Registry Dashboard — Book IV: Categorical Microcosm

Filter rule on this page: dashboard_display — counts the five display types (definition, lemma, proposition, remark, theorem). The full registry root for IV has 1864 objects across all types; this dashboard enumerates 1292 of them. See the Filter Rules manifest for the authoritative definitions, or the Release Manifest for the per-book reconciliation across all surfaces.

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

</tr> </tbody> </table>

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 + ιτ) 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 time ``flows'' 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	Why ``electromagnetic'' and not just ``electric''	tau-e	—
IV.R346	remark	Null = no fiber winding, not ``zero''	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: ιτ⁻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	</td>	^2	tau-e	form
IV.R398	remark	NLO Correction Factor for m_μ/m_e — Open Structural Identification (OQ-C5a)	conje	form
IV.D348	definition	Higgs Mass omega-Sector Formula	tau-e	form
IV.T150	theorem	Factor-4 from Non-omega Generator Count	tau-e	form
IV.T151	theorem	Higgs Mass Sub-1000 ppm Formula via Window Universality	tau-e	form
IV.P188	proposition	m_H/m_W Ratio [auto-upgrades with IV.P199]	tau-e	form
IV.R399	remark	Open: omega Self-Energy Correction and Coefficient-6 Identification	conje	form
IV.D349	definition	Cabibbo Angle from T² Holonomy: λ_C = ιτ·κ_D	tau-e	form
IV.T152	theorem	Cabibbo Angle: sin(θ_C) = ιτ·(1−ιτ) at −2327 ppm	tau-e	form
IV.T153	theorem	PMNS Large Mixing Requires A-Sector Flavor Rotation Beyond σ-Matrix	tau-e	form
IV.T154	theorem	GIM-Analog Cancelation from σ-Equivariance	tau-e	form
IV.P189	proposition	Quark-Lepton Complementarity: θ₁₂+θ_C ≈ π/4 from Fiber-Base Duality	tau-e	form
IV.P190	proposition	Wolfenstein Parameters: ρ̄=1/(2π) at +974 ppm (τ-effective); A and η̄ open	tau-e	form
IV.R400	remark	PMNS CP Phase δ_CP from σ-Breaking; New OQ-C6/C7	tau-e	form
IV.T155	theorem	Higgs Bonus Coefficient n=6 [SUPERSEDED by IV.T166 n=7]	conje	not_
IV.T156	theorem	m_μ/m_e NNLO Correction: δ = 1/W₃(4)² = 1/25	tau-e	form
IV.P191	proposition	Window RG Period: W₃(4)^k Governs k-th Perturbative Order	tau-e	form
IV.R401	remark	Cross-Check: 1/21 in p-n Mass Difference = 1/(4·W₃(4)+1); n=7 Higgs at +8 ppm	tau-e	form
IV.D355	definition	Strong CP Resolution: SA-i Forbids Topological C-Sector Charge	estab	skel
IV.T160	theorem	θ_QCD = 0 from C-Sector SA-i Admissibility	tau-e	skel
IV.T161	theorem	Neutron EDM = 0 Exactly from τ-Framework	tau-e	skel
IV.P195	proposition	No Axion Required: Strong CP Solved Structurally by SA-i	tau-e	skel
IV.R405	remark	Comparison to Peccei-Quinn: SA-i as τ-Native PQ Mechanism	tau-e	skel
IV.D356	definition	A-Sector PMNS Rotation: pi-Generator Flavor Rotation on tau^1	tau-e	form
IV.T162	theorem	sin(theta_23) = 1/(1+iota_tau): Atmospheric Mixing from A-Sector	tau-e	form
IV.T163	theorem	QLC-Exact: sin(theta_12) = (sqrt(1-lambda_C^2) - lambda_C)/sqrt(2)	tau-e	form
IV.P196	proposition	PMNS Large Mixing from A-Sector Rotation and QLC Complementarity	tau-e	form
IV.R406	remark	CP Phase delta_CP from A-Sector Rotation: Connects IV.T152 and V.T174	tau-e	form
IV.D357	definition	Wolfenstein CP Parameters from omega-Sector Holonomy Period	tau-e	form
IV.T164	theorem	rho_bar = 1/(2*pi): omega-Period Structural Derivation (tau-effective)	tau-e	form
IV.T165	theorem	A = 1 - (3/2)*iota_tau^2 at -887 ppm from PDG 0.826 (tau-effective)	tau-e	form
IV.P197	proposition	eta_bar from CP Phase Structure: Best tau-Candidate sqrt(5)/(2*pi)	tau-e	form
IV.P198	proposition	CKM Unitarity Triangle Angles from tau-Framework	tau-e	form
IV.R407	remark	OQ-CKM1 Status after Sprint 5C: rho_bar Derived, A Improved, eta_bar Open	tau-e	form
IV.D358	definition	Higgs n=7: Structural Identification as 2*lobes + sectors	tau-e	skel
IV.T166	theorem	Higgs n=7 Formula at +8.0 ppm from PDG 125.20 GeV (tau-effective)	tau-e	form
IV.P199	proposition	n=7 Structural Uniqueness: Coherence Functional Samples Lobe+Sector	tau-e	form
IV.R408	remark	m_mu/m_e Sub-100 ppm NNLO Status and Higgs n Comparison Table	tau-e	skel
IV.D359	definition	Wolfenstein η̄ from 5-Generator Pentagon: CP Violation from ω-Period	tau-e	form
IV.T167	theorem	Jarlskog Invariant J from τ-Parameters	tau-e	form
IV.T168	theorem	η̄ Best τ-Candidate: ιτ^(−1/4)·κ_D^(5/4)/√5 at −2285 ppm	tau-e	form
IV.P200	proposition	CP Violation from ω-Period Pentagon: Pentagon-Angle Derivation	tau-e	form
IV.R409	remark	OQ-CKM1 Status after Sprint 6B: All Four Wolfenstein τ-Effective	tau-e	form
IV.D360	definition	m_μ/m_e NNLO k=23/3: W₃(4)+W₃(3)+1 Window-Algebra Exponent	tau-e	form
IV.T169	theorem	m_μ/m_e NNLO: k=23/3 Correction at +43 ppm (τ-effective)	tau-e	form
IV.T170	theorem	p-n Mass NLO: C.5 Two-Sector Formula Structural Derivation	tau-e	form
IV.P201	proposition	C.5 Coefficients: 3/16 = N_c/2⁴ and 3/20 = N_c/(4·W₃(4))	tau-e	form
IV.R410	remark	m_μ NNLO + p-n NLO Precision Status after Sprint 6D	tau-e	form
IV.D361	definition	Fiber-Base Homology of τ³	estab	form
IV.D362	definition	Solenoidal Generator–Force Map	tau-e	form
IV.T171	theorem	Fourth Generation Excluded (Topological)	tau-e	form
IV.T172	theorem	Generation Mass Hierarchy from Eigenvalue Ordering	tau-e	form
IV.P202	proposition	15 = 3 × 5 Boundary Mode Decomposition	tau-e	form
IV.R411	remark	IV.OP3 Status: SOLVED (τ-effective) after Sprint 7A	tau-e	not_
IV.D363	definition	Quarter-Lobe Holonomy: ιτ Exponent -1/4 = -1/(2·	lobes	)	tau-e	form
IV.D364	definition	Pentagon Dark Coupling: κ_D Exponent 5/4 =	gen	/(2·	lobes	)	tau-e	form
IV.T173	theorem	η̄ Exponent Derivation: All from	lobes	=2,	gen	=5 (τ-effective)	tau-e	form
IV.P203	proposition	Jarlskog J Full-τ Consistency: J_τ = A_τ²·λ_τ⁶·η̄_τ	tau-e	form
IV.R412	remark	OQ η̄ Exponents: SOLVED (τ-effective) after Sprint 7F	tau-e	not_
IV.D365	definition	A-Sector NLO PMNS Rotation	tau-e	form
IV.T174	theorem	θ₂₃ NLO via Window Algebra at +8604 ppm	tau-e	form
IV.T175	theorem	θ₁₂ from QLC + Higgs NLO at +3106 ppm	tau-e	form
IV.P204	proposition	δ_CP = π + arctan(ιτ) at +9365 ppm	tau-e	form
IV.R413	remark	OQ-C7 Status: PARTIAL-IMPROVED after Sprint 7C	tau-e	not_
IV.D366	definition	k=15/2 Baryogenesis-Lepton Duality: Shared Exponent 15 = dim(τ³)·W₃(4)	tau-e	form
IV.D367	definition	NNLO Exponent Catalog: 7 Window-Universal Corrections	tau-e	form
IV.T176	theorem	m_μ/m_e NNLO: k=15/2 at −8.2 ppm (τ-effective, best)	tau-e	form
IV.P205	proposition	Window Universality for All 7 NNLO Exponents	tau-e	form
IV.R414	remark	OQ k=7.5: SOLVED (τ-effective) after Sprint 7G	tau-e	not_
IV.T177	theorem	Precision W Boson Mass from Window Universality at −0.5 ppm	tau-e	skel
IV.R415	remark	CDF–ATLAS W Mass Tension and τ³ Prediction	tau-e	not_
IV.T178	theorem	Z Boson Mass from Window Universality at +1039 ppm	tau-e	skel
IV.R416	remark	On-Shell vs MS-bar Weinberg Angle Scheme Distinction	tau-e	not_
IV.P206	proposition	Electroweak Boson Mass Ratio Consistency	tau-e	skel
IV.P207	proposition	Higgs-to-W Mass Ratio at +8.8 ppm (n=7 upgrade)	tau-e	skel
IV.D368	definition	Anomalous Magnetic Moment Definition	estab	skel
IV.T179	theorem	Schwinger Term from L-Holonomy at −3 ppm (NLO)	tau-e	skel
IV.P208	proposition	Electron g-2 QED Series with α_τ: −3 ppm at NLO	tau-e	skel
IV.R417	remark	Structural vs Perturbative Content in g-2	tau-e	not_
IV.T180	theorem	Muon g−2 from τ³ at +8.8 ppm	tau-e	skel
IV.R418	remark	Muon g-2 Anomaly Resolution Requires Sub-ppm α	tau-e	not_
IV.P209	proposition	g-2 Falsification Criteria for τ³	tau-e	skel
IV.T181	theorem	CKM Matrix Elements from Wolfenstein Parameters	tau-e	skel
IV.P210	proposition	First-Row CKM Unitarity (Cabibbo Anomaly Test)	tau-e	skel
IV.R419	remark	Jarlskog Invariant from Wolfenstein at −8750 ppm	tau-e	not_
IV.T182	theorem	Axial Coupling g_A from CF Window Algebra at 5.5 ppm	tau-e	skel
IV.R420	remark	Physical Meaning of g_A = κ_D²/ιτ	tau-e	not_
IV.D369	definition	Quark Mass Winding Exponents	tau-e	skel
IV.T183	theorem	Top Quark Mass from Fiber Amplitude at +7200 ppm	tau-e	skel
IV.P211	proposition	Quark-Lepton Exponent Symmetry	tau-e	skel
IV.R421	remark	Light Quark Mass Ratio m_u/m_d Candidates	conje	not_
IV.R422	remark	IV.OP5 Status: Quark Mass Problem (Wave 32)	conje	not_
IV.T184	theorem	PMNS θ₁₃ from Cubic Crossing Amplitude at +5000 ppm	tau-e	skel
IV.R423	remark	PMNS Angle Summary (All Three Predicted)	tau-e	not_
IV.P212	proposition	PMNS Unitarity Triangle Consistency	tau-e	skel
IV.T185	theorem	Proton Charge Radius from Compton Scale at +440 ppm	tau-e	skel
IV.T186	theorem	Proton g-Factor from Quark Winding at +2000 ppm	tau-e	skel
IV.P213	proposition	Lamb Shift Structural Estimate (Proton-Size Contribution)	tau-e	skel
IV.R424	remark	Precision Atomic Physics Assessment (Wave 34)	tau-e	not_
IV.D370	definition	Quark Triplet σ-Matrix Parameters	estab	skel
IV.T187	theorem	Koide Deviation for Quark Triplets	tau-e	skel
IV.R425	remark	Why σ-Equivariance Fails for Current Masses	tau-e	not_
IV.T189	theorem	m_t/m_b Ratio from ιτ at +99 ppm	tau-e	skel
IV.P215	proposition	Intra-Triplet Ratio Exponents	tau-e	skel
IV.P216	proposition	Step-Exponent Confinement Correction	tau-e	skel
IV.D371	definition	Confinement-Corrected Mass Scale	estab	skel
IV.T188	theorem	Koide-Optimal Confinement Shift	tau-e	skel
IV.R426	remark	Koide Restoration Landscape	tau-e	not_
IV.D372	definition	m_u/m_d Isospin NLO Structure	conje	skel
IV.P217	proposition	m_u/m_d NLO Candidate at +29 ppm	conje	skel
IV.R427	remark	m_u/m_d Status Assessment	conje	not_
IV.D373	definition	κ(C;n) Confinement Depth Spectrum	estab	skel
IV.T190	theorem	Corrected Top Mass at −1449 ppm	tau-e	skel
IV.P218	proposition	Confinement-Free Top Quark	tau-e	skel
IV.D374	definition	Epstein Zeta Quark Mode Spectrum	tau-e	skel
IV.R428	remark	Chowla-Selberg Integrality and Confinement	conje	not_
IV.P214	proposition	IV.OP5 Status (Wave 35)	conje	skel
IV.T191	theorem	Charm Mass from τ-Chain at +1150 ppm	tau-e	form
IV.T192	theorem	Strange Mass from τ-Chain at +1559 ppm	tau-e	form
IV.D375	definition	m_c/m_s Cross-Check	tau-e	form
IV.T193	theorem	Bottom Mass from τ-Chain at −1351 ppm	tau-e	form
IV.R429	remark	m_u/m_d NLO Structural Analysis	conje	not_
IV.P219	proposition	m_d/m_s Prediction at −1921 ppm	tau-e	form
IV.D376	definition	Higgs Self-Coupling from τ-Chain	tau-e	form
IV.T194	theorem	τ-Chain Higgs Self-Coupling at +16 ppm	tau-e	form
IV.P220	proposition	HL-LHC and FCC-hh Falsification Window	estab	form
IV.D377	definition	Six-Quark τ-Chain Mass Table	tau-e	form
IV.T195	theorem	Jarlskog Invariant from τ-CKM at −35000 ppm	tau-e	form
IV.R430	remark	IV.OP5 Status Update (Wave 37)	conje	not_
IV.D378	definition	Winding Transition Matrix on T² Primitive Modes	tau-e	form
IV.D379	definition	Generation Eigenvalue Spectrum on Anisotropic T²	estab	form
IV.T196	theorem	m_t/m_b Exponent from Winding Algebra	tau-e	form
IV.P221	proposition	m_c/m_t Exponent from Higgs-Weighted Transition	tau-e	form
IV.R431	remark	Exponent Universality: All Ratios from W	tau-e	form
IV.D380	definition	Top–Up Exponent Duality on T²	tau-e	form
IV.T197	theorem	Up Quark Direct Mass from Exponent Duality at +395 ppm	tau-e	form
IV.P222	proposition	m_u/m_d Isospin Ratio from Winding Algebra at +82 ppm	tau-e	form
IV.R432	remark	Up Quark Status Update (Wave 44)	tau-e	not_
IV.D381	definition	Wolfenstein A NLO: Confinement Correction	tau-e	form
IV.T198	theorem	Jarlskog NLO at +2624 ppm	tau-e	form
IV.R433	remark	CKM Precision Assessment (Wave 44)	tau-e	not_
IV.T199	theorem	m_s/m_b Exponent from Confinement-Weighted Mode Count	tau-e	form
IV.T200	theorem	m_d/m_s Exponent from Winding Phase Space	tau-e	form
IV.R434	remark	Down-Type Sector Exponent Completeness	tau-e	not_
IV.T201	theorem	m_u/m_d Isospin Exponent from Higgs-Mediated Splitting	tau-e	form
IV.R435	remark	IV.OP9 Status: SOLVED (7/7 Exponents Derived)	tau-e	not_
IV.D382	definition	Updated Six-Quark Table (Wave 45)	tau-e	form
IV.P223	proposition	Lobe-Power Hierarchy of Quark Exponents	tau-e	form
IV.R436	remark	IV.OP5 Status Update (Wave 45)	tau-e	not_
IV.T202	theorem	Proton Charge Radius NLO at +12 ppm	tau-e	form
IV.R437	remark	Precision Atomic Physics Assessment Update (Wave 46)	tau-e	not_
IV.D383	definition	Neutron Lifetime Input Table (Wave 46)	tau-e	form
IV.T203	theorem	Neutron Lifetime Precision Prediction	tau-e	form
IV.P224	proposition	Cancelled-Form Error Budget (77 ppm RSS)	tau-e	form
IV.R438	remark	Nuclear Sector Assessment (Wave 46)	tau-e	not_
IV.D384	definition	Two-Loop Window Coefficient c₂	conje	form
IV.T204	theorem	Window Depth–Loop Order Correspondence	conje	form
IV.P225	proposition	c₂ Numerical Bound from W₄(3)	conje	form
IV.R439	remark	OQ-B4 Status: PARTIAL	tau-e	—
IV.D385	definition	α NLO Correction Candidate Catalog	conje	form
IV.T205	theorem	α Precision Barrier at 9.8 ppm	tau-e	form
IV.R440	remark	α Precision Assessment	tau-e	—
IV.T206	theorem	θ₂₃ NNLO from Holonomy-at-Window-Squared	conje	form
IV.T207	theorem	δ_CP NLO from Fiber Correction	conje	form
IV.D386	definition	High-ppm Structural Limit Catalog	tau-e	form
IV.R441	remark	High-ppm Audit Summary (Wave 49)	tau-e	—
IV.D387	definition	Magnetic Moment on T²	tau-e	form
IV.D388	definition	τ-Ising Hamiltonian on T²	tau-e	form
IV.P226	proposition	Spontaneous Magnetization on T²	tau-e	form
IV.T208	theorem	No Magnetic Monopoles on T²	estab	form
IV.R443	remark	Electric vs Magnetic Duality	estab	—
IV.D389	definition	Magnetic Domain Wall on T²	tau-e	form
IV.P227	proposition	Domain Wall Energy from T² Winding	tau-e	form
IV.T209	theorem	Curie Transition as T² Symmetry Breaking	tau-e	form
IV.P228	proposition	Magnetic Orders as Defect-Tuple Signatures	tau-e	form
IV.P230	proposition	He-4 Minimal Donut Criterion	tau-e	not_
IV.T212	theorem	Cooper Pairing from EW-Mixed Normalization	tau-e	not_
IV.R444	remark	Upgrade of BCS to Corollary	tau-e	—
IV.P231	proposition	Type-I/II Classification from Defect-Tuple Inequality	tau-e	not_
IV.D391	definition	Dislocation as Holonomy Defect	tau-e	not_
IV.P232	proposition	Dislocation Classification	tau-e	not_
IV.D392	definition	Glass Threshold K_glass	tau-e	not_
IV.D393	definition	EM-Glass	tau-e	not_
IV.D394	definition	CheckCrystal Decision Procedure	tau-e	not_
IV.D395	definition	CheckGlass Decision Procedure	tau-e	not_
IV.P233	proposition	Decidability of Crystal and Glass Classification	tau-e	not_
IV.P234	proposition	Two-Channel Framework	tau-e	not_
IV.D390	definition	Non-Dissipative Endomorphism	tau-e	form
IV.T210	theorem	NFL-Boundary Theorem	tau-e	form
IV.R442	remark	NFL-Depth Corollary	tau-e	—
IV.T211	theorem	Decidable Phase Classification Meta-Theorem	tau-e	form
IV.P229	proposition	Ten Regime Instantiations	tau-e	form
IV.D396	definition	Anomaly as Naturality Obstruction	tau-e	—
IV.T213	theorem	ABJ Chiral Anomaly	estab	—
IV.T214	theorem	Anomaly Cancellation as Naturality Requirement	tau-e	—
IV.R445	remark	Feynman Diagram Composition Rules	tau-e	—
IV.R446	remark	Vertex Types as Gauge-Equivariant Composition	tau-e	—
IV.D397	definition	S-Matrix Decomposition	estab	—
IV.T215	theorem	Unitarity from Dagger Structure	tau-e	—
IV.R447	remark	Optical Theorem	estab	—
IV.D398	definition	Ontic Entity Criterion	tau-e	—
IV.D399	definition	Non-Ontic Entity Classification	tau-e	—
IV.T216	theorem	Complete Ontic Classification	tau-e	—
IV.R448	remark	τ as the ℕ of Holomorphy	metap	—