IV Reflection Summary | tau-e | — |
| IV.D144 | definition | The C-sector | tau-e | form |
| IV.R46 | remark | The (1 + ι_τ) contrast | tau-e | — |
| IV.P80 | proposition | Spectral tightening in the C-sector | tau-e | form |
| IV.R47 | remark | Comparison with the B-sector | tau-e | — |
| IV.R48 | remark | Why two types of holonomy | tau-e | — |
| IV.D145 | definition | Strong loop class | tau-e | form |
| IV.P81 | proposition | Finiteness and decidability | tau-e | form |
| IV.P82 | proposition | Loop class inclusion | tau-e | form |
| IV.R49 | remark | Why new loops appear | tau-e | — |
| IV.D146 | definition | Strong holonomy defect | tau-e | form |
| IV.D147 | definition | Strong defect functional | tau-e | form |
| IV.P83 | proposition | Properties of \Delta_n^s | tau-e | form |
| IV.D148 | definition | Strong admissibility | tau-e | form |
| IV.P84 | proposition | Non-emptiness of $\mathrm{Adm_s{[n | tau-e | form |
| IV.D149 | definition | Finite-stage strong vacuum | tau-e | form |
| IV.P85 | proposition | Existence and uniqueness at each stage | tau-e | form |
| IV.D150 | definition | Strong vacuum | tau-e | form |
| IV.T68 | theorem | Truncation coherence for Γ*_s | tau-e | form |
| IV.R50 | remark | Physical meaning of truncation coherence | tau-e | — |
| IV.D151 | definition | \mathrm{HolEnd_τ(s) | tau-e | form |
| IV.D152 | definition | Fix(s) | tau-e | form |
| IV.P86 | proposition | Structure of Fix(s) | tau-e | form |
| IV.D153 | definition | Canonical strong lift | tau-e | form |
| IV.P87 | proposition | Properties of the canonical strong lift | tau-e | form |
| IV.R51 | remark | Why the coupling is deferred | tau-e | — |
| IV.R52 | remark | The gauge group is derived, not postulated | tau-e | — |
| IV.P88 | proposition | Ternary Decomposition of the η-Circle | tau-e | form |
| IV.R53 | remark | Why three and not two | tau-e | — |
| IV.D154 | definition | Color Charge | tau-e | form |
| IV.R54 | remark | Labeling convention | tau-e | — |
| IV.D155 | definition | Anti-Color | tau-e | form |
| IV.D156 | definition | Color Neutrality | tau-e | form |
| IV.P89 | proposition | Color Quantization | tau-e | form |
| IV.R55 | remark | Contrast with electric charge | tau-e | — |
| IV.P90 | proposition | Color-Charged Modes Have n \neq 0 \pmod{3 | tau-e | form |
| IV.P91 | proposition | χ₋-Dominance Forces Non-Commutativity | tau-e | form |
| IV.P92 | proposition | Tracelessness from Color-Neutral Vacuum | tau-e | form |
| IV.T69 | theorem | SU(3) Gauge Algebra | tau-e | form |
| IV.R56 | remark | Eight generators = eight gluon types | tau-e | — |
| IV.R57 | remark | Connection to III.D43 | tau-e | — |
| IV.P93 | proposition | Gluon Self-Interaction | tau-e | form |
| IV.R58 | remark | In τ language | tau-e | — |
| IV.R59 | remark | Asymptotic freedom preview | tau-e | — |
| IV.T70 | theorem | Color Number Theorem | tau-e | form |
| IV.R60 | remark | Asymptotic formula | tau-e | — |
| IV.D157 | definition | Wilson Loop | tau-e | form |
| IV.D158 | definition | Fractional CR-sublattice | tau-e | form |
| IV.R61 | remark | Electric charge analogy | tau-e | — |
| IV.D159 | definition | Color-confined mode | tau-e | form |
| IV.T71 | theorem | Confinement Theorem | tau-e | form |
| IV.R62 | remark | Physical interpretation | tau-e | — |
| IV.R63 | remark | Comparison with the mass gap | tau-e | — |
| IV.D160 | definition | Color singlet | tau-e | form |
| IV.P94 | proposition | Singlet Stability | tau-e | form |
| IV.R64 | remark | Multi-quark states | tau-e | — |
| IV.P95 | proposition | Singlet Classification | tau-e | form |
| IV.P96 | proposition | Linear Confinement Potential | tau-e | form |
| IV.R65 | remark | Physical string tension | tau-e | — |
| IV.D161 | definition | Baryon number | tau-e | form |
| IV.L8 | lemma | Winding Preservation | tau-e | form |
| IV.T72 | theorem | Proton Stability | tau-e | form |
| IV.R66 | remark | Comparison with GUTs | tau-e | — |
| IV.R67 | remark | What about the electroweak sector? | tau-e | — |
| IV.R68 | remark | The pattern of maturation | tau-e | — |
| IV.R69 | remark | The role of NF Discreteness | tau-e | — |
| IV.D162 | definition | τ-Holonomy sector | tau-e | form |
| IV.D163 | definition | Canonical vacuum at stage~n | tau-e | form |
| IV.P97 | proposition | Well-definedness | tau-e | form |
| IV.D164 | definition | Localized perturbations | tau-e | form |
| IV.D165 | definition | Finite-difference quadratic form | tau-e | form |
| IV.D166 | definition | Excitation cost | tau-e | form |
| IV.D167 | definition | Canonical smallest excitation | tau-e | form |
| IV.P98 | proposition | Properties of h_n | tau-e | form |
| IV.D168 | definition | Three kernel hypotheses | tau-e | form |
| IV.R70 | remark | Verification status | tau-e | — |
| IV.T73 | theorem | τ-Gap Meta-Theorem (III.T26) | tau-e | form |
| IV.L9 | lemma | Finite-stage spectral problem | tau-e | form |
| IV.L10 | lemma | Positive gap at each stage | tau-e | form |
| IV.L11 | lemma | Vacuum coherence | tau-e | form |
| IV.L12 | lemma | Excitation coherence | tau-e | form |
| IV.R71 | remark | Why α-Idx prevents gap closure | tau-e | — |
| IV.P99 | proposition | Higgs sector as τ-holonomy sector | tau-e | form |
| IV.P100 | proposition | Higgs sector satisfies (KH-1)–(KH-3) | tau-e | form |
| IV.P101 | proposition | Strong sector as τ-holonomy sector | tau-e | form |
| IV.P102 | proposition | Strong sector kernel hypotheses | tau-e | form |
| IV.R72 | remark | Glueball mass scale | conje | — |
| IV.R73 | remark | The Millennium Problem, revisited | conje | — |
| IV.R74 | remark | The simplest non-trivial case | tau-e | — |
| IV.R75 | remark | The lattice evidence | tau-e | — |
| IV.D169 | definition | Strong configuration space | tau-e | form |
| IV.D170 | definition | Strong connection assignment | tau-e | form |
| IV.R76 | remark | Analogy with lattice gauge theory | tau-e | — |
| IV.D171 | definition | Strong curvature | tau-e | form |
| IV.D172 | definition | Plaquette-aggregated strong defect | tau-e | form |
| IV.P103 | proposition | Equivalence of defect formulations | tau-e | form |
| IV.D173 | definition | Canonical strong vacuum, plaquette form | tau-e | form |
| IV.P104 | proposition | Refinement coherence | tau-e | form |
| IV.D174 | definition | Strong quadratic form | tau-e | form |
| IV.P105 | proposition | Properties of Q_n^s | tau-e | form |
| IV.D175 | definition | Spectral gap at stage~n | tau-e | form |
| IV.D176 | definition | YM sector coupling | tau-e | form |
| IV.R77 | remark | Lean verification | tau-e | — |
| IV.P106 | proposition | Gap mode coherence | tau-e | form |
| IV.D177 | definition | Gap quantum | tau-e | form |
| IV.P107 | proposition | Gap positivity at each finite stage | tau-e | form |
| IV.P108 | proposition | Tower monotonicity | tau-e | form |
| IV.R78 | remark | Physical interpretation | tau-e | — |
| IV.T74 | theorem | Profinite spectral preservation | tau-e | form |
| IV.R79 | remark | The key structural point | tau-e | — |
| IV.R80 | remark | No continuum parameter to tune | tau-e | — |
| IV.T75 | theorem | τ-Yang–Mills Mass Gap Theorem | tau-e | form |
| IV.R81 | remark | Physical meaning | tau-e | — |
| IV.D178 | definition | Readout functor (conjectural) | conje | form |
| IV.D179 | definition | Orthodox Bridge Conjecture | conje | form |
| IV.R82 | remark | Comparison with lattice QCD | tau-e | — |
| IV.R83 | remark | No claim to the prize | tau-e | — |
| IV.R84 | remark | Why π-typed, not η-typed | tau-e | — |
| IV.D180 | definition | π-lift at stage n | tau-e | form |
| IV.D181 | definition | π-lift ω-limit | tau-e | form |
| IV.D182 | definition | The τ-strong coupling constant | tau-e | form |
| IV.T76 | theorem | Uniqueness of the strong coupling | tau-e | form |
| IV.P109 | proposition | No primitive mixing | tau-e | form |
| IV.R85 | remark | Mixing at the cross-coupling level | tau-e | — |
| IV.D183 | definition | π-support penalty | tau-e | form |
| IV.R86 | remark | Lean translatability | tau-e | — |
| IV.P110 | proposition | The argmin is the π-lift | tau-e | form |
| IV.R87 | remark | Range proof from Lean | tau-e | — |
| IV.R88 | remark | What varies | tau-e | — |
| IV.P111 | proposition | Λ_QCD as readout saturation | tau-e | form |
| IV.D184 | definition | Ontic coupling | tau-e | form |
| IV.D185 | definition | Regime selector | tau-e | form |
| IV.D186 | definition | Regime readout map | tau-e | form |
| IV.T77 | theorem | No ontic running — strong sector | tau-e | form |
| IV.P112 | proposition | Asymptotic freedom as spectral tightening | tau-e | form |
| IV.R89 | remark | The physical picture | tau-e | — |
| IV.R90 | remark | Unification in the τ picture | tau-e | — |
| IV.R91 | remark | The denominator matters | tau-e | — |
| IV.D187 | definition | Quark mode | tau-e | form |
| IV.P113 | proposition | Quark electric charges | tau-e | form |
| IV.R92 | remark | Fractional charges are not inputs | tau-e | — |
| IV.D188 | definition | Antiquark mode | tau-e | form |
| IV.R93 | remark | Why only three, not more | tau-e | — |
| IV.D189 | definition | Quark generations from \Lemniscate | tau-e | form |
| IV.P114 | proposition | Generation mass ordering | tau-e | form |
| IV.R94 | remark | Leptons share the same structure | tau-e | — |
| IV.P115 | proposition | Gluon count | tau-e | form |
| IV.R95 | remark | Why \mathrm{SU not \mathrm{U | tau-e | — |
| IV.P116 | proposition | Gluon self-interaction vertices | tau-e | form |
| IV.R96 | remark | Self-interaction and confinement | tau-e | — |
| IV.P117 | proposition | Structural asymptotic freedom | tau-e | form |
| IV.R97 | remark | The χ₋/χ₊ duality | tau-e | — |
| IV.P118 | proposition | Asymptotic freedom from N_c and N_f | tau-e | form |
| IV.D190 | definition | Meson state | tau-e | form |
| IV.D191 | definition | Baryon state | tau-e | form |
| IV.R98 | remark | Neutron stability in nuclei | tau-e | — |
| IV.R99 | remark | The worst prediction | metap | — |
| IV.D192 | definition | Boundary-first normalization | tau-e | form |
| IV.P119 | proposition | No uncountable factorization | tau-e | form |
| IV.R100 | remark | The key distinction | tau-e | — |
| IV.P120 | proposition | Canonical vacuum uniqueness | tau-e | form |
| IV.R101 | remark | No vacuum energy offset | tau-e | — |
| IV.D193 | definition | Earned vs.\ unearned mode count | tau-e | form |
| IV.T78 | theorem | No vacuum catastrophe in τ | tau-e | form |
| IV.R102 | remark | Largest but finite | tau-e | — |
| IV.T79 | theorem | Tail stabilization of vacuum energy | tau-e | form |
| IV.R103 | remark | Global stabilization | tau-e | — |
| IV.R104 | remark | No double counting | tau-e | — |
| IV.R105 | remark | Open problem: numerical value | tau-e | — |
| IV.R106 | remark | Book III template vs Book IV instantiation | tau-e | — |
| IV.T80 | theorem | Exactly four primitive forces (physical reading) | tau-e | form |
| IV.T81 | theorem | Exactly one derived sector | tau-e | form |
| IV.R107 | remark | Topological rigidity | tau-e | — |
| IV.D194 | definition | 9-element canonical generator set | tau-e | form |
| IV.T82 | theorem | Generator adequacy and minimality | tau-e | form |
| IV.D195 | definition | tau-Yukawa coupling | tau-e | form |
| IV.R108 | remark | Yukawa as readout not parameter | tau-e | form |
| IV.R109 | remark | SM parameter count comparison | tau-e | form |
| IV.R110 | remark | No BSM particles | tau-e | form |
| IV.D196 | definition | Three mode classes on lemniscate | tau-e | form |
| IV.T83 | theorem | Exactly three generations | tau-e | form |
| IV.R111 | remark | Fourth generation excluded | tau-e | form |
| IV.R112 | remark | Winding number classes | tau-e | — |
| IV.D197 | definition | Quark winding classes | tau-e | form |
| IV.P121 | proposition | Quark mass pattern | tau-e | form |
| IV.R113 | remark | Honest assessment of quark masses | conje | form |
| IV.P122 | proposition | Muon mass exponent | tau-e | form |
| IV.R114 | remark | Non-integer exponents are physical | tau-e | form |
| IV.P123 | proposition | Tau lepton mass exponent | tau-e | form |
| IV.D198 | definition | Koide parameter | tau-e | form |
| IV.T84 | theorem | Koide relation Q=2/3 | tau-e | form |
| IV.R115 | remark | The 45-degree crossing angle | tau-e | form |
| IV.R116 | remark | Why 0.0009% and not exact | tau-e | form |
| IV.P124 | proposition | Neutrino mass scale | tau-e | form |
| IV.R117 | remark | Normal ordering predicted | tau-e | form |
| IV.R118 | remark | Individual neutrino masses | conje | form |
| IV.R119 | remark | Scope gradient across mass table | tau-e | form |
| IV.R120 | remark | One constant one anchor zero parameters | tau-e | form |
| IV.R121 | remark | Neutron as parent of atomic matter | tau-e | form |
| IV.P125 | proposition | Beta-decay Q-value | tau-e | form |
| IV.R122 | remark | Structural lifetime estimate | tau-e | form |
| IV.P126 | proposition | Bohr radius from iota_tau | tau-e | form |
| IV.R123 | remark | No classical trajectory | tau-e | — |
| IV.T85 | theorem | Hydrogen energy levels | tau-e | form |
| IV.D199 | definition | Rydberg constant | tau-e | form |
| IV.T86 | theorem | Rydberg prediction at 0.025 ppm | tau-e | form |
| IV.R124 | remark | A testable prediction | tau-e | form |
| IV.P127 | proposition | Spectral transition as mode-switching | tau-e | form |
| IV.R125 | remark | Forbidden transitions | tau-e | form |
| IV.T87 | theorem | Fine structure from holonomy corrections | tau-e | form |
| IV.R126 | remark | Lamb shift in tau-framework | conje | form |
| IV.R127 | remark | All roads lead through m_e | tau-e | form |
| IV.D200 | definition | Meson classification | tau-e | form |
| IV.R128 | remark | Eta-eta prime splitting | conje | form |
| IV.D201 | definition | Glueball | tau-e | form |
| IV.R129 | remark | Glueballs and the mass gap | tau-e | form |
| IV.P128 | proposition | Nucleon mass decomposition | tau-e | form |
| IV.R130 | remark | Mass from nothing | tau-e | — |
| IV.R131 | remark | Isospin splitting from polarity | tau-e | form |
| IV.R132 | remark | Proton lighter but ontologically later | tau-e | — |
| IV.D202 | definition | Nuclear force | tau-e | form |
| IV.R133 | remark | Deuteron binding in tau-language | tau-e | form |
| IV.P129 | proposition | Nuclear force saturation | tau-e | form |
| IV.P130 | proposition | Nuclear shell structure | tau-e | form |
| IV.R134 | remark | Spin-orbit from omega-sector | tau-e | — |
| IV.R135 | remark | Why He-4 is tightly bound | tau-e | form |
| IV.P131 | proposition | Iron peak from competing sectors | tau-e | form |
| IV.R136 | remark | Nucleosynthesis forward to Book V | tau-e | form |
| IV.R137 | remark | Alpha-decay as mode cluster ejection | tau-e | form |
| IV.R138 | remark | Neutron stability inside nuclei | tau-e | form |
| IV.R139 | remark | Gamma-decay as mode transition | tau-e | form |
| IV.P132 | proposition | Decay channels from sector admissibility | tau-e | form |
| IV.D203 | definition | Atom as dressed nuclear mode | tau-e | form |
| IV.D204 | definition | Electron quantum numbers | tau-e | form |
| IV.R140 | remark | Madelung rule from T^2 geometry | tau-e | form |
| IV.T88 | theorem | Period length sequence | tau-e | form |
| IV.R141 | remark | Topological not accidental | tau-e | form |
| IV.D205 | definition | Covalent bond | tau-e | form |
| IV.R142 | remark | Molecules as mode-sharing graphs | metap | — |
| IV.D206 | definition | Ionic bond | tau-e | form |
| IV.D207 | definition | Metallic bond | tau-e | form |
| IV.R143 | remark | No new parameters for chemistry | tau-e | — |
| IV.R144 | remark | Mode-repulsion geometry | tau-e | form |
| IV.D208 | definition | Hybrid modes | tau-e | form |
| IV.R145 | remark | Homochirality and parity violation | conje | — |
| IV.R146 | remark | Structural vs quantitative chemistry | tau-e | — |
| IV.R147 | remark | The donut ladder | tau-e | — |
| IV.R148 | remark | Comparison with orthodox physics | tau-e | form |
| IV.R149 | remark | Parameter count | tau-e | form |
| IV.D209 | definition | Ontic entity | tau-e | form |
| IV.R150 | remark | Ontic entities list | tau-e | — |
| IV.R151 | remark | Non-ontic entities list | tau-e | form |
| IV.R152 | remark | Where the ontological line falls | tau-e | form |
| IV.R153 | remark | Dictionary limits | tau-e | form |
| IV.R154 | remark | Temperature is not fundamental | tau-e | form |
| IV.P133 | proposition | Topological integrality of theta | tau-e | form |
| IV.R155 | remark | Euler budget recap | tau-e | form |
| IV.D210 | definition | Macroscopic defect tuple | tau-e | form |
| IV.R156 | remark | Why interaction corrections matter | tau-e | form |
| IV.D211 | definition | Macroscopic mobility | tau-e | form |
| IV.D212 | definition | Macroscopic vorticity | tau-e | form |
| IV.D213 | definition | Macroscopic compression | tau-e | form |
| IV.D214 | definition | Total topological charge | tau-e | form |
| IV.P134 | proposition | Topological charge conservation | tau-e | form |
| IV.D215 | definition | Clopen cylinder at depth n | tau-e | form |
| IV.R157 | remark | Why clopen matters | tau-e | — |
| IV.D216 | definition | Level-n defect functional | tau-e | form |
| IV.R158 | remark | Finitely specified at each depth | tau-e | — |
| IV.T89 | theorem | Tower compatibility | tau-e | form |
| IV.R159 | remark | Ultrametric locality | tau-e | — |
| IV.D217 | definition | Universal defect functional | tau-e | form |
| IV.P135 | proposition | Existence and uniqueness of limit | tau-e | form |
| IV.T90 | theorem | Sector additivity | tau-e | form |
| IV.R160 | remark | No continuum limit needed | tau-e | — |
| IV.D218 | definition | Defect tuple space | tau-e | form |
| IV.D219 | definition | Critical mobility threshold | tau-e | form |
| IV.D220 | definition | Crystal regime | tau-e | form |
| IV.D221 | definition | Glass regime | tau-e | form |
| IV.D222 | definition | Euler fluid regime | tau-e | form |
| IV.P136 | proposition | tau-Euler equation | tau-e | form |
| IV.D223 | definition | Navier-Stokes regime | tau-e | form |
| IV.R161 | remark | Navier-Stokes regularity | conje | — |
| IV.D224 | definition | MHD regime | tau-e | form |
| IV.R162 | remark | Why MHD lives in B-sector | tau-e | — |
| IV.D225 | definition | Plasma regime | tau-e | form |
| IV.D226 | definition | Superfluid regime | tau-e | form |
| IV.P137 | proposition | Superfluid vortex quantization | tau-e | form |
| IV.R163 | remark | Helium-4 and beyond | tau-e | form |
| IV.D227 | definition | Superconductor regime | tau-e | form |
| IV.P138 | proposition | Flux quantization | tau-e | form |
| IV.R164 | remark | Cooper pairing is topological | tau-e | form |
| IV.R165 | remark | Transitions as inequality crossings | tau-e | — |
| IV.D228 | definition | Temperature as defect gradient | tau-e | form |
| IV.R166 | remark | Comparison with orthodox thermodynamics | tau-e | — |
| IV.P139 | proposition | Status of Boltzmann constant | tau-e | form |
| IV.R167 | remark | Parallel with hbar and c | tau-e | — |
| IV.T91 | theorem | Second law via defect functional | tau-e | form |
| IV.R168 | remark | No negative temperatures | tau-e | — |
| IV.D229 | definition | First-order phase transition | tau-e | form |
| IV.D230 | definition | Second-order phase transition | tau-e | form |
| IV.T92 | theorem | Phase transition as regime crossing | tau-e | form |
| IV.R169 | remark | Universality and critical exponents | conje | form |
| IV.D231 | definition | tau-Euler flow | tau-e | form |
| IV.R170 | remark | Kelvin theorem as budget law | tau-e | form |
| IV.D232 | definition | tau-Navier-Stokes flow | tau-e | form |
| IV.P140 | proposition | Finite at every primorial level | tau-e | form |
| IV.R171 | remark | Viscosity coefficient eta_tau | tau-e | — |
| IV.T93 | theorem | τ-NS Regularity on T² Fiber (C3 Defect Contractivity) | conje | form |
| IV.R172 | remark | Honesty about the Clay problem | conje | — |
| IV.D233 | definition | Superfluid regime | tau-e | form |
| IV.P141 | proposition | Quantized circulation | tau-e | form |
| IV.D234 | definition | Superconductor regime | tau-e | form |
| IV.R173 | remark | BCS gap as spectral gap | tau-e | — |
| IV.D235 | definition | Crystal regime | tau-e | form |
| IV.R174 | remark | Crystal symmetry from torus subgroups | tau-e | form |
| IV.D236 | definition | Glass regime | tau-e | form |
| IV.R175 | remark | Glass transition not a true phase transition | tau-e | — |
| IV.D237 | definition | Quasicrystal regime | tau-e | form |
| IV.R176 | remark | Penrose tilings on the torus | metap | form |
| IV.D238 | definition | First-order phase transition | tau-e | form |
| IV.D239 | definition | Second-order phase transition | tau-e | form |
| IV.P142 | proposition | Defect tuple as universal order parameter | tau-e | form |
| IV.R177 | remark | Universality from sector structure | tau-e | — |
| IV.P143 | proposition | Melting sequence monotone mobility | tau-e | form |
| IV.D240 | definition | Topological branch | tau-e | form |
| IV.T94 | theorem | Fiber-base factorization | tau-e | form |
| IV.R178 | remark | Why the separation is clean | tau-e | — |
| IV.P144 | proposition | Fiber-level physics is complete | tau-e | form |
| IV.R179 | remark | The base is not the rest | tau-e | — |
| IV.D241 | definition | Tower-natural transformation | tau-e | form |
| IV.R180 | remark | Noether theorem as corollary | tau-e | form |
| IV.R181 | remark | Why not a larger gauge group | tau-e | — |
| IV.R182 | remark | Individual C P CP violations | tau-e | — |
| IV.P145 | proposition | UV finiteness | tau-e | form |
| IV.R183 | remark | Vacuum catastrophe resolved | tau-e | — |
| IV.R184 | remark | Why C1 is conjectural | conje | form |
| IV.R185 | remark | Comparison with ch15 ledger | tau-e | form |
| IV.R186 | remark | The formalization frontier | tau-e | form |
| IV.R187 | remark | Open vs wrong problems | tau-e | form |
| IV.D242 | definition | tau-sphaleron question | conje | form |
| IV.R188 | remark | Why sphaleron is deferred | conje | — |
| IV.D243 | definition | Book V import list | tau-e | form |
| IV.D244 | definition | Book VI import list | tau-e | form |
| IV.D245 | definition | Book VII import list | tau-e | form |
| IV.R189 | remark | One-way flow of export contracts | tau-e | — |
| IV.R190 | remark | Why neutron not electron as anchor | tau-e | form |
| IV.R191 | remark | Scope of the change | tau-e | — |
| IV.R192 | remark | This is not a textbook order | tau-e | — |
| IV.R193 | remark | What no free parameters'' means | tau-e | — |
| IV.R194 | remark | The circularity of substrate ontology | tau-e | — |
| IV.D246 | definition | Coherence Kernel --- Physics Presentation | tau-e | form |
| IV.D247 | definition | Generator--Sector Assignment | tau-e | form |
| IV.P146 | proposition | Uniqueness of Assignment | tau-e | form |
| IV.R195 | remark | No arbitrary choices | tau-e | — |
| IV.R196 | remark | Time is computation | tau-e | — |
| IV.D248 | definition | Ontic Minimality | tau-e | form |
| IV.R197 | remark | Necessity of each component | tau-e | — |
| IV.R198 | remark | It from Coherence | tau-e | — |
| IV.R199 | remark | Forward reference: the refinement tower IS the block universe | tau-e | — |
| IV.D249 | definition | Refinement Tower mathcalR | tau-e | form |
| IV.R200 | remark | The tower is the fundamental object | tau-e | — |
| IV.D250 | definition | Profinite Limit hatalpha | tau-e | form |
| IV.R201 | remark | Profinite, not sequential | tau-e | — |
| IV.R202 | remark | Presentism and the growing block | tau-e | — |
| IV.R203 | remark | No passage of time | tau-e | — |
| IV.P147 | proposition | Subsystem Horizon | tau-e | form |
| IV.R204 | remark | The view from within | tau-e | — |
| IV.R205 | remark | Preview: quantum measurement | tau-e | — |
| IV.D251 | definition | Proto-Time t_p | tau-e | form |
| IV.P148 | proposition | NNO from the alpha-Orbit | tau-e | form |
| IV.T95 | theorem | Structural Arrow of Time | tau-e | form |
| IV.R206 | remark | No Past Hypothesis needed | tau-e | — |
| IV.R207 | remark | Why timeflows’’ and space does not | tau-e | — |
| IV.R208 | remark | Ultrametric refinement | tau-e | — |
| IV.R209 | remark | Forward references | tau-e | — |
| IV.D252 | definition | Base tau^1 — physical reading | tau-e | form |
| IV.D253 | definition | Fiber T^2 — physical reading | tau-e | form |
| IV.D254 | definition | Fibered product arena tau^3 | tau-e | form |
| IV.R210 | remark | Fibered, not Cartesian | tau-e | — |
| IV.D255 | definition | Master constant iota_tau | tau-e | form |
| IV.P149 | proposition | Quasi-ergodicity | tau-e | form |
| IV.R211 | remark | One constant to rule them all | tau-e | form |
| IV.R212 | remark | Lean formalization | tau-e | form |
| IV.P150 | proposition | Four dimensions earned | tau-e | form |
| IV.R213 | remark | CR-structure | tau-e | form |
| IV.D256 | definition | Lemniscate boundary — physical reading | tau-e | form |
| IV.R214 | remark | The self-intersection point | tau-e | — |
| IV.R215 | remark | The algebraic lemniscate — no topology assumed | tau-e | — |
| IV.P151 | proposition | Micro/Macro decomposition | tau-e | form |
| IV.R216 | remark | The coupling connects them | tau-e | form |
| IV.D257 | definition | Chart readout homomorphism | tau-e | form |
| IV.R217 | remark | The analogy | tau-e | — |
| IV.R218 | remark | What tau^3 is and is not | tau-e | — |
| IV.D258 | definition | Yoneda self-image | tau-e | form |
| IV.R219 | remark | Self-description, not metaphor | tau-e | — |
| IV.T96 | theorem | Central Theorem — physical form | tau-e | form |
| IV.R220 | remark | Hartogs extension as physical completeness | tau-e | — |
| IV.D259 | definition | Boundary character | tau-e | form |
| IV.D260 | definition | Bipolar decomposition of characters | tau-e | form |
| IV.P152 | proposition | Master constant is sigma-fixed | tau-e | form |
| IV.D261 | definition | Physical-constants core | tau-e | form |
| IV.D262 | definition | Canonical sector lifts | tau-e | form |
| IV.R221 | remark | Why all lifts are rational | tau-e | form |
| IV.D263 | definition | Chart readout homomorphism | tau-e | form |
| IV.P153 | proposition | Smooth manifold from coherent readouts | tau-e | form |
| IV.R222 | remark | Why 2 + 2 gives 1+ | tau-e | form |
| IV.T97 | theorem | Boundary Triad Theorem | tau-e | form |
| IV.R223 | remark | Scope: tau-effective | tau-e | — |
| IV.D264 | definition | Generator–Sector Correspondence | tau-e | form |
| IV.T98 | theorem | Uniqueness of Phi | tau-e | form |
| IV.R224 | remark | Lean formalization | tau-e | — |
| IV.D265 | definition | Coupling Ledger | tau-e | form |
| IV.T99 | theorem | Temporal Complement | tau-e | form |
| IV.R225 | remark | Physical meaning | tau-e | form |
| IV.P154 | proposition | Temporal Multiplicative Closure | tau-e | form |
| IV.P155 | proposition | Multiplicative Closure | tau-e | form |
| IV.P156 | proposition | Power Hierarchy | tau-e | form |
| IV.R226 | remark | Power structure | tau-e | form |
| IV.T100 | theorem | No-Running Principle | tau-e | form |
| IV.R227 | remark | The readout functor | tau-e | — |
| IV.R228 | remark | Asymptotic freedom reinterpreted | tau-e | — |
| IV.D266 | definition | Boundary holonomy generators | tau-e | form |
| IV.T101 | theorem | Generator Adequacy | tau-e | form |
| IV.R229 | remark | The mirror of the kernel | tau-e | — |
| IV.D267 | definition | Defect bundle (ontic particle) | tau-e | form |
| IV.D268 | definition | Radiation | tau-e | form |
| IV.D269 | definition | Virtual particle | tau-e | form |
| IV.R230 | remark | Lean formalization | tau-e | form |
| IV.D270 | definition | Five primary invariants | tau-e | form |
| IV.P157 | proposition | Second-law inversion | tau-e | form |
| IV.R231 | remark | No external clock | tau-e | — |
| IV.D271 | definition | Mass as fiber stiffness | tau-e | form |
| IV.R232 | remark | Lean formalization | tau-e | — |
| IV.R233 | remark | Why gravity is weak | tau-e | form |
| IV.D272 | definition | Propagation operator | tau-e | form |
| IV.P158 | proposition | Schr”odinger shadow | tau-e | form |
| IV.R234 | remark | No quantization needed | tau-e | — |
| IV.D273 | definition | Planck character | tau-e | form |
| IV.T102 | theorem | tau-Heisenberg inequality | tau-e | form |
| IV.R235 | remark | Preview of Part~III | tau-e | — |
| IV.R236 | remark | Lean formalization | tau-e | form |
| IV.D274 | definition | Defect functional | tau-e | form |
| IV.T103 | theorem | Euler budget conservation | tau-e | form |
| IV.R237 | remark | Lean formalization | tau-e | form |
| IV.R238 | remark | Nothing is postulated | tau-e | — |
| IV.R239 | remark | Not metaphor | tau-e | — |
| IV.R240 | remark | The complete physics in two books | tau-e | — |
| IV.R241 | remark | On the emptiness of the stage | tau-e | — |
| IV.R242 | remark | Part~II in perspective | tau-e | form |
| IV.R243 | remark | The ladder is not a hierarchy of importance | tau-e | form |
| IV.R244 | remark | Neither language is more fundamental | tau-e | — |
| IV.P159 | proposition | Calibration is structural | tau-e | form |
| IV.R245 | remark | The honest timing | tau-e | form |
| IV.R246 | remark | The Lean criterion | tau-e | — |
| IV.R247 | remark | Origin of the No Knobs Principle | tau-e | form |
| IV.D275 | definition | Gravitational self-coupling | tau-e | form |
| IV.P160 | proposition | Numerical value | tau-e | form |
| IV.D276 | definition | Weak self-coupling | tau-e | form |
| IV.P161 | proposition | Numerical value | tau-e | form |
| IV.D277 | definition | Electromagnetic self-coupling | tau-e | form |
| IV.P162 | proposition | Numerical value | tau-e | form |
| IV.D278 | definition | Strong self-coupling | tau-e | form |
| IV.P163 | proposition | Numerical value | tau-e | form |
| IV.R248 | remark | Lean verification | tau-e | form |
| IV.D279 | definition | Omega self-coupling | tau-e | form |
| IV.P164 | proposition | Numerical value | tau-e | form |
| IV.D280 | definition | Weak–gravity cross-coupling | tau-e | form |
| IV.D281 | definition | Electroweak cross-coupling | tau-e | form |
| IV.D282 | definition | Weak–strong cross-coupling | tau-e | form |
| IV.D283 | definition | EM–strong cross-coupling | tau-e | form |
| IV.D284 | definition | EM–gravity cross-coupling | tau-e | form |
| IV.D285 | definition | Strong–gravity cross-coupling | tau-e | form |
| IV.R249 | remark | Electroweak–strong-gravity coincidence | tau-e | — |
| IV.R250 | remark | Lean verification | tau-e | form |
| IV.T104 | theorem | Temporal Complement | tau-e | form |
| IV.R251 | remark | Lean formalization | tau-e | — |
| IV.T105 | theorem | Temporal Multiplicative Closure | tau-e | form |
| IV.R252 | remark | Lean formalization | tau-e | form |
| IV.T106 | theorem | Power Hierarchy | tau-e | form |
| IV.R253 | remark | Lean formalization | tau-e | — |
| IV.P165 | proposition | Hierarchy Resolution | tau-e | form |
| IV.R254 | remark | No landscape | tau-e | form |
| IV.D286 | definition | Spectral Fine-Structure Formula | tau-e | form |
| IV.R255 | remark | The meaning of 0.6% | tau-e | form |
| IV.R256 | remark | Lean verification | tau-e | form |
| IV.D287 | definition | Five Relational Units | tau-e | form |
| IV.T107 | theorem | Holonomy Fine-Structure Formula | tau-e | form |
| IV.R257 | remark | Origin of the formula | tau-e | form |
| IV.R258 | remark | The three holonomy circles | tau-e | form |
| IV.D288 | definition | Holonomy Correction Factor | tau-e | form |
| IV.R259 | remark | An analogy | tau-e | — |
| IV.R260 | remark | The value of being wrong | tau-e | form |
| IV.R261 | remark | What Lean cannot verify | tau-e | — |
| IV.D289 | definition | Five Relational Units | tau-e | form |
| IV.R262 | remark | What the paper got right | tau-e | form |
| IV.T108 | theorem | tau-Collapse: Five to One | tau-e | form |
| IV.T109 | theorem | Level~0 mass ratio formula | tau-e | form |
| IV.T110 | theorem | Level~1+ mass ratio formula | tau-e | form |
| IV.R263 | remark | Not a numerical fit | tau-e | form |
| IV.D290 | definition | Unpolarized defect bundle | tau-e | form |
| IV.P166 | proposition | Neutron Minimality | tau-e | form |
| IV.R264 | remark | The Planck mass in tau-physics | tau-e | form |
| IV.D291 | definition | Calibration Anchor | tau-e | form |
| IV.R265 | remark | One input, not zero | tau-e | form |
| IV.R266 | remark | Lean formalization | tau-e | form |
| IV.D292 | definition | tau-to-SI conversion | tau-e | form |
| IV.T111 | theorem | Parameter Count | tau-e | form |
| IV.R267 | remark | Falsifiability | tau-e | form |
| IV.D293 | definition | Speed of light in relational units | tau-e | form |
| IV.R268 | remark | Why 0.07% and not exact | tau-e | form |
| IV.D294 | definition | Planck’s constant in relational units | tau-e | form |
| IV.D295 | definition | Coulomb constant in relational units | tau-e | form |
| IV.D296 | definition | Vacuum permittivity in relational units | tau-e | form |
| IV.D297 | definition | Vacuum permeability in relational units | tau-e | form |
| IV.R269 | remark | Consistency check | tau-e | form |
| IV.R270 | remark | The tier boundary is sharp | tau-e | form |
| IV.D298 | definition | Planck character | tau-e | form |
| IV.T112 | theorem | sigma-Fixed Planck Character | tau-e | form |
| IV.P167 | proposition | Attained minimum | tau-e | form |
| IV.R271 | remark | Connection to the derivation chain | tau-e | — |
| IV.R272 | remark | Lean formalization | tau-e | — |
| IV.R273 | remark | The D-sector derivation | tau-e | — |
| IV.R274 | remark | The pi-corrected distance | tau-e | form |
| IV.R275 | remark | Counting parameters | tau-e | form |
| IV.R276 | remark | R~formula independence | tau-e | form |
| IV.R277 | remark | The sqrt3 | tau-e | form |
| IV.D299 | definition | Beta function | tau-e | form |
| IV.R278 | remark | Honest acknowledgment | tau-e | — |
| IV.D300 | definition | Coupling Ledger and Observable Ledger | tau-e | form |
| IV.D301 | definition | Readout Functor | tau-e | form |
| IV.P168 | proposition | Readout Properties | tau-e | form |
| IV.P169 | proposition | Beta Function as Readout Derivative | tau-e | form |
| IV.R279 | remark | Asymptotic freedom revisited | tau-e | form |
| IV.R280 | remark | Scheme dependence resolved | tau-e | form |
| IV.R281 | remark | The common thread | tau-e | — |
| IV.D302 | definition | Entropy splitting | tau-e | form |
| IV.P170 | proposition | Total Entropy Invariance | tau-e | form |
| IV.D303 | definition | Regime transition | tau-e | form |
| IV.R282 | remark | Lean formalization | tau-e | form |
| IV.D304 | definition | Readout landscape | tau-e | form |
| IV.T113 | theorem | Readout Landscape Theorem | tau-e | form |
| IV.D305 | definition | Coupling Constants Table | tau-e | form |
| IV.R283 | remark | Lean verification | tau-e | form |
| IV.D306 | definition | Fundamental Scales Table | tau-e | form |
| IV.R284 | remark | tau-native Planck character | tau-e | — |
| IV.D307 | definition | Particle Mass Table | tau-e | form |
| IV.R285 | remark | Honest deviations | tau-e | form |
| IV.D308 | definition | Structural Constants Table | tau-e | form |
| IV.R286 | remark | 5, 4, 3, 2, 1 | tau-e | form |
| IV.R287 | remark | Honest fraction | tau-e | form |
| IV.R288 | remark | Circular dependencies excluded | tau-e | — |
| IV.R289 | remark | Where the postulates dissolve | tau-e | — |
| IV.R290 | remark | Historical resonance: Penrose twistors | tau-e | — |
| IV.R291 | remark | The flat torus is not a deficiency | tau-e | — |
| IV.R292 | remark | CR-type (1,1) and three-dimensionality | tau-e | form |
| IV.R293 | remark | Connection to Book~II | tau-e | — |
| IV.R294 | remark | Spin-frac1 | tau-e | form |
| IV.R295 | remark | Bosons and fermions | tau-e | form |
| IV.R296 | remark | Abelianization and the torus projection | tau-e | — |
| IV.R297 | remark | The fiber–base reading | tau-e | — |
| IV.R298 | remark | Connection to neutron beta decay | tau-e | — |
| IV.R299 | remark | The Segal–Bargmann precedent | tau-e | — |
| IV.R300 | remark | Why not L^2(tau^3) directly? | tau-e | — |
| IV.R301 | remark | Holography in physics and in tau | tau-e | — |
| IV.R302 | remark | Connection to Book~III: the spectral philosophy | tau-e | — |
| IV.R303 | remark | Bell’s theorem and the tau-substrate | tau-e | — |
| IV.R304 | remark | Forward reference: the address-obstruction theorem | tau-e | — |
| IV.R305 | remark | Comparison with geometric quantization | tau-e | form |
| IV.R306 | remark | Why gamma and eta? | tau-e | — |
| IV.R307 | remark | Preview of the address-obstruction theorem | tau-e | — |
| IV.R308 | remark | Connection to Book~III: | tau-e | — |
| IV.R309 | remark | Quantization as address discretization | tau-e | — |
| IV.R310 | remark | Connection to Book~II spectral gap | tau-e | form |
| IV.R311 | remark | Connection to the iota_tau-Generator Lemma | tau-e | — |
| IV.R312 | remark | Born rule derived, not postulated | tau-e | form |
| IV.R313 | remark | No particles yet | tau-e | — |
| IV.R314 | remark | Why standard QM works | tau-e | form |
| IV.R315 | remark | Why clopen, not just open? | tau-e | — |
| IV.R316 | remark | The mechanism in plain language | tau-e | form |
| IV.R317 | remark | Comparison with the Fourier uncertainty principle | tau-e | — |
| IV.R318 | remark | Boundary witness budget | tau-e | form |
| IV.R319 | remark | Comparison with EPR and Bell | tau-e | — |
| IV.R320 | remark | Physical interpretation | tau-e | form |
| IV.R321 | remark | Common origin | tau-e | — |
| IV.R322 | remark | Why squared''? | tau-e | — |
| IV.R323 | remark | Decoherence is not collapse | tau-e | form |
| IV.R324 | remark | Classicality is the exception | tau-e | — |
| IV.R325 | remark | Generator roles | tau-e | — |
| IV.R326 | remark | One Hamiltonian rules all | tau-e | form |
| IV.R327 | remark | Ground state energy | tau-e | — |
| IV.R328 | remark | Why |barpartial | tau-e | form |
| IV.R329 | remark | The cosmological constant problem | tau-e | — |
| IV.R330 | remark | Massless particles | tau-e | form |
| IV.R331 | remark | Mixed configurations | tau-e | — |
| IV.R332 | remark | Local redistribution | tau-e | form |
| IV.R333 | remark | Comparison with Boltzmann | tau-e | form |
| IV.R334 | remark | Connection to Landauer's principle | tau-e | — |
| IV.R335 | remark | Connection to Book~III: | tau-e | — |
| IV.R336 | remark | Ten orders of magnitude from one constant | tau-e | form |
| IV.D309 | definition | Breathing operator | tau-e | form |
| IV.P171 | proposition | Breathing spectrum | tau-e | form |
| IV.D310 | definition | Epstein zeta structure textnormal(IV.D40) | tau-e | form |
| IV.R337 | remark | Toroidal dominance | tau-e | form |
| IV.D311 | definition | Chowla--Selberg decomposition textnormal(IV.D41) | tau-e | form |
| IV.T114 | theorem | Leading exponent -7 textnormal(IV.T10) | tau-e | form |
| IV.R338 | remark | Structural origin of the exponent 7 | tau-e | form |
| IV.D312 | definition | Lemniscate three-fold textnormal(IV.D42) | tau-e | form |
| IV.D313 | definition | Spectral distance sqrt3 | tau-e | form |
| IV.T115 | theorem | Three-fold distance squared textnormal(IV.T11) | tau-e | form |
| IV.R339 | remark | Physical interpretation of the correction | tau-e | form |
| IV.R340 | remark | sqrt3 | tau-e | form |
| IV.R341 | remark | Why the same sqrt3 | tau-e | form |
| IV.R342 | remark | Factorizations | tau-e | form |
| IV.R343 | remark | Massive versus massless | tau-e | form |
| IV.D314 | definition | Triple Holonomy --- IV.D44 | tau-e | form |
| IV.R344 | remark | Charge Conjugation Selection Rule --- IV.R12 | tau-e | — |
| IV.T116 | theorem | Correction Smallness --- IV.T12 | tau-e | form |
| IV.D315 | definition | Holonomy Correction Data --- IV.D45 | tau-e | form |
| IV.T117 | theorem | Derivation Chain Complete --- IV.T15 | tau-e | form |
| IV.P172 | proposition | All Links tau-Effective --- IV.P07 | tau-e | form |
| IV.D316 | definition | Mass Ratio Bulk Term --- IV.D46 | tau-e | form |
| IV.T118 | theorem | Bulk Overshoots --- IV.T13 | tau-e | form |
| IV.D317 | definition | Level 0 Formula --- IV.D47 | tau-e | form |
| IV.T119 | theorem | Level 0 Range --- IV.T14 | tau-e | form |
| IV.D318 | definition | Level 1+ Formula --- IV.D48 | tau-e | form |
| IV.R345 | remark | Whyelectromagnetic’’ and not just electric'' | tau-e | — |
| IV.R346 | remark | Null = no fiber winding, notzero’’ | tau-e | — |
| IV.R347 | remark | Masslessness is geometric, not postulated | tau-e | form |
| IV.R348 | remark | Experimental limits on photon mass | tau-e | form |
| IV.R349 | remark | The Planck–Einstein relation in tau^3 | tau-e | — |
| IV.R350 | remark | Massless spin-1: two, not three | tau-e | form |
| IV.R351 | remark | Double slit in tau^3 | tau-e | — |
| IV.R352 | remark | Charge is geometry, not substance | tau-e | — |
| IV.T120 | theorem | Charge quantization | tau-e | form |
| IV.R353 | remark | No need for magnetic monopoles | tau-e | — |
| IV.R354 | remark | Beta decay verification | tau-e | — |
| IV.R355 | remark | Conservation versus Noether | tau-e | form |
| IV.R356 | remark | No fundamental electromagnetic field | tau-e | — |
| IV.R357 | remark | Why alpha is small | tau-e | form |
| IV.R358 | remark | The historical path | tau-e | — |
| IV.T121 | theorem | Gauge covariance of D_mu | tau-e | form |
| IV.R359 | remark | Physical content lives in the curvature | tau-e | form |
| IV.P173 | proposition | AB interference shift | tau-e | form |
| IV.R360 | remark | Experimental confirmation | tau-e | form |
| IV.R361 | remark | Why the AB effect is necessary, | tau-e | — |
| IV.R362 | remark | The AB effect as an observable | tau-e | form |
| IV.R363 | remark | Why EM is simpler | tau-e | — |
| IV.R364 | remark | Gauge invariance at Elayer1 | tau-e | — |
| IV.R365 | remark | Why alpha_mathrmem | tau-e | form |
| IV.R366 | remark | Measuring alpha_mathrmem | tau-e | — |
| IV.R367 | remark | No external input | tau-e | form |
| IV.R368 | remark | Why (1,0) winding | tau-e | — |
| IV.R369 | remark | Compactness is essential | tau-e | — |
| IV.R370 | remark | The iota_tau^2 cancellation | tau-e | form |
| IV.R371 | remark | The meaning of 0.6% | tau-e | form |
| IV.R372 | remark | Lean verification | tau-e | form |
| IV.R373 | remark | pi^3 approx 31: not a Mersenne prime | tau-e | form |
| IV.R374 | remark | Origin of the formula | tau-e | form |
| IV.R375 | remark | An analogy | tau-e | — |
| IV.R376 | remark | The defect functional interpretation | tau-e | — |
| IV.R377 | remark | Grand unification reconsidered | tau-e | — |
| IV.R378 | remark | The smallness of alpha_mathrmem | tau-e | form |
| IV.R379 | remark | Comparison with other theories | tau-e | form |
| IV.T122 | theorem | Parity Bridge — recalled from III.T07 | tau-e | form |
| IV.R380 | remark | Physical consequence | tau-e | form |
| IV.D319 | definition | Polarity-switching transition | tau-e | form |
| IV.P174 | proposition | Polarity-switching requires balanced polarity | tau-e | form |
| IV.R381 | remark | Physical interpretation | tau-e | — |
| IV.R382 | remark | Mass hierarchy: M_Z > M_W | tau-e | form |
| IV.T123 | theorem | Chirality Selection | tau-e | form |
| IV.R383 | remark | Definition versus theorem | tau-e | form |
| IV.C01 | corollary | Right-handed decoupling | tau-e | form |
| IV.R384 | remark | Historical context: Wu, Lee, and Yang | tau-e | — |
| IV.D320 | definition | Electroweak scale in Category~tau | tau-e | form |
| IV.R385 | remark | Contrast with the Standard Model | tau-e | form |
| IV.T124 | theorem | The sqrt3 | tau-e | form |
| IV.P175 | proposition | Three-Fold Structure of~mathbbL | tau-e | form |
| IV.T34p | theorem | Solenoid Pitch Theorem | tau-e | form |
| IV.D321 | definition | Tick Morphism | tau-e | form |
| IV.D322 | definition | Tick Kind | tau-e | form |
| IV.D323 | definition | Internal Identity | tau-e | form |
| IV.D324 | definition | Equation Layer | estab | form |
| IV.D325 | definition | Measurement Procedure | tau-e | form |
| IV.D326 | definition | Readout Functor | tau-e | form |
| IV.D327 | definition | Readout Anchor | tau-e | form |
| IV.T125 | theorem | Tick-Sector Bijection | tau-e | form |
| IV.T126 | theorem | Tick Exhaustion | tau-e | form |
| IV.T127 | theorem | Mass Ratio as Internal Identity | tau-e | form |
| IV.T128 | theorem | Readout Preserves Identities | tau-e | form |
| IV.T129 | theorem | Single-Anchor Sufficiency | tau-e | form |
| IV.P176 | proposition | Internal Equations Dimensionless | tau-e | form |
| IV.P177 | proposition | Codomain Is Operational | tau-e | form |
| IV.R386 | remark | Layer Discipline | estab | form |
| IV.D328 | definition | Generator Carrier Assignment | tau-e | form |
| IV.D329 | definition | Generator Polarity Assignment | tau-e | form |
| IV.D330 | definition | Structural EM Activity | tau-e | form |
| IV.T130 | theorem | Structural–Physics Census Equivalence | tau-e | form |
| IV.P178 | proposition | SM-Independent Census | tau-e | form |
| IV.C02 | theorem | Twin Prime Residue Theorem | tau-e | form |
| IV.R387 | remark | OQ.11 Resolution Status | tau-e | not |
| IV.D331 | definition | Tensor-Square Character Algebra | tau-e | form |
| IV.T133 | theorem | EM Tensor Density Theorem | tau-e | form |
| IV.P179 | proposition | E₁ Page Derivation of α-Coefficient | tau-e | form |
| IV.R388 | remark | OQ-A1 Status: RESOLVED | tau-e | not_ |
| IV.D334 | definition | NLO Weinberg Correction | tau-e | form |
| IV.T134 | theorem | Window Algebra Origin | tau-e | form |
| IV.P180 | proposition | Exponent-Width Coincidence | tau-e | form |
| IV.R389 | remark | Scale Consistency | tau-e | form |
| IV.T135 | theorem | Fermi Form w-Independence | tau-e | form |
| IV.P181 | proposition | Mode Interpretation of EW Coefficients | tau-e | form |
| IV.R390 | remark | OQ-A3 Resolution Status | tau-e | form |
| IV.R391 | remark | OQ-B2 Partial Resolution Status | tau-e | form |
| IV.D335 | definition | EW-Active Mode | tau-e | form |
| IV.D336 | definition | EW 3-Way Partition | tau-e | form |
| IV.T136 | theorem | EW Partition Theorem | tau-e | form |
| IV.T137 | theorem | EW Density = 5/7 | tau-e | form |
| IV.T138 | theorem | EW–CF Bridge | tau-e | form |
| IV.T139 | theorem | NLO from EW Projection | tau-e | form |
| IV.P182 | proposition | Complement Characterization | tau-e | form |
| IV.R392 | remark | OQ-B2 Resolved | tau-e | form |
| IV.D337 | definition | sin²θ_W NNLO Formula | tau-e | form |
| IV.D338 | definition | M_W NLO Formula | tau-e | form |
| IV.D339 | definition | alpha_s NLO Formula | tau-e | form |
| IV.T140 | theorem | Window Universality — W₃(4)=5 | tau-e | form |
| IV.R393 | remark | NNLO Precision Summary | tau-e | form |
| IV.D340 | definition | Nucleon Boundary Mode | estab | form |
| IV.D341 | definition | QCD Contribution to p-n splitting | tau-e | form |
| IV.D342 | definition | EM Coulomb Contribution to p-n splitting | tau-e | form |
| IV.T141 | theorem | Proton-Neutron Mass Difference — Tree Level [SUPERSEDED] | conje | form |
| IV.T142 | theorem | Proton-Neutron Mass Difference — Two-Sector | tau-e | form |
| IV.P183 | proposition | Sign of p-n splitting: QCD > EM | tau-e | form |
| IV.P184 | proposition | NLO Color-Generator Correction 6/5 | tau-e | form |
| IV.R394 | remark | Comparison to Cottingham decomposition | estab | form |
| IV.D343 | definition | Neutrino Exponent Best-Fit | tau-e | form |
| IV.R395 | remark | Normal Ordering from sigma-Structure | tau-e | form |
| IV.D344 | definition | sigma-Equivariant Lepton Mass Matrix | tau-e | form |
| IV.D345 | definition | Koide Angle delta=2/9 | tau-e | form |
| IV.T143 | theorem | Koide Q=2/3 from sigma-Symmetry | tau-e | form |
| IV.T144 | theorem | m_mu/m_e Leading-Order Formula | tau-e | form |
| IV.P185 | proposition | Three-Generation Winding Closure | tau-e | form |
| IV.R396 | remark | Quark-Lepton Universality Conjecture | tau-e | not_ |
| IV.D346 | definition | tau-Charge Conjugation C_tau = sigma | estab | form |
| IV.T145 | theorem | Uniqueness: C_tau = sigma (lobe-swap uniquely determines C) | tau-e | form |
| IV.T146 | theorem | Majorana Theorem: zero-U(1) modes are Majorana | tau-e | form |
| IV.P186 | proposition | beta-Decay nu/nu-bar as Helicity Labels (not particle/antiparticle) | tau-e | form |
| IV.D347 | definition | Three Primitive Winding Classes on T² | estab | form |
| IV.T147 | theorem | Three-Generation Closure: Spectral Gap on T² | tau-e | form |
| IV.T148 | theorem | m_μ/m_e Best NLO Formula: ι_τ⁻⁴·⁹⁶ at +307 ppm | tau-e | form |
| IV.T149 | theorem | Lepton σ-Matrix Exponents from PDG Back-Solve | tau-e | form |
| IV.P187 | proposition | Quark-Lepton Universality: Exponent Step ≈ −2.7 | tau-e | form |
| IV.R397 | remark | 0vbb Prediction: must exist; rate proportional to |