Registry — Book IV: Categorical Microcosm

IV.C01 Right-handed decoupling corollary IV.C02 Twin Prime Residue Theorem theorem IV.C3 Ontological Identity of Laws and Matter corollary IV.C4 Absence of God's-Eye View corollary IV.C5 The -Matrix as Composed Morphism corollary IV.C6 Spacetime Symmetries corollary IV.D01 Sector Physics Template definition IV.D02 EM Sector at E₁ definition IV.D03 Strong Sector at E₁ definition IV.D04 Higgs Sector at E₁ definition IV.D05 Gravity Sector at E₁ definition IV.D06 Weak Sector at E₁ definition IV.D07 Coupling Formula Map definition IV.D08 Spectral Fine Structure definition IV.D09 Primary Invariant definition IV.D10 Carrier Type definition IV.D100 Electromagnetic Field Tensor definition IV.D101 Electric and Magnetic Field Extraction definition IV.D102 Hodge Dual of F definition IV.D103 Electromagnetic Current 4-Vector definition IV.D104 tau-Native Fine-Structure Constant definition IV.D105 Null Transport Mode definition IV.D106 Holonomy Correction Factor definition IV.D107 Sector Polarity definition IV.D108 Polarity-Preserving Interaction definition IV.D109 Channel-Switching Defect Bundle definition IV.D11 Physical Quantity Template definition IV.D110 W-pm Defect Bundles definition IV.D111 Z0 Defect Bundle definition IV.D112 Chirality definition IV.D113 A-Sector Sigma-Admissibility definition IV.D114 Parity Operator definition IV.D115 Crossing-Point Action Space definition IV.D116 Weak Isospin Generators definition IV.D117 Left-Handed Doublets definition IV.D118 Weak Gauge Bosons Before Mixing definition IV.D119 Weak Holonomy Loop definition IV.D12 Particle Kind definition IV.D120 W Boson Parameters definition IV.D121 Weinberg Angle from Tau-Couplings definition IV.D122 Z Boson Parameters definition IV.D123 The Rho Parameter definition IV.D124 Neutrino as Time-Eigenmode definition IV.D125 Neutrino Flavor Eigenstates definition IV.D126 PMNS Matrix definition IV.D127 τ-Hypercharge definition IV.D128 Pre-Mixing Electroweak Gauge Group definition IV.D129 Charged W Bosons definition IV.D13 Planck Character definition IV.D130 Weinberg Angle definition IV.D131 Mixing Compatibility definition IV.D132 Electroweak Coherence State definition IV.D133 Coherence Fixing (Ch33) definition IV.D134 Coherence Fixing (Formal) definition IV.D135 Finite-Stage Coherence Functional definition IV.D136 Physical Vacuum Ω*[ω] definition IV.D137 Minimality Condition definition IV.D138 EM-Nullity Condition definition IV.D139 σ-Polarity Condition definition IV.D14 Uncertainty Product definition IV.D140 Vacuum Hessian at Crossing Point definition IV.D141 τ-Higgs Mass definition IV.D142 τ-Yukawa Coupling (Ch34) definition IV.D143 τ-Yukawa Coupling (Overlap Integral) definition IV.D144 The C-sector definition IV.D145 Strong loop class definition IV.D146 Strong holonomy defect definition IV.D147 Strong defect functional definition IV.D148 Strong admissibility definition IV.D149 Finite-stage strong vacuum definition IV.D15 Sector Lift Functor definition IV.D150 Strong vacuum definition IV.D151 \mathrm{HolEnd_τ(s) definition IV.D152 Fix(s) definition IV.D153 Canonical strong lift definition IV.D154 Color Charge definition IV.D155 Anti-Color definition IV.D156 Color Neutrality definition IV.D157 Wilson Loop definition IV.D158 Fractional CR-sublattice definition IV.D159 Color-confined mode definition IV.D16 Defect Component definition IV.D160 Color singlet definition IV.D161 Baryon number definition IV.D162 τ-Holonomy sector definition IV.D163 Canonical vacuum at stage~n definition IV.D164 Localized perturbations definition IV.D165 Finite-difference quadratic form definition IV.D166 Excitation cost definition IV.D167 Canonical smallest excitation definition IV.D168 Three kernel hypotheses definition IV.D169 Strong configuration space definition IV.D17 Defect Tuple definition IV.D170 Strong connection assignment definition IV.D171 Strong curvature definition IV.D172 Plaquette-aggregated strong defect definition IV.D173 Canonical strong vacuum, plaquette form definition IV.D174 Strong quadratic form definition IV.D175 Spectral gap at stage~n definition IV.D176 YM sector coupling definition IV.D177 Gap quantum definition IV.D178 Readout functor (conjectural) definition IV.D179 Orthodox Bridge Conjecture definition IV.D18 Fluid Regime definition IV.D180 π-lift at stage n definition IV.D181 π-lift ω-limit definition IV.D182 The τ-strong coupling constant definition IV.D183 π-support penalty definition IV.D184 Ontic coupling definition IV.D185 Regime selector definition IV.D186 Regime readout map definition IV.D187 Quark mode definition IV.D188 Antiquark mode definition IV.D189 Quark generations from \Lemniscate definition IV.D19 Regime Signature definition IV.D190 Meson state definition IV.D191 Baryon state definition IV.D192 Boundary-first normalization definition IV.D193 Earned vs.\ unearned mode count definition IV.D194 9-element canonical generator set definition IV.D195 tau-Yukawa coupling definition IV.D196 Three mode classes on lemniscate definition IV.D197 Quark winding classes definition IV.D198 Koide parameter definition IV.D199 Rydberg constant definition IV.D20 Mass Index definition IV.D200 Meson classification definition IV.D201 Glueball definition IV.D202 Nuclear force definition IV.D203 Atom as dressed nuclear mode definition IV.D204 Electron quantum numbers definition IV.D205 Covalent bond definition IV.D206 Ionic bond definition IV.D207 Metallic bond definition IV.D208 Hybrid modes definition IV.D209 Ontic entity definition IV.D21 Energy Index definition IV.D210 Macroscopic defect tuple definition IV.D211 Macroscopic mobility definition IV.D212 Macroscopic vorticity definition IV.D213 Macroscopic compression definition IV.D214 Total topological charge definition IV.D215 Clopen cylinder at depth n definition IV.D216 Level-n defect functional definition IV.D217 Universal defect functional definition IV.D218 Defect tuple space definition IV.D219 Critical mobility threshold definition IV.D22 Speed Constant definition IV.D220 Crystal regime definition IV.D221 Glass regime definition IV.D222 Euler fluid regime definition IV.D223 Navier-Stokes regime definition IV.D224 MHD regime definition IV.D225 Plasma regime definition IV.D226 Superfluid regime definition IV.D227 Superconductor regime definition IV.D228 Temperature as defect gradient definition IV.D229 First-order phase transition definition IV.D23 Mass-Energy Relation definition IV.D230 Second-order phase transition definition IV.D231 tau-Euler flow definition IV.D232 tau-Navier-Stokes flow definition IV.D233 Superfluid regime definition IV.D234 Superconductor regime definition IV.D235 Crystal regime definition IV.D236 Glass regime definition IV.D237 Quasicrystal regime definition IV.D238 First-order phase transition definition IV.D239 Second-order phase transition definition IV.D24 Entropy Splitting definition IV.D240 Topological branch definition IV.D241 Tower-natural transformation definition IV.D242 tau-sphaleron question definition IV.D243 Book V import list definition IV.D244 Book VI import list definition IV.D245 Book VII import list definition IV.D246 Coherence Kernel --- Physics Presentation definition IV.D247 Generator--Sector Assignment definition IV.D248 Ontic Minimality definition IV.D249 Refinement Tower mathcalR definition IV.D25 Defect Budget definition IV.D250 Profinite Limit hatalpha definition IV.D251 Proto-Time t_p definition IV.D252 Base tau^1 --- physical reading definition IV.D253 Fiber T^2 --- physical reading definition IV.D254 Fibered product arena tau^3 definition IV.D255 Master constant iota_tau definition IV.D256 Lemniscate boundary --- physical reading definition IV.D257 Chart readout homomorphism definition IV.D258 Yoneda self-image definition IV.D259 Boundary character definition IV.D26 SI Reference Table definition IV.D260 Bipolar decomposition of characters definition IV.D261 Physical-constants core definition IV.D262 Canonical sector lifts definition IV.D263 Chart readout homomorphism definition IV.D264 Generator--Sector Correspondence definition IV.D265 Coupling Ledger definition IV.D266 Boundary holonomy generators definition IV.D267 Defect bundle (ontic particle) definition IV.D268 Radiation definition IV.D269 Virtual particle definition IV.D27 SI Measured Constants definition IV.D270 Five primary invariants definition IV.D271 Mass as fiber stiffness definition IV.D272 Propagation operator definition IV.D273 Planck character definition IV.D274 Defect functional definition IV.D275 Gravitational self-coupling definition IV.D276 Weak self-coupling definition IV.D277 Electromagnetic self-coupling definition IV.D278 Strong self-coupling definition IV.D279 Omega self-coupling definition IV.D28 Weinberg Near-Match definition IV.D280 Weak--gravity cross-coupling definition IV.D281 Electroweak cross-coupling definition IV.D282 Weak--strong cross-coupling definition IV.D283 EM--strong cross-coupling definition IV.D284 EM--gravity cross-coupling definition IV.D285 Strong--gravity cross-coupling definition IV.D286 Spectral Fine-Structure Formula definition IV.D287 Five Relational Units definition IV.D288 Holonomy Correction Factor definition IV.D289 Five Relational Units definition IV.D29 Strong Near-Match definition IV.D290 Unpolarized defect bundle definition IV.D291 Calibration Anchor definition IV.D292 tau-to-SI conversion definition IV.D293 Speed of light in relational units definition IV.D294 Planck's constant in relational units definition IV.D295 Coulomb constant in relational units definition IV.D296 Vacuum permittivity in relational units definition IV.D297 Vacuum permeability in relational units definition IV.D298 Planck character definition IV.D299 Beta function definition IV.D30 Calibration Anchor definition IV.D300 Coupling Ledger and Observable Ledger definition IV.D301 Readout Functor definition IV.D302 Entropy splitting definition IV.D303 Regime transition definition IV.D304 Readout landscape definition IV.D305 Coupling Constants Table definition IV.D306 Fundamental Scales Table definition IV.D307 Particle Mass Table definition IV.D308 Structural Constants Table definition IV.D309 Breathing operator definition IV.D31 τ-to-SI Conversion definition IV.D310 Epstein zeta structure definition IV.D311 Chowla--Selberg decomposition definition IV.D312 Lemniscate three-fold definition IV.D313 Spectral distance sqrt3 definition IV.D314 Triple Holonomy --- IV.D44 definition IV.D315 Holonomy Correction Data --- IV.D45 definition IV.D316 Mass Ratio Bulk Term --- IV.D46 definition IV.D317 Level 0 Formula --- IV.D47 definition IV.D318 Level 1+ Formula --- IV.D48 definition IV.D319 Polarity-switching transition definition IV.D32 Tau Physical Scale definition IV.D320 Electroweak scale in Category~tau definition IV.D321 Tick Morphism definition IV.D322 Tick Kind definition IV.D323 Internal Identity definition IV.D324 Equation Layer definition IV.D325 Measurement Procedure definition IV.D326 Readout Functor definition IV.D327 Readout Anchor definition IV.D328 Generator Carrier Assignment definition IV.D329 Generator Polarity Assignment definition IV.D33 Speed of Light definition IV.D330 Structural EM Activity definition IV.D331 Tensor-Square Character Algebra definition IV.D334 NLO Weinberg Correction definition IV.D335 EW-Active Mode definition IV.D336 EW 3-Way Partition definition IV.D337 sin²θ_W NNLO Formula definition IV.D338 M_W NLO Formula definition IV.D339 alpha_s NLO Formula definition IV.D34 Planck Constant definition IV.D340 Nucleon Boundary Mode definition IV.D341 QCD Contribution to p-n splitting definition IV.D342 EM Coulomb Contribution to p-n splitting definition IV.D343 Neutrino Exponent Best-Fit definition IV.D344 sigma-Equivariant Lepton Mass Matrix definition IV.D345 Koide Angle delta=2/9 definition IV.D346 tau-Charge Conjugation C_tau = sigma definition IV.D347 Three Primitive Winding Classes on T² definition IV.D348 Higgs Mass omega-Sector Formula definition IV.D349 Cabibbo Angle from T² Holonomy: λ_C = ι_τ·κ_D definition IV.D35 Coulomb Constant definition IV.D355 Strong CP Resolution: SA-i Forbids Topological C-Sector Charge definition IV.D356 A-Sector PMNS Rotation: pi-Generator Flavor Rotation on tau^1 definition IV.D357 Wolfenstein CP Parameters from omega-Sector Holonomy Period definition IV.D358 Higgs n=7: Structural Identification as 2*lobes + sectors definition IV.D359 Wolfenstein η̄ from 5-Generator Pentagon: CP Violation from ω-Period definition IV.D36 Vacuum Permittivity definition IV.D360 m_μ/m_e NNLO k=23/3: W₃(4)+W₃(3)+1 Window-Algebra Exponent definition IV.D361 Fiber-Base Homology of τ³ definition IV.D362 Solenoidal Generator–Force Map definition IV.D363 Quarter-Lobe Holonomy: ι_τ Exponent -1/4 = -1/(2·|lobes|) definition IV.D364 Pentagon Dark Coupling: κ_D Exponent 5/4 = |gen|/(2·|lobes|) definition IV.D365 A-Sector NLO PMNS Rotation definition IV.D366 k=15/2 Baryogenesis-Lepton Duality: Shared Exponent 15 = dim(τ³)·W₃(4) definition IV.D367 NNLO Exponent Catalog: 7 Window-Universal Corrections definition IV.D368 Anomalous Magnetic Moment Definition definition IV.D369 Quark Mass Winding Exponents definition IV.D37 Vacuum Permeability definition IV.D370 Quark Triplet σ-Matrix Parameters definition IV.D371 Confinement-Corrected Mass Scale definition IV.D372 m_u/m_d Isospin NLO Structure definition IV.D373 κ(C;n) Confinement Depth Spectrum definition IV.D374 Epstein Zeta Quark Mode Spectrum definition IV.D375 m_c/m_s Cross-Check definition IV.D376 Higgs Self-Coupling from τ-Chain definition IV.D377 Six-Quark τ-Chain Mass Table definition IV.D378 Winding Transition Matrix on T² Primitive Modes definition IV.D379 Generation Eigenvalue Spectrum on Anisotropic T² definition IV.D38 Ledger Entry definition IV.D380 Top–Up Exponent Duality on T² definition IV.D381 Wolfenstein A NLO: Confinement Correction definition IV.D382 Updated Six-Quark Table (Wave 45) definition IV.D383 Neutron Lifetime Input Table (Wave 46) definition IV.D384 Two-Loop Window Coefficient c₂ definition IV.D385 α NLO Correction Candidate Catalog definition IV.D386 High-ppm Structural Limit Catalog definition IV.D387 Magnetic Moment on T² definition IV.D388 τ-Ising Hamiltonian on T² definition IV.D389 Magnetic Domain Wall on T² definition IV.D39 Complete Constants Ledger definition IV.D390 Non-Dissipative Endomorphism definition IV.D391 Dislocation as Holonomy Defect definition IV.D392 Glass Threshold K_glass definition IV.D393 EM-Glass definition IV.D394 CheckCrystal Decision Procedure definition IV.D395 CheckGlass Decision Procedure definition IV.D396 Anomaly as Naturality Obstruction definition IV.D397 S-Matrix Decomposition definition IV.D398 Ontic Entity Criterion definition IV.D399 Non-Ontic Entity Classification definition IV.D40 Epstein Zeta Structure definition IV.D400 Boundary Holonomy Algebra definition IV.D401 Cross-Coupling Ledger definition IV.D402 Force as Boundary Mode definition IV.D403 21\,cm Hydrogen Line definition IV.D404 2nd Edition Corrections definition IV.D405 Activation Energy definition IV.D406 Advanced Burning definition IV.D407 Fine-Structure Constant definition IV.D408 alpha-Capture to Oxygen definition IV.D409 alpha-Clustering definition IV.D41 Chowla-Selberg Decomposition definition IV.D410 alpha-Decay definition IV.D411 alpha-Decay definition IV.D412 Antibonding Molecular Orbital definition IV.D413 Antineutrino definition IV.D414 Adenosine Triphosphate (ATP) definition IV.D415 Baryon as Triple Micro-Donut definition IV.D416 Decay definition IV.D417 -Decay definition IV.D418 Bi-Rotation definition IV.D419 Binding Energy Per Nucleon definition IV.D42 Lemniscate Three-Fold definition IV.D420 Bond Dipole Moment definition IV.D421 Bonding Molecular Orbital definition IV.D422 What Book~V Will Provide definition IV.D423 Boson Atlas definition IV.D424 Bounded Sector Extraction definition IV.D425 Breathing Dynamics definition IV.D426 Calibration Tiers definition IV.D427 Catalyst definition IV.D428 Catenation definition IV.D429 CF of the Mass Ratio definition IV.D43 Spectral Distance √3 definition IV.D430 CF of the Master Constant definition IV.D431 CF of the Weinberg Value definition IV.D432 Character Path definition IV.D433 Charged Current Interaction definition IV.D434 -Dominant Sector definition IV.D435 Chlorophyll definition IV.D436 Circulation on T2 definition IV.D437 CKM Matrix definition IV.D438 CKM Matrix definition IV.D439 CNO Cycle definition IV.D44 Triple Holonomy definition IV.D440 Color Holonomy definition IV.D441 Complexity Hierarchy definition IV.D442 Conductor Classification definition IV.D443 Contractive Defect Functional definition IV.D444 Cooper Pair definition IV.D445 Coulomb Binding as Character Overlap definition IV.D446 CR-Function definition IV.D447 Radioactive Decay Law definition IV.D448 Natural Decay Series definition IV.D449 Defect Components definition IV.D45 Holonomy Correction definition IV.D450 Defect Phase Space definition IV.D451 Inner Radiative Correction definition IV.D452 NLO Radiative Correction definition IV.D453 Diophantine Near-Resonance definition IV.D454 DNA Bases definition IV.D455 The Donut Ladder definition IV.D456 Donut Ladder definition IV.D457 Doubly Magic Nuclei definition IV.D458 Nuclear Drip Lines definition IV.D459 Electric Field as Mode definition IV.D46 Mass Ratio Bulk Term definition IV.D460 Electron definition IV.D461 Electron Linkage Principle definition IV.D462 Pauling Electronegativity definition IV.D463 Electromagnetic Gauge Transformation definition IV.D464 Enriched Category definition IV.D465 Enthalpy definition IV.D466 Entropy definition IV.D467 Evaluation Morphism definition IV.D468 Physical Electroweak Bosons definition IV.D469 The Two Electroweak Forces definition IV.D47 Level 0 Formula definition IV.D470 Electroweak Gauge Group definition IV.D471 Electroweak Calibration Tiers definition IV.D472 Book~IV Export Contract definition IV.D473 Fe-56 Voxel-Donut Structure definition IV.D474 Fundamental Fermion Atlas definition IV.D475 Color Flux Tube definition IV.D476 Force Atlas definition IV.D477 Four-Layer Architecture definition IV.D478 Functional Group definition IV.D479 Axial Coupling definition IV.D48 Level 1+ Formula definition IV.D480 -Decay definition IV.D481 Holonomy definition IV.D482 Gibbs Free Energy definition IV.D483 Gluon Color Content definition IV.D484 Gluon Exchange definition IV.D485 Gluon Field Strength definition IV.D486 GW170817 definition IV.D487 Hadron Classification definition IV.D488 Hadron Classification definition IV.D489 Valley of Stability definition IV.D49 CR-Manifold definition IV.D490 Gold, Lead, Uranium definition IV.D491 Heterocyclic Aromatic definition IV.D492 Higgs Decay Channels definition IV.D493 Higgs Measured Properties definition IV.D494 HOMO and LUMO definition IV.D495 Hydrocarbon Classes definition IV.D496 Hydrogen Atom definition IV.D497 Incompressible -Euler definition IV.D498 Instantons on tau3 definition IV.D499 Iron Group Elements definition IV.D50 CR-Structure on τ³ definition IV.D500 Island of Stability definition IV.D501 Lemniscate Junction definition IV.D502 Junction Crossing definition IV.D503 Koide Formula definition IV.D504 LCAO Molecular Orbital definition IV.D505 Lepton Mode Ontology definition IV.D506 Nuclear Magic Numbers definition IV.D507 Magnetic Reynolds Number definition IV.D508 Magnetic Field as Spatial Curvature definition IV.D509 Majorana vs.\ Dirac Neutrinos definition IV.D51 Character Modes definition IV.D510 Mass-5 and Mass-8 Gaps definition IV.D511 Mass as Resistance definition IV.D512 Meson as Double Micro-Donut definition IV.D513 -MHD Coupling definition IV.D514 Micro/Macro Decomposition definition IV.D515 The Microcosm definition IV.D516 T2 Mode Friction definition IV.D517 Molecular Dipole Moment definition IV.D518 Natural Charge definition IV.D519 Natural Transformation definition IV.D52 CR-Address definition IV.D520 Neutral Current Interaction definition IV.D521 Neutrino definition IV.D522 Neutrino Generations definition IV.D523 Neutrino Oscillations definition IV.D524 Neutrino Exponent Strategy: Torus Winding Census --- V.D235 definition IV.D525 NF-Discrete Tower definition IV.D526 No-Free-Parameters Principle definition IV.D527 Noble Gases definition IV.D528 Nuclear Structure definition IV.D529 Nucleosynthesis Origin Classification definition IV.D53 Address Precision (ch16) definition IV.D530 Nucleotide definition IV.D531 -Crossing definition IV.D532 Onion Shell Structure definition IV.D533 Ontological Levels in tau3 definition IV.D534 Peptide Bond definition IV.D535 Periodic Table as Mode Map definition IV.D536 Potential Energy Surface definition IV.D537 Phase Space Integral definition IV.D538 Photon Absorption definition IV.D539 Photon Emission definition IV.D54 CR-Admissible Sublattice definition IV.D540 Physical Character definition IV.D541 Bond definition IV.D542 PMNS Matrix definition IV.D543 PMNS Matrix definition IV.D544 pp-Chain definition IV.D545 Proton definition IV.D546 Proton Derivation Principle definition IV.D547 Pseudo-Dense Coverage definition IV.D548 Fermion Quadrinity definition IV.D549 Six Quark Flavours definition IV.D55 Character on a Space definition IV.D550 Quark Mass Spectrum definition IV.D551 r-Process definition IV.D552 Reynolds Number definition IV.D553 s-Process definition IV.D554 Saturation definition IV.D555 Electron Shell Capacities definition IV.D556 Silicon Burning definition IV.D557 Bond definition IV.D558 Standard Model Electroweak Parameters definition IV.D559 Named Spectral Series definition IV.D56 Character Variety of T² definition IV.D560 Spontaneous Symmetry Breaking definition IV.D561 Standard Table Structure definition IV.D562 Stereoisomers definition IV.D563 Structural Isomers definition IV.D564 -Euler Regime definition IV.D565 -Table Layout definition IV.D566 The tau3 Manifold definition IV.D567 Three Ontological Layers definition IV.D568 Reaction Coordinate and Transition State definition IV.D569 Transition Surface definition IV.D57 Address Precision (ch17) definition IV.D570 Translation Dictionary definition IV.D571 Two Capture Processes definition IV.D572 Vacuum Energy from Defect Functional definition IV.D573 Valence and Sea Quarks definition IV.D574 CKM Element definition IV.D575 Weak Interaction as Rearrangement definition IV.D576 Weinberg Coupling Value definition IV.D577 Tree-Level Deviation definition IV.D578 Mass Ratio definition IV.D579 Yang--Mills on tau3 definition IV.D58 Geometric Charge definition IV.D580 Master Constant definition IV.D581 Microcosm and Macrocosm definition IV.D582 No-Running Principle definition IV.D583 Particle as Boundary Mode definition IV.D584 Readout definition IV.D585 Readout Drift definition IV.D586 Scope Labels definition IV.D587 Self-Coupling Ledger definition IV.D588 The Two-Arc Structure definition IV.D589 Neutrino Exponent Ratio from Lemniscate --- V.D237 definition IV.D59 Sharp and Spread States definition IV.D590 Neutrino CF-Asymmetry Candidate --- V.D236 definition IV.D60 Space of CR-Functions definition IV.D61 Canonical Inner Product definition IV.D62 Holomorphic Hilbert Space definition IV.D63 Physical State Space definition IV.D64 Entanglement definition IV.D65 Holomorphic Vector Field definition IV.D66 Quantum Operator definition IV.D67 Observable definition IV.D68 Address Uncertainty definition IV.D69 τ-Normal Form for Joint Address definition IV.D70 Phase Transport Witness definition IV.D71 Clopen Position Localization definition IV.D72 σ-Equivariant Crossing-Point Mediator definition IV.D73 Canonical Saturation State definition IV.D74 Address Resolution definition IV.D75 Decoherence definition IV.D76 Energy as CR-Tension definition IV.D77 Graph Energy Density definition IV.D78 Mass as Fiber Stiffness definition IV.D79 Frequency as Base Circulation definition IV.D80 Holomorphic Entropy definition IV.D81 Temporal Direction definition IV.D82 Photon Mode definition IV.D83 U(1) Holonomy on T^2 definition IV.D84 Electric Charge definition IV.D85 EM Principal Bundle definition IV.D86 Local Trivialization definition IV.D87 Transition Function definition IV.D88 Section of the EM Bundle definition IV.D89 Gauge Connection definition IV.D90 Covariant Derivative definition IV.D91 Parallel Transport definition IV.D92 Electromagnetic Field Strength definition IV.D93 Aharonov-Bohm Phase definition IV.D94 EM Loop Space definition IV.D95 Sigma-Equivariance definition IV.D96 Wilson Loop definition IV.D97 Non-Abelian Gauge Field definition IV.D98 EM Gauge Bundle definition IV.D99 EM Connection 1-Form definition IV.L01 Wedge Holonomy lemma IV.L02 Holonomy Normalization lemma IV.L03 Photon Phase Quantum lemma IV.L04 Mediator Ratio lemma IV.L05 Sigma_A-Admissibility Selects Chirality lemma IV.L06 Crossing-Point Mode Classification lemma IV.L07 Discrete Hessian Spectrum lemma IV.L10 Positive gap at each stage lemma IV.L11 Vacuum coherence lemma IV.L12 Excitation coherence lemma IV.L8 Winding Preservation lemma IV.L9 Finite-stage spectral problem lemma IV.P01 All Couplings Positive proposition IV.P02 α Numerical Range proposition IV.P03 Power Hierarchy proposition IV.P04 No-Running Principle proposition IV.P05 All Near-Matches in Range proposition IV.P06 √3 Approximation Quality proposition IV.P07 All Links Tau-Effective proposition IV.P08 Integrability Criterion proposition IV.P09 Integrability of τ³ CR-Structure proposition IV.P10 Density Halving proposition IV.P100 Higgs sector satisfies (KH-1)--(KH-3) proposition IV.P101 Strong sector as τ-holonomy sector proposition IV.P102 Strong sector kernel hypotheses proposition IV.P103 Equivalence of defect formulations proposition IV.P104 Refinement coherence proposition IV.P105 Properties of Q_n^s proposition IV.P106 Gap mode coherence proposition IV.P107 Gap positivity at each finite stage proposition IV.P108 Tower monotonicity proposition IV.P109 No primitive mixing proposition IV.P11 Characters on T² proposition IV.P110 The argmin is the π-lift proposition IV.P111 Λ_QCD as readout saturation proposition IV.P112 Asymptotic freedom as spectral tightening proposition IV.P113 Quark electric charges proposition IV.P114 Generation mass ordering proposition IV.P115 Gluon count proposition IV.P116 Gluon self-interaction vertices proposition IV.P117 Structural asymptotic freedom proposition IV.P118 Asymptotic freedom from N_c and N_f proposition IV.P119 No uncountable factorization proposition IV.P12 Automatic Quantization proposition IV.P120 Canonical vacuum uniqueness proposition IV.P121 Quark mass pattern proposition IV.P122 Muon mass exponent proposition IV.P123 Tau lepton mass exponent proposition IV.P124 Neutrino mass scale proposition IV.P125 Beta-decay Q-value proposition IV.P126 Bohr radius from iota_tau proposition IV.P127 Spectral transition as mode-switching proposition IV.P128 Nucleon mass decomposition proposition IV.P129 Nuclear force saturation proposition IV.P13 Charge Quantization from Winding proposition IV.P130 Nuclear shell structure proposition IV.P131 Iron peak from competing sectors proposition IV.P132 Decay channels from sector admissibility proposition IV.P133 Topological integrality of theta proposition IV.P134 Topological charge conservation proposition IV.P135 Existence and uniqueness of limit proposition IV.P136 tau-Euler equation proposition IV.P137 Superfluid vortex quantization proposition IV.P138 Flux quantization proposition IV.P139 Status of Boltzmann constant proposition IV.P14 Energy Duality proposition IV.P140 Finite at every primorial level proposition IV.P141 Quantized circulation proposition IV.P142 Defect tuple as universal order parameter proposition IV.P143 Melting sequence monotone mobility proposition IV.P144 Fiber-level physics is complete proposition IV.P145 UV finiteness proposition IV.P146 Uniqueness of Assignment proposition IV.P147 Subsystem Horizon proposition IV.P148 NNO from the alpha-Orbit proposition IV.P149 Quasi-ergodicity proposition IV.P15 Conjugate Precision Trade-off proposition IV.P150 Four dimensions earned proposition IV.P151 Micro/Macro decomposition proposition IV.P152 Master constant is sigma-fixed proposition IV.P153 Smooth manifold from coherent readouts proposition IV.P154 Temporal Multiplicative Closure proposition IV.P155 Multiplicative Closure proposition IV.P156 Power Hierarchy proposition IV.P157 Second-law inversion proposition IV.P158 Schr"odinger shadow proposition IV.P159 Calibration is structural proposition IV.P16 Algebraic Properties of CR(τ³) proposition IV.P160 Numerical value proposition IV.P161 Numerical value proposition IV.P162 Numerical value proposition IV.P163 Numerical value proposition IV.P164 Numerical value proposition IV.P165 Hierarchy Resolution proposition IV.P166 Neutron Minimality proposition IV.P167 Attained minimum proposition IV.P168 Readout Properties proposition IV.P169 Beta Function as Readout Derivative proposition IV.P17 Inner Product Properties proposition IV.P170 Total Entropy Invariance proposition IV.P171 Breathing spectrum proposition IV.P172 All Links tau-Effective --- IV.P07 proposition IV.P173 AB interference shift proposition IV.P174 Polarity-switching requires balanced polarity proposition IV.P175 Three-Fold Structure of~mathbbL proposition IV.P176 Internal Equations Dimensionless proposition IV.P177 Codomain Is Operational proposition IV.P178 SM-Independent Census proposition IV.P179 E₁ Page Derivation of α-Coefficient proposition IV.P18 Inner Product Uniqueness proposition IV.P180 Exponent-Width Coincidence proposition IV.P181 Mode Interpretation of EW Coefficients proposition IV.P182 Complement Characterization proposition IV.P183 Sign of p-n splitting: QCD > EM proposition IV.P184 NLO Color-Generator Correction 6/5 proposition IV.P185 Three-Generation Winding Closure proposition IV.P186 beta-Decay nu/nu-bar as Helicity Labels (not particle/antiparticle) proposition IV.P187 Quark-Lepton Universality: Exponent Step ≈ −2.7 proposition IV.P188 m_H/m_W Ratio [auto-upgrades with IV.P199] proposition IV.P189 Quark-Lepton Complementarity: θ₁₂+θ_C ≈ π/4 from Fiber-Base Duality proposition IV.P19 Central Theorem Implies Boundary Determination proposition IV.P190 Wolfenstein Parameters: ρ̄=1/(2π) at +974 ppm (τ-effective); A and η̄ open proposition IV.P191 Window RG Period: W₃(4)^k Governs k-th Perturbative Order proposition IV.P195 No Axion Required: Strong CP Solved Structurally by SA-i proposition IV.P196 PMNS Large Mixing from A-Sector Rotation and QLC Complementarity proposition IV.P197 eta_bar from CP Phase Structure: Best tau-Candidate sqrt(5)/(2*pi) proposition IV.P198 CKM Unitarity Triangle Angles from tau-Framework proposition IV.P199 n=7 Structural Uniqueness: Coherence Functional Samples Lobe+Sector proposition IV.P20 Spectral Completeness proposition IV.P200 CP Violation from ω-Period Pentagon: Pentagon-Angle Derivation proposition IV.P201 C.5 Coefficients: 3/16 = N_c/2⁴ and 3/20 = N_c/(4·W₃(4)) proposition IV.P202 15 = 3 × 5 Boundary Mode Decomposition proposition IV.P203 Jarlskog J Full-τ Consistency: J_τ = A_τ²·λ_τ⁶·η̄_τ proposition IV.P204 δ_CP = π + arctan(ι_τ) at +9365 ppm proposition IV.P205 Window Universality for All 7 NNLO Exponents proposition IV.P206 Electroweak Boson Mass Ratio Consistency proposition IV.P207 Higgs-to-W Mass Ratio at +8.8 ppm (n=7 upgrade) proposition IV.P208 Electron g-2 QED Series with α_τ: −3 ppm at NLO proposition IV.P209 g-2 Falsification Criteria for τ³ proposition IV.P21 Generic Entanglement proposition IV.P210 First-Row CKM Unitarity (Cabibbo Anomaly Test) proposition IV.P211 Quark-Lepton Exponent Symmetry proposition IV.P212 PMNS Unitarity Triangle Consistency proposition IV.P213 Lamb Shift Structural Estimate (Proton-Size Contribution) proposition IV.P214 IV.OP5 Status (Wave 35) proposition IV.P215 Intra-Triplet Ratio Exponents proposition IV.P216 Step-Exponent Confinement Correction proposition IV.P217 m_u/m_d NLO Candidate at +29 ppm proposition IV.P218 Confinement-Free Top Quark proposition IV.P219 m_d/m_s Prediction at −1921 ppm proposition IV.P22 Superposition from Linearity proposition IV.P220 HL-LHC and FCC-hh Falsification Window proposition IV.P221 m_c/m_t Exponent from Higgs-Weighted Transition proposition IV.P222 m_u/m_d Isospin Ratio from Winding Algebra at +82 ppm proposition IV.P223 Lobe-Power Hierarchy of Quark Exponents proposition IV.P224 Cancelled-Form Error Budget (77 ppm RSS) proposition IV.P225 c₂ Numerical Bound from W₄(3) proposition IV.P226 Spontaneous Magnetization on T² proposition IV.P227 Domain Wall Energy from T² Winding proposition IV.P228 Magnetic Orders as Defect-Tuple Signatures proposition IV.P229 Ten Regime Instantiations proposition IV.P23 Commutator Equals Lifted Lie Bracket proposition IV.P230 He-4 Minimal Donut Criterion proposition IV.P231 Type-I/II Classification from Defect-Tuple Inequality proposition IV.P232 Dislocation Classification proposition IV.P233 Decidability of Crystal and Glass Classification proposition IV.P234 Two-Channel Framework proposition IV.P235 Self-Coupling Order proposition IV.P236 Holonomy Correction proposition IV.P237 C1 for -NS proposition IV.P238 C2 for -NS proposition IV.P239 C3 for -NS proposition IV.P24 X and P Are Self-Adjoint proposition IV.P240 Fine Structure Constant proposition IV.P241 Physical Interpretations proposition IV.P242 Route A Accuracy proposition IV.P243 Strong Coupling Near-Match proposition IV.P244 Structural Decomposition proposition IV.P245 Gravitational Fine Structure Constant proposition IV.P246 Antineutrino Helicity proposition IV.P248 Automorphism Action proposition IV.P249 Baryon Number as Topological Charge proposition IV.P25 Saturation Equality proposition IV.P250 Baryon Spectrum from T2 proposition IV.P251 BBN Abundances proposition IV.P252 BBN Sensitivity proposition IV.P253 BCS Theory as T2 Mode Condensation proposition IV.P254 Bravais Lattices from T2 proposition IV.P255 Proton--Electron Complementarity proposition IV.P256 Burning Stage Durations proposition IV.P257 C-Sector Contractivity proposition IV.P258 Root Cause of the Catastrophe proposition IV.P259 The Vacuum Catastrophe Is a Category Error proposition IV.P26 Measurement Repeatability proposition IV.P260 The Partial Quotient proposition IV.P261 Structural Reading of the Partial Quotients proposition IV.P262 Structural Reading proposition IV.P263 -Dominance Consequences proposition IV.P264 tau3 Origin of CKM Mixing proposition IV.P265 Constitutive Relations from Defect Tuple proposition IV.P266 Cooper Pairing from T2 Modes proposition IV.P267 Gauge Coupling Hierarchy proposition IV.P268 Properties of CR-Space proposition IV.P269 Critical Exponents proposition IV.P27 Classical Limit proposition IV.P270 Cross-Scale Coherence proposition IV.P271 Derivation Audit proposition IV.P272 Consequences of the Derivation proposition IV.P273 Down-Type Quark Hierarchy proposition IV.P274 Electron Mass proposition IV.P275 Electron Spin proposition IV.P276 No Fundamental EM Field proposition IV.P277 Energy as Tension Integral proposition IV.P278 -Generator Properties proposition IV.P279 Precision Measurements proposition IV.P28 Determinism-Probability Reconciliation proposition IV.P280 Problems with External Laws proposition IV.P281 Extra Cross-Couplings from Multiplicative Closure proposition IV.P282 Fermi Constant from iota_tau proposition IV.P283 Fermi Ingredient Scorecard (8/8) proposition IV.P284 Five Lessons from the Electroweak Arc proposition IV.P285 Frozen-In vs.\ Diffusive Regimes proposition IV.P286 Structural Nature of the Gap proposition IV.P287 Boundary Mode Census proposition IV.P288 Glass Transition proposition IV.P289 Glueball Mass Prediction proposition IV.P29 Energy-Localization Bound proposition IV.P290 Gluon Exchange Conserves Color proposition IV.P291 Gold Production proposition IV.P292 Why He Does Not Bond proposition IV.P293 He-4 Superfluid from T2 proposition IV.P294 Hess's Law proposition IV.P295 SM Hierarchy Problem proposition IV.P296 -Higgs Vacuum proposition IV.P297 Higher Windings Unstable proposition IV.P298 Holonomy and Confinement proposition IV.P299 Holonomy Structure proposition IV.P30 Entropy-Mode-Count Bound proposition IV.P300 Life's Homochirality proposition IV.P301 Hypercharge as Parallel-Resistor Combination proposition IV.P302 Independence from proposition IV.P303 Inflaton Exclusion proposition IV.P304 Topological Inverse-Square Law proposition IV.P305 Ionisation as Surface Transition proposition IV.P306 Numerical Value of proposition IV.P307 Landau Criterion proposition IV.P308 Landau Theory from Defect Free Energy proposition IV.P309 Leading Exponent proposition IV.P31 Reversibility-Irreversibility Resolution proposition IV.P310 Lemniscate Capacity proposition IV.P311 Characters on L proposition IV.P312 EM Length Hierarchy proposition IV.P313 Charged Lepton Hierarchy proposition IV.P314 Light Reactions proposition IV.P315 Magic Numbers proposition IV.P316 Scope of the Mass Gap Result proposition IV.P317 Mass-5 and Mass-8 Gaps proposition IV.P318 Meson Spectrum and the Pion proposition IV.P319 Bond Order from MO Counting proposition IV.P32 No Rest Frame proposition IV.P320 Boundary Mode Census proposition IV.P321 Boson Mass Ratio proposition IV.P322 Neutrino from Decay proposition IV.P323 Three Neutrino Flavours proposition IV.P324 Non-Zero Neutrino Mass proposition IV.P325 Neutron Primacy in the Atlas proposition IV.P326 Error Budget: RSS = 77~ppm proposition IV.P327 The Neutron as Minimal Ontic Defect proposition IV.P328 Neutron Pseudo-Density proposition IV.P329 Neutron Pseudo-Density proposition IV.P33 Photon Spin and Polarization proposition IV.P330 Neutron Quark Composition proposition IV.P331 Neutron Superiority as Calibration Standard proposition IV.P332 Uniqueness of the Neutron Mode proposition IV.P333 No Color Singlet Gluon proposition IV.P334 No-Slip from Topological Anchoring proposition IV.P335 Non-Abelian vs Abelian Field Strength proposition IV.P336 Navier--Stokes as Non-Ideal Defect Flow proposition IV.P337 Structural Parallel proposition IV.P338 First-Order vs.\ Second-Order proposition IV.P339 Neutrino Oscillations from Geometry proposition IV.P34 Charge of Fundamental Modes proposition IV.P340 Periodic Trends from proposition IV.P341 Photon as B-Sector Carrier proposition IV.P342 Why the Photon Remains Massless proposition IV.P343 Photon Momentum proposition IV.P344 Point Vortex System on T2 proposition IV.P345 Two Canonical Projections proposition IV.P346 Proton Mass proposition IV.P347 Proton Spin proposition IV.P348 QFT as Readout Approximation proposition IV.P349 QLC from Fiber-Base Duality proposition IV.P35 Photon as Boundary Character proposition IV.P350 Key Experimental Confirmations proposition IV.P351 Radiative Corrections Close the Gap proposition IV.P352 Real/Virtual Distinction in tau3 proposition IV.P353 Reconnection on T2 proposition IV.P354 Repulsion Hierarchy proposition IV.P355 Reynolds Regimes proposition IV.P356 Aromatic Stability and Ring Fusion proposition IV.P357 Saturation Drives Transitions proposition IV.P358 Saturation Principle for Rung Transitions proposition IV.P359 What Running Is proposition IV.P36 Photon Coupling Strength proposition IV.P360 Saturation Implies Stability proposition IV.P361 Sector Applicability proposition IV.P362 Force Sector Comparison proposition IV.P363 Sector Geography proposition IV.P364 Standard Electroweak Breaking proposition IV.P365 Stationary States proposition IV.P366 Strong Coupling Summary proposition IV.P367 Strong CP and proposition IV.P368 Strong Sector Contribution proposition IV.P369 tau3 vs.\ Standard Symmetry Breaking proposition IV.P37 EM Bundle Topology proposition IV.P370 Symmetry and Polarity proposition IV.P371 tau3 Oscillation Origin proposition IV.P372 Temperature--Equilibrium Coupling proposition IV.P373 Classification of Transitions proposition IV.P374 Complete Transition Map proposition IV.P375 Completeness of the Tuple proposition IV.P376 Turbulence as Defect Cascade proposition IV.P377 Unitary Evolution proposition IV.P378 Up-Type Quark Hierarchy proposition IV.P379 Open Questions proposition IV.P38 Parallel Transport is Gauge-Covariant proposition IV.P380 Structural Smallness of proposition IV.P381 tau3 Interpretation of Valence and Sea proposition IV.P382 Weinberg Angle Interpretation proposition IV.P383 Why tau3 Succeeds proposition IV.P384 Z Properties proposition IV.P385 Zero Free Parameters proposition IV.P386 No-GUT Principle proposition IV.P387 Projection Artifact proposition IV.P388 Running Is Readout Drift proposition IV.P389 Lemniscate Spectral Capacity proposition IV.P39 Explicit Form of F_mu_nu proposition IV.P390 Neutrino Span ; Ratio from --- V.P129 proposition IV.P391 Normal Mass Ordering from -Polarity --- V.P127 proposition IV.P392 Asymmetry --- V.P128 proposition IV.P40 Observable Hierarchy proposition IV.P41 Self-Interaction from Non-Commutativity proposition IV.P42 Non-Abelian Holonomy proposition IV.P43 Categorical Derivation of Gauge Structure proposition IV.P44 Current Conservation from Gauge Invariance proposition IV.P45 Charge Density as Winding-Number Density proposition IV.P46 Current as Defect Transport proposition IV.P47 Photon-Wave Identity proposition IV.P48 Magnetic Force Does No Work proposition IV.P49 Perturbative Expansion in alpha proposition IV.P50 Photon Uniqueness proposition IV.P51 alpha_em versus iota_tau proposition IV.P52 Polarity Assignments proposition IV.P53 Mass of Channel-Switching Defect Bundles proposition IV.P54 Parity in Non-Weak Sectors proposition IV.P55 SU(2)_L from Lemniscate Automorphisms proposition IV.P56 Charged and Neutral Combinations proposition IV.P57 Fermi Constant from Weak Coupling proposition IV.P58 EM-Weak Coupling Hierarchy proposition IV.P59 W as Crossing Process proposition IV.P60 Z Field Composition proposition IV.P61 Measured Rho proposition IV.P62 Z Heavier Than W proposition IV.P63 Beta-Decay Rate from Tau-Units proposition IV.P64 Invisible Width and N_nu proposition IV.P65 Electroweak Coupling Summary proposition IV.P66 Mass Eigenvalue Ratios proposition IV.P67 CP Violation from Sigma-Polarity proposition IV.P68 Coupling Relations proposition IV.P69 Weinberg Angle Residual Analysis proposition IV.P70 No Grand Unification proposition IV.P71 Parity Bridge and Mixing proposition IV.P72 Crossing-Point Degeneracy proposition IV.P73 Finite-Stage Vacuum Existence and Uniqueness proposition IV.P74 Finite-Stage Hessian Properties proposition IV.P75 Higgs Mass Readout proposition IV.P76 Goldstone Modes at Crossing Point proposition IV.P77 Decay Mode Consistency proposition IV.P78 τ-Yukawa Hierarchy proposition IV.P79 Framework Comparison proposition IV.P80 Spectral tightening in the C-sector proposition IV.P81 Finiteness and decidability proposition IV.P82 Loop class inclusion proposition IV.P83 Properties of \Delta_n^s proposition IV.P84 Non-emptiness of $\mathrm{Adm_s{[n proposition IV.P85 Existence and uniqueness at each stage proposition IV.P86 Structure of Fix(s) proposition IV.P87 Properties of the canonical strong lift proposition IV.P88 Ternary Decomposition of the η-Circle proposition IV.P89 Color Quantization proposition IV.P90 Color-Charged Modes Have n \neq 0 \pmod{3 proposition IV.P91 χ₋-Dominance Forces Non-Commutativity proposition IV.P92 Tracelessness from Color-Neutral Vacuum proposition IV.P93 Gluon Self-Interaction proposition IV.P94 Singlet Stability proposition IV.P95 Singlet Classification proposition IV.P96 Linear Confinement Potential proposition IV.P97 Well-definedness proposition IV.P98 Properties of h_n proposition IV.P99 Higgs sector as τ-holonomy sector proposition IV.R01 Holonomy vs Spectral remark IV.R02 Wrong Formula Correction remark IV.R03 Physical Constants Core remark IV.R04 Neutron First Ontic Particle remark IV.R07 Ontological Priority remark IV.R08 G Frontier remark IV.R09 Self-Assessment remark IV.R10 Normalization remark IV.R106 Book III template vs Book IV instantiation remark IV.R107 Topological rigidity remark IV.R108 Yukawa as readout not parameter remark IV.R109 SM parameter count comparison remark IV.R11 √3 Triad remark IV.R110 No BSM particles remark IV.R111 Fourth generation excluded remark IV.R112 Winding number classes remark IV.R113 Honest assessment of quark masses remark IV.R114 Non-integer exponents are physical remark IV.R115 The 45-degree crossing angle remark IV.R116 Why 0.0009% and not exact remark IV.R117 Normal ordering predicted remark IV.R118 Individual neutrino masses remark IV.R119 Scope gradient across mass table remark IV.R12 Charge Conjugation remark IV.R120 One constant one anchor zero parameters remark IV.R121 Neutron as parent of atomic matter remark IV.R122 Structural lifetime estimate remark IV.R123 No classical trajectory remark IV.R124 A testable prediction remark IV.R125 Forbidden transitions remark IV.R126 Lamb shift in tau-framework remark IV.R127 All roads lead through m_e remark IV.R128 Eta-eta prime splitting remark IV.R129 Glueballs and the mass gap remark IV.R130 Mass from nothing remark IV.R131 Isospin splitting from polarity remark IV.R132 Proton lighter but ontologically later remark IV.R133 Deuteron binding in tau-language remark IV.R134 Spin-orbit from omega-sector remark IV.R135 Why He-4 is tightly bound remark IV.R136 Nucleosynthesis forward to Book V remark IV.R137 Alpha-decay as mode cluster ejection remark IV.R138 Neutron stability inside nuclei remark IV.R139 Gamma-decay as mode transition remark IV.R14 Fractional Charges and Confinement remark IV.R140 Madelung rule from T^2 geometry remark IV.R141 Topological not accidental remark IV.R142 Molecules as mode-sharing graphs remark IV.R143 No new parameters for chemistry remark IV.R144 Mode-repulsion geometry remark IV.R145 Homochirality and parity violation remark IV.R146 Structural vs quantitative chemistry remark IV.R147 The donut ladder remark IV.R148 Comparison with orthodox physics remark IV.R149 Parameter count remark IV.R15 Quasi-Ergodic Coverage remark IV.R150 Ontic entities list remark IV.R151 Non-ontic entities list remark IV.R152 Where the ontological line falls remark IV.R153 Dictionary limits remark IV.R154 Temperature is not fundamental remark IV.R155 Euler budget recap remark IV.R156 Why interaction corrections matter remark IV.R157 Why clopen matters remark IV.R158 Finitely specified at each depth remark IV.R159 Ultrametric locality remark IV.R160 No continuum limit needed remark IV.R161 Navier-Stokes regularity remark IV.R162 Why MHD lives in B-sector remark IV.R163 Helium-4 and beyond remark IV.R164 Cooper pairing is topological remark IV.R165 Transitions as inequality crossings remark IV.R166 Comparison with orthodox thermodynamics remark IV.R167 Parallel with hbar and c remark IV.R168 No negative temperatures remark IV.R169 Universality and critical exponents remark IV.R170 Kelvin theorem as budget law remark IV.R171 Viscosity coefficient eta_tau remark IV.R172 Honesty about the Clay problem remark IV.R173 BCS gap as spectral gap remark IV.R174 Crystal symmetry from torus subgroups remark IV.R175 Glass transition not a true phase transition remark IV.R176 Penrose tilings on the torus remark IV.R177 Universality from sector structure remark IV.R178 Why the separation is clean remark IV.R179 The base is not the rest remark IV.R180 Noether theorem as corollary remark IV.R181 Why not a larger gauge group remark IV.R182 Individual C P CP violations remark IV.R183 Vacuum catastrophe resolved remark IV.R184 Why C1 is conjectural remark IV.R185 Comparison with ch15 ledger remark IV.R186 The formalization frontier remark IV.R187 Open vs wrong problems remark IV.R188 Why sphaleron is deferred remark IV.R189 One-way flow of export contracts remark IV.R190 Why neutron not electron as anchor remark IV.R21 Energy-Entropy Complementarity remark IV.R211 One constant to rule them all remark IV.R212 Lean formalization remark IV.R213 CR-structure remark IV.R216 The coupling connects them remark IV.R22 No Past Hypothesis Needed remark IV.R221 Why all lifts are rational remark IV.R222 Why 2 + 2 gives 1+ remark IV.R225 Physical meaning remark IV.R226 Power structure remark IV.R230 Lean formalization remark IV.R233 Why gravity is weak remark IV.R236 Lean formalization remark IV.R237 Lean formalization remark IV.R242 Part~II in perspective remark IV.R243 The ladder is not a hierarchy of importance remark IV.R245 The honest timing remark IV.R247 Origin of the No Knobs Principle remark IV.R248 Lean verification remark IV.R25 Gauge Invariance Is Geometric Not Dynamic remark IV.R250 Lean verification remark IV.R252 Lean formalization remark IV.R254 No landscape remark IV.R255 The meaning of 0.6% remark IV.R256 Lean verification remark IV.R257 Origin of the formula remark IV.R258 The three holonomy circles remark IV.R260 The value of being wrong remark IV.R262 What the paper got right remark IV.R263 Not a numerical fit remark IV.R264 The Planck mass in tau-physics remark IV.R265 One input, not zero remark IV.R266 Lean formalization remark IV.R267 Falsifiability remark IV.R268 Why 0.07% and not exact remark IV.R269 Consistency check remark IV.R27 sqrt(3) Triad remark IV.R270 The tier boundary is sharp remark IV.R274 The pi-corrected distance remark IV.R275 Counting parameters remark IV.R276 R~formula independence remark IV.R277 The sqrt3 remark IV.R279 Asymptotic freedom revisited remark IV.R28 CP Violation and the Crossing Phase remark IV.R280 Scheme dependence resolved remark IV.R282 Lean formalization remark IV.R283 Lean verification remark IV.R285 Honest deviations remark IV.R286 5, 4, 3, 2, 1 remark IV.R287 Honest fraction remark IV.R29 Hierarchy Problem in Category Tau remark IV.R292 CR-type (1,1) and three-dimensionality remark IV.R293 Connection to Book~II remark IV.R294 Spin-frac1 remark IV.R295 Bosons and fermions remark IV.R301 Holography in physics and in tau remark IV.R302 Connection to Book~III: the spectral philosophy remark IV.R303 Bell's theorem and the tau-substrate remark IV.R305 Comparison with geometric quantization remark IV.R308 Connection to Book~III: remark IV.R31 Weinberg Angle Gap Assessment remark IV.R310 Connection to Book~II spectral gap remark IV.R312 Born rule derived, not postulated remark IV.R314 Why standard QM works remark IV.R316 The mechanism in plain language remark IV.R318 Boundary witness budget remark IV.R320 Physical interpretation remark IV.R323 Decoherence is not collapse remark IV.R325 Generator roles remark IV.R326 One Hamiltonian rules all remark IV.R328 Why |barpartial remark IV.R330 Massless particles remark IV.R332 Local redistribution remark IV.R333 Comparison with Boltzmann remark IV.R336 Ten orders of magnitude from one constant remark IV.R337 Toroidal dominance remark IV.R338 Structural origin of the exponent 7 remark IV.R339 Physical interpretation of the correction remark IV.R34 Coherence Fixing vs Mass Generation remark IV.R347 Masslessness is geometric, not postulated remark IV.R348 Experimental limits on photon mass remark IV.R349 The Planck--Einstein relation in tau^3 remark IV.R35 Structural Resolution of Hierarchy remark IV.R350 Massless spin-1: two, not three remark IV.R351 Double slit in tau^3 remark IV.R359 Physical content lives in the curvature remark IV.R36 Where τ and SM Could Diverge remark IV.R360 Experimental confirmation remark IV.R362 The AB effect as an observable remark IV.R365 Why alpha_mathrmem remark IV.R366 Measuring alpha_mathrmem remark IV.R367 No external input remark IV.R368 Why (1,0) winding remark IV.R369 Compactness is essential remark IV.R37 Explicit Overlap Integrals remark IV.R370 The iota_tau^2 cancellation remark IV.R371 The meaning of 0.6% remark IV.R372 Lean verification remark IV.R373 pi^3 approx 31: not a Mersenne prime remark IV.R379 Comparison with other theories remark IV.R38 Precision Gap Lesson remark IV.R380 Physical consequence remark IV.R382 Mass hierarchy: M_Z > M_W remark IV.R383 Definition versus theorem remark IV.R386 Layer Discipline remark IV.R387 OQ.11 Resolution Status remark IV.R388 OQ-A1 Status: RESOLVED remark IV.R389 Scale Consistency remark IV.R39 √3 as Category τ Signature remark IV.R390 OQ-A3 Resolution Status remark IV.R391 OQ-B2 Partial Resolution Status remark IV.R392 OQ-B2 Resolved remark IV.R393 NNLO Precision Summary remark IV.R394 Comparison to Cottingham decomposition remark IV.R395 Normal Ordering from sigma-Structure remark IV.R396 Quark-Lepton Universality Conjecture remark IV.R397 0vbb Prediction: must exist; rate proportional to ||^2 remark IV.R398 NLO Correction Factor for m_μ/m_e — Open Structural Identification (OQ-C5a) remark IV.R399 Open: omega Self-Energy Correction and Coefficient-6 Identification remark IV.R40 Honesty About Precision remark IV.R400 PMNS CP Phase δ_CP from σ-Breaking; New OQ-C6/C7 remark IV.R401 Cross-Check: 1/21 in p-n Mass Difference = 1/(4·W₃(4)+1); n=7 Higgs at +8 ppm remark IV.R405 Comparison to Peccei-Quinn: SA-i as τ-Native PQ Mechanism remark IV.R406 CP Phase delta_CP from A-Sector Rotation: Connects IV.T152 and V.T174 remark IV.R407 OQ-CKM1 Status after Sprint 5C: rho_bar Derived, A Improved, eta_bar Open remark IV.R408 m_mu/m_e Sub-100 ppm NNLO Status and Higgs n Comparison Table remark IV.R409 OQ-CKM1 Status after Sprint 6B: All Four Wolfenstein τ-Effective remark IV.R410 m_μ NNLO + p-n NLO Precision Status after Sprint 6D remark IV.R411 IV.OP3 Status: SOLVED (τ-effective) after Sprint 7A remark IV.R413 OQ-C7 Status: PARTIAL-IMPROVED after Sprint 7C remark IV.R415 CDF–ATLAS W Mass Tension and τ³ Prediction remark IV.R416 On-Shell vs MS-bar Weinberg Angle Scheme Distinction remark IV.R417 Structural vs Perturbative Content in g-2 remark IV.R418 Muon g-2 Anomaly Resolution Requires Sub-ppm α remark IV.R419 Jarlskog Invariant from Wolfenstein at −8750 ppm remark IV.R420 Physical Meaning of g_A = κ_D²/ι_τ remark IV.R421 Light Quark Mass Ratio m_u/m_d Candidates remark IV.R422 IV.OP5 Status: Quark Mass Problem (Wave 32) remark IV.R423 PMNS Angle Summary (All Three Predicted) remark IV.R424 Precision Atomic Physics Assessment (Wave 34) remark IV.R425 Why σ-Equivariance Fails for Current Masses remark IV.R426 Koide Restoration Landscape remark IV.R427 m_u/m_d Status Assessment remark IV.R428 Chowla-Selberg Integrality and Confinement remark IV.R429 m_u/m_d NLO Structural Analysis remark IV.R430 IV.OP5 Status Update (Wave 37) remark IV.R431 Exponent Universality: All Ratios from W remark IV.R432 Up Quark Status Update (Wave 44) remark IV.R433 CKM Precision Assessment (Wave 44) remark IV.R434 Down-Type Sector Exponent Completeness remark IV.R435 IV.OP9 Status: SOLVED (7/7 Exponents Derived) remark IV.R436 IV.OP5 Status Update (Wave 45) remark IV.R437 Precision Atomic Physics Assessment Update (Wave 46) remark IV.R438 Nuclear Sector Assessment (Wave 46) remark IV.R439 OQ-B4 Status: PARTIAL remark IV.R440 α Precision Assessment remark IV.R441 High-ppm Audit Summary (Wave 49) remark IV.R442 NFL-Depth Corollary corollary IV.R443 Electric vs Magnetic Duality remark IV.R444 Upgrade of BCS to Corollary remark IV.R445 Feynman Diagram Composition Rules remark IV.R446 Vertex Types as Gauge-Equivariant Composition remark IV.R447 Optical Theorem corollary IV.R448 τ as the ℕ of Holomorphy remark IV.R449 One Torus, Two Modes remark IV.R450 Helium as Boson remark IV.R451 Ch48 Hund Example remark IV.R452 Nuclear Magic Numbers remark IV.R453 Noble Gas Closures remark IV.R454 T2 Interpretation remark IV.R455 Delocalisation Dominance remark IV.R456 Bond as Graph Transformation remark IV.R457 Carbon Versatility remark IV.R458 Frontier Orbitals in tau3 Language remark IV.R459 Fukui's Contribution remark IV.R46 The (1 + ι_τ) contrast remark IV.R460 HF and CO remark IV.R461 Paramagnetism of remark IV.R462 Variational Principle remark IV.R463 mode tension remark IV.R464 mode complementarity remark IV.R465 Quantum Coherence in Light Harvesting remark IV.R466 Ch55 No New Forces remark IV.R467 The Governing Quotation remark IV.R468 Economy remark IV.R469 The Approximation remark IV.R470 The Approximation remark IV.R471 Lattice encoding remark IV.R472 Comparison with alpha remark IV.R473 tau3 View of Aromaticity remark IV.R474 tau3 Interpretation of Catalysis remark IV.R475 From iota_tau to remark IV.R476 The CF as NLO Dictionary remark IV.R477 tau3 Interpretation of Chirality remark IV.R478 Classical 2D regularity remark IV.R479 DNA Bases as Templates remark IV.R480 Euler vs.\ Navier--Stokes: The Algebraic Divide remark IV.R481 Preview: The Fermi Formula remark IV.R482 Functional Groups as Mode Decorations remark IV.R483 tau3 Thermodynamic Identity remark IV.R484 78 Improvement from Cancellation remark IV.R485 The Residual remark IV.R486 Ontological Levels remark IV.R487 tau3 View of the PES remark IV.R488 Quasicrystal and Ferromagnet remark IV.R489 Readout Functor and Tick Units remark IV.R490 On-Shell vs.\ Schemes remark IV.R491 Comparison with Standard Model remark IV.R492 : Genuinely Open Frontier remark IV.R493 Why Three Spatial Dimensions remark IV.R494 Forward Reference: The Master Constant remark IV.R495 Structural Realism remark IV.R53 Why three and not two remark IV.R61 Electric charge analogy remark IV.R69 The role of NF Discreteness remark IV.R73 The Millennium Problem, revisited remark IV.R74 The simplest non-trivial case remark IV.R84 Why π-typed, not η-typed remark IV.R92 Fractional charges are not inputs remark IV.R99 The worst prediction remark IV.T01 Temporal Complement theorem IV.T02 Temporal Multiplicative Closure theorem IV.T03 Sector Lifts σ-Equivariant theorem IV.T04 Euler Budget Conservation theorem IV.T05 Parameter Count theorem IV.T06 τ-Collapse (5→1) theorem IV.T07 Maxwell Relation theorem IV.T08 Coulomb-Permittivity theorem IV.T09 Ledger Count theorem IV.T10 Leading Exponent theorem IV.T100 No-Running Principle theorem IV.T101 Generator Adequacy theorem IV.T102 tau-Heisenberg inequality theorem IV.T103 Euler budget conservation theorem IV.T104 Temporal Complement theorem IV.T105 Temporal Multiplicative Closure theorem IV.T106 Power Hierarchy theorem IV.T107 Holonomy Fine-Structure Formula theorem IV.T108 tau-Collapse: Five to One theorem IV.T109 Level~0 mass ratio formula theorem IV.T11 Three-Fold Distance Squared theorem IV.T110 Level~1+ mass ratio formula theorem IV.T111 Parameter Count theorem IV.T112 sigma-Fixed Planck Character theorem IV.T113 Readout Landscape Theorem theorem IV.T114 Leading exponent -7 theorem IV.T115 Three-fold distance squared theorem IV.T116 Correction Smallness --- IV.T12 theorem IV.T117 Derivation Chain Complete --- IV.T15 theorem IV.T118 Bulk Overshoots --- IV.T13 theorem IV.T119 Level 0 Range --- IV.T14 theorem IV.T12 Correction Smallness theorem IV.T120 Charge quantization theorem IV.T121 Gauge covariance of D_mu theorem IV.T122 Parity Bridge --- recalled from III.T07 theorem IV.T123 Chirality Selection theorem IV.T124 The sqrt3 theorem IV.T125 Tick-Sector Bijection theorem IV.T126 Tick Exhaustion theorem IV.T127 Mass Ratio as Internal Identity theorem IV.T128 Readout Preserves Identities theorem IV.T129 Single-Anchor Sufficiency theorem IV.T13 Bulk Overshoots theorem IV.T130 Structural–Physics Census Equivalence theorem IV.T133 EM Tensor Density Theorem theorem IV.T134 Window Algebra Origin theorem IV.T135 Fermi Form w-Independence theorem IV.T136 EW Partition Theorem theorem IV.T137 EW Density = 5/7 theorem IV.T138 EW–CF Bridge theorem IV.T139 NLO from EW Projection theorem IV.T14 Level 0 Range theorem IV.T140 Window Universality — W₃(4)=5 theorem IV.T141 Proton-Neutron Mass Difference — Tree Level [SUPERSEDED] theorem IV.T142 Proton-Neutron Mass Difference — Two-Sector theorem IV.T143 Koide Q=2/3 from sigma-Symmetry theorem IV.T144 m_mu/m_e Leading-Order Formula theorem IV.T145 Uniqueness: C_tau = sigma (lobe-swap uniquely determines C) theorem IV.T146 Majorana Theorem: zero-U(1) modes are Majorana theorem IV.T147 Three-Generation Closure: Spectral Gap on T² theorem IV.T148 m_μ/m_e Best NLO Formula: ι_τ⁻⁴·⁹⁶ at +307 ppm theorem IV.T149 Lepton σ-Matrix Exponents from PDG Back-Solve theorem IV.T15 Derivation Chain Complete theorem IV.T150 Factor-4 from Non-omega Generator Count theorem IV.T151 Higgs Mass Sub-1000 ppm Formula via Window Universality theorem IV.T152 Cabibbo Angle: sin(θ_C) = ι_τ·(1−ι_τ) at −2327 ppm theorem IV.T153 PMNS Large Mixing Requires A-Sector Flavor Rotation Beyond σ-Matrix theorem IV.T154 GIM-Analog Cancelation from σ-Equivariance theorem IV.T155 Higgs Bonus Coefficient n=6 [SUPERSEDED by IV.T166 n=7] theorem IV.T156 m_μ/m_e NNLO Correction: δ = 1/W₃(4)² = 1/25 theorem IV.T16 CR Parity Constraint theorem IV.T160 θ_QCD = 0 from C-Sector SA-i Admissibility theorem IV.T161 Neutron EDM = 0 Exactly from τ-Framework theorem IV.T162 sin(theta_23) = 1/(1+iota_tau): Atmospheric Mixing from A-Sector theorem IV.T163 QLC-Exact: sin(theta_12) = (sqrt(1-lambda_C^2) - lambda_C)/sqrt(2) theorem IV.T164 rho_bar = 1/(2*pi): omega-Period Structural Derivation (tau-effective) theorem IV.T165 A = 1 - (3/2)*iota_tau^2 at -887 ppm from PDG 0.826 (tau-effective) theorem IV.T166 Higgs n=7 Formula at +8.0 ppm from PDG 125.20 GeV (tau-effective) theorem IV.T167 Jarlskog Invariant J from τ-Parameters theorem IV.T168 η̄ Best τ-Candidate: ι_τ^(−1/4)·κ_D^(5/4)/√5 at −2285 ppm theorem IV.T169 m_μ/m_e NNLO: k=23/3 Correction at +43 ppm (τ-effective) theorem IV.T17 Emergence of Spin-1/2 theorem IV.T170 p-n Mass NLO: C.5 Two-Sector Formula Structural Derivation theorem IV.T171 Fourth Generation Excluded (Topological) theorem IV.T172 Generation Mass Hierarchy from Eigenvalue Ordering theorem IV.T173 η̄ Exponent Derivation: All from |lobes|=2, |gen|=5 (τ-effective) theorem IV.T174 θ₂₃ NLO via Window Algebra at +8604 ppm theorem IV.T175 θ₁₂ from QLC + Higgs NLO at +3106 ppm theorem IV.T176 m_μ/m_e NNLO: k=15/2 at −8.2 ppm (τ-effective, best) theorem IV.T177 Precision W Boson Mass from Window Universality at −0.5 ppm theorem IV.T178 Z Boson Mass from Window Universality at +1039 ppm theorem IV.T179 Schwinger Term from L-Holonomy at −3 ppm (NLO) theorem IV.T18 Hilbert Space Properties theorem IV.T180 Muon g−2 from τ³ at +8.8 ppm theorem IV.T181 CKM Matrix Elements from Wolfenstein Parameters theorem IV.T182 Axial Coupling g_A from CF Window Algebra at 5.5 ppm theorem IV.T183 Top Quark Mass from Fiber Amplitude at +7200 ppm theorem IV.T184 PMNS θ₁₃ from Cubic Crossing Amplitude at +5000 ppm theorem IV.T185 Proton Charge Radius from Compton Scale at +440 ppm theorem IV.T186 Proton g-Factor from Quark Winding at +2000 ppm theorem IV.T187 Koide Deviation for Quark Triplets theorem IV.T188 Koide-Optimal Confinement Shift theorem IV.T189 m_t/m_b Ratio from ι_τ at +99 ppm theorem IV.T19 Orthonormal Basis theorem IV.T190 Corrected Top Mass at −1449 ppm theorem IV.T191 Charm Mass from τ-Chain at +1150 ppm theorem IV.T192 Strange Mass from τ-Chain at +1559 ppm theorem IV.T193 Bottom Mass from τ-Chain at −1351 ppm theorem IV.T194 τ-Chain Higgs Self-Coupling at +16 ppm theorem IV.T195 Jarlskog Invariant from τ-CKM at −35000 ppm theorem IV.T196 m_t/m_b Exponent from Winding Algebra theorem IV.T197 Up Quark Direct Mass from Exponent Duality at +395 ppm theorem IV.T198 Jarlskog NLO at +2624 ppm theorem IV.T199 m_s/m_b Exponent from Confinement-Weighted Mode Count theorem IV.T20 Topological Quantization theorem IV.T200 m_d/m_s Exponent from Winding Phase Space theorem IV.T201 m_u/m_d Isospin Exponent from Higgs-Mediated Splitting theorem IV.T202 Proton Charge Radius NLO at +12 ppm theorem IV.T203 Neutron Lifetime Precision Prediction theorem IV.T204 Window Depth–Loop Order Correspondence theorem IV.T205 α Precision Barrier at 9.8 ppm theorem IV.T206 θ₂₃ NNLO from Holonomy-at-Window-Squared theorem IV.T207 δ_CP NLO from Fiber Correction theorem IV.T208 No Magnetic Monopoles on T² theorem IV.T209 Curie Transition as T² Symmetry Breaking theorem IV.T21 Canonical Commutation Relation theorem IV.T210 NFL-Boundary Theorem theorem IV.T211 Decidable Phase Classification Meta-Theorem theorem IV.T212 Cooper Pairing from EW-Mixed Normalization theorem IV.T213 ABJ Chiral Anomaly theorem IV.T214 Anomaly Cancellation as Naturality Requirement theorem IV.T215 Unitarity from Dagger Structure theorem IV.T216 Complete Ontic Classification theorem IV.T217 Five-Fold Initiality of tau3 theorem IV.T218 Electron Mass theorem IV.T219 Generator--Sector Correspondence theorem IV.T22 Phase Transport Monotonicity theorem IV.T220 A-Sector Rotation Mechanism theorem IV.T221 A--B Sector Cooperation theorem IV.T222 Alfv\'en Waves on T2 theorem IV.T223 Three-Route Convergence theorem IV.T224 alpha Derivation theorem IV.T225 alpha as Holonomy Mediator theorem IV.T226 Spectral Formula for alpha theorem IV.T227 Tower Formula for alpha theorem IV.T228 Tower Formula Accuracy theorem IV.T229 Arrhenius Equation theorem IV.T23 No-Joint-Minimum Theorem theorem IV.T230 Aufbau Principle from T2 Breathing Modes theorem IV.T231 Aufbau Principle theorem IV.T232 Energy Budget theorem IV.T233 Decay as Junction Passage theorem IV.T234 Bianchi Identity --- Homogeneous Maxwell Equations theorem IV.T235 Binding Energy Curve Shape theorem IV.T236 Bi-Rotation Energy Identity theorem IV.T237 Bloch's Theorem on T2 theorem IV.T238 Bohr Radius theorem IV.T239 T2 Holonomy Inner Product Derivation of theorem IV.T24 Planck Character Uniqueness theorem IV.T240 Calibration Cascade theorem IV.T241 Cascade Closure theorem IV.T242 CF Compression theorem IV.T243 Charge-Holonomy Identification theorem IV.T244 Chemistry from Character Algebra theorem IV.T245 CKM from Generation Overlap theorem IV.T246 Compositeness Signatures theorem IV.T247 Conservation as Naturality theorem IV.T248 Constants as Structural Outputs theorem IV.T249 Core-Collapse Supernova (Type II) theorem IV.T25 Heisenberg Uncertainty (Position-Momentum) theorem IV.T250 Coulomb Emergence theorem IV.T251 Universality of the Defect Functional theorem IV.T252 The Derivation Chain theorem IV.T253 Diagrams Are Morphisms theorem IV.T254 Dual Magic Number Structure theorem IV.T255 Ehrenfest Classification from theorem IV.T256 Eight Gluon Types theorem IV.T257 Electron Charge from Conservation theorem IV.T258 EM Calibration Cascade theorem IV.T259 Wave Equation from -Maxwell theorem IV.T26 Heisenberg Uncertainty (Time-Energy) theorem IV.T260 Electroweak Coupling Derivations theorem IV.T261 Electroweak Cross-Coupling theorem IV.T262 Electroweak Precision Agreement theorem IV.T263 tau3 Electroweak Scale theorem IV.T264 Zero Free Parameters theorem IV.T265 Existence of Stable Modes theorem IV.T266 Export Contracts theorem IV.T267 Fe-56 Half-Filling theorem IV.T268 Iron-56 Maximum Stability theorem IV.T269 Fermi Constant from W Mass theorem IV.T27 Born Rule theorem IV.T270 Fermion Count from tau3 theorem IV.T271 Finite Structural Vacuum Energy theorem IV.T272 Force-Particle Duality theorem IV.T273 Address-Obstruction Theorem (Formal Heisenberg) theorem IV.T274 The Four Principles of Chemistry theorem IV.T275 Fractional Quark Charges theorem IV.T276 Frontier Orbital Reactivity theorem IV.T277 Fusion--Fission Crossover theorem IV.T278 Galactic Chemical Evolution theorem IV.T279 Gamow Tunneling theorem IV.T28 Schrödinger Equation theorem IV.T280 Gauge Invariance from Lemniscate Structure theorem IV.T281 Topological Generation Bound theorem IV.T282 Generations from Lemniscate Sectors theorem IV.T283 Generation Mass Scaling theorem IV.T284 Geometric Origin of Spin-Orbit theorem IV.T285 Gibbs--Equilibrium Relation theorem IV.T286 Gluon Mass and Spin theorem IV.T287 Gluon Self-Interaction theorem IV.T288 Emergence of theorem IV.T289 Helicity from tau1 Orientation theorem IV.T29 Energy Duality theorem IV.T290 The Hermetic Principle theorem IV.T291 tau3 Hierarchy Resolution theorem IV.T292 Higgs as Collective Excitation theorem IV.T293 Holonomy Correction theorem IV.T294 Hoyle Resonance theorem IV.T295 Hybrid Orbital Construction theorem IV.T296 Hydrogen Ground State theorem IV.T297 Hydrogen Ground State theorem IV.T298 Hyperfine Splitting of Hydrogen theorem IV.T299 The Internal Law Thesis theorem IV.T30 Energy Conservation theorem IV.T300 Island of Stability theorem IV.T301 Koide Relation from L theorem IV.T302 Lepton Mass Scaling theorem IV.T303 Magic Number Stability Enhancement theorem IV.T304 tau3 Majorana Prediction theorem IV.T305 tau1 Self-Duality: Neutrinos Are Majorana theorem IV.T306 tau3 Mass Fundamentality theorem IV.T307 Neutron-to-Electron Mass Ratio theorem IV.T308 Meissner Effect from T2 Topology theorem IV.T309 Mode Classification theorem IV.T31 Second Law of Thermodynamics theorem IV.T310 Neutrino Charge Neutrality theorem IV.T311 Neutrino Conservation Laws theorem IV.T312 Neutrino Propagation Direction theorem IV.T313 Neutrino Helicity theorem IV.T314 Neutrino Lightness from tau1 Dimensionality theorem IV.T315 Neutrino Mass Suppression theorem IV.T316 Neutrino Mass Bound theorem IV.T317 Exact Cancellation of and theorem IV.T318 Cancelled Form of the Neutron Lifetime theorem IV.T319 Neutron Lifetime as Diophantine Invariant theorem IV.T32 Structural Arrow of Time theorem IV.T320 Neutron Lifetime from iota_tau theorem IV.T321 Neutron Ontological Priority Confirmed theorem IV.T322 NLO Weinberg Angle theorem IV.T323 Absence of External Lawgiver theorem IV.T324 No Fine-Tuning in Category~ theorem IV.T325 Navier--Stokes Regularity from the -Gap theorem IV.T326 Nucleon Mass from Binding Energy theorem IV.T327 Ontic Sequence theorem IV.T328 Oscillation Mechanism theorem IV.T329 Physical Boson Emergence theorem IV.T33 Photon Masslessness theorem IV.T330 Holonomy Correction theorem IV.T331 Positive Regularity (III.T25) theorem IV.T332 Predictive Record of Book~IV theorem IV.T333 Proton Charge from Conservation theorem IV.T334 tau3 as Pure Process theorem IV.T335 Quark--Lepton Complementarity theorem IV.T336 Quadrinity Unity theorem IV.T337 Regime Classification theorem IV.T338 Running as Scale-Dependent Holonomy theorem IV.T339 Rutherford Scattering Formula theorem IV.T34 Photon Propagation Speed theorem IV.T340 Rydberg Formula theorem IV.T341 Scale-Independent Classification theorem IV.T342 Sector Coupling Formulas theorem IV.T343 Sector Exhaustion theorem IV.T344 Sector Force Formula theorem IV.T345 The Self-Describing Universe theorem IV.T346 tau3 Self-Enrichment theorem IV.T347 Shell Capacity theorem IV.T348 Featureless to Harmonic theorem IV.T349 SU(2) from Lemniscate Topology theorem IV.T35 Dissolution of Wave-Particle Duality theorem IV.T350 Symmetries as Endofunctors theorem IV.T351 -Bernoulli theorem IV.T352 -Euler Equation theorem IV.T353 -Gap Meta-Theorem theorem IV.T354 -Kelvin Theorem theorem IV.T355 -Maxwell Inhomogeneous Equations theorem IV.T356 -MHD System theorem IV.T357 -Navier--Stokes Equation theorem IV.T358 -NS Regularity on T2 theorem IV.T359 T2 Saturation in Cosmic Evolution theorem IV.T36 Charge Conservation theorem IV.T360 T2 Gap Explanation theorem IV.T361 T2 Saturation Optimum theorem IV.T362 tau3 Clustering Principle theorem IV.T363 tau3 Decay Classification theorem IV.T364 tau3 Magic Number Origin theorem IV.T365 tau3 Mass Origin theorem IV.T366 tau3 Neutron Capture theorem IV.T367 Thermonuclear Supernova (Type Ia) theorem IV.T368 Three Colors from Topology theorem IV.T369 Three Generations from theorem IV.T37 Gauge Invariance Kernel Theorem theorem IV.T370 Three Generations from tau3 Topology theorem IV.T371 Three Generations from Topology theorem IV.T372 Topological Charge Quantization theorem IV.T373 Translation Completeness theorem IV.T374 Unchanged Core theorem IV.T375 Unified Magic Number Origin theorem IV.T376 Vortex Quantization on T2 theorem IV.T377 VSEPR Geometry Classification theorem IV.T378 Weak Interaction Violations theorem IV.T379 tau3 Weinberg Angle Derivation theorem IV.T38 Gauge Invariance of F_mu_nu theorem IV.T380 Level 0 Mass Ratio theorem IV.T381 Level 1+ Mass Ratio theorem IV.T382 Shared Ontological Layer theorem IV.T383 Majorana CP Phases from --- V.T176 theorem IV.T384 from Lemniscate Counting --- V.T178 theorem IV.T385 One-Parameter Neutrino Family --- V.T173 theorem IV.T386 Best-Integer Approximation --- V.D343 theorem IV.T387 PMNS: Shared Eigenvector Structure --- V.T174 theorem IV.T388 Grid-Optimal -Exponents at --- V.T175 theorem IV.T39 tau-AB Kernel Theorem theorem IV.T40 AB Phase Quantization theorem IV.T41 Ambrose-Singer Reconstruction theorem IV.T42 Homogeneous Maxwell Equations theorem IV.T43 Inhomogeneous Maxwell Equations theorem IV.T44 Complete tau-Maxwell System theorem IV.T45 Source-Defect Correspondence theorem IV.T46 Coulomb's Law from tau-Maxwell theorem IV.T47 EM Wave Equation theorem IV.T48 Non-Trivial Strong Vacuum theorem IV.T49 Holonomy Formula for alpha_em theorem IV.T50 No-Running Principle for alpha_em theorem IV.T51 Parity Violation in the A-Sector theorem IV.T52 Weak Gauge Group theorem IV.T53 Weak Coupling from Sector Parameters theorem IV.T54 W Mass from Coherence Fixing theorem IV.T55 Z Mass from Coherence Fixing theorem IV.T56 Tree-Level Rho = 1 theorem IV.T57 Low-Energy Limit of W Exchange (Fermi Limit) theorem IV.T58 Spectral Gap Exponent theorem IV.T59 Neutrino Interaction Channels theorem IV.T60 Neutral Boson Mixing theorem IV.T61 Weinberg Angle Prediction theorem IV.T62 Mixing Uniqueness Theorem theorem IV.T63 Vacuum Construction theorem IV.T64 Higgs Scalarity Theorem theorem IV.T65 No Hierarchy Problem in Category τ theorem IV.T66 Electroweak Prediction Table theorem IV.T67 Zero Free Parameters in Electroweak Sector theorem IV.T68 Truncation coherence for Γ*_s theorem IV.T69 SU(3) Gauge Algebra theorem IV.T70 Color Number Theorem theorem IV.T71 Confinement Theorem theorem IV.T72 Proton Stability theorem IV.T73 τ-Gap Meta-Theorem (III.T26) theorem IV.T74 Profinite spectral preservation theorem IV.T75 τ-Yang--Mills Mass Gap Theorem theorem IV.T76 Uniqueness of the strong coupling theorem IV.T77 No ontic running --- strong sector theorem IV.T78 No vacuum catastrophe in τ theorem IV.T79 Tail stabilization of vacuum energy theorem IV.T80 Exactly four primitive forces (physical reading) theorem IV.T81 Exactly one derived sector theorem IV.T82 Generator adequacy and minimality theorem IV.T83 Exactly three generations theorem IV.T84 Koide relation Q=2/3 theorem IV.T85 Hydrogen energy levels theorem IV.T86 Rydberg prediction at 0.025 ppm theorem IV.T87 Fine structure from holonomy corrections theorem IV.T88 Period length sequence theorem IV.T89 Tower compatibility theorem IV.T90 Sector additivity theorem IV.T91 Second law via defect functional theorem IV.T92 Phase transition as regime crossing theorem IV.T93 τ-NS Regularity on T² Fiber (C3 Defect Contractivity) theorem IV.T94 Fiber-base factorization theorem IV.T95 Structural Arrow of Time theorem IV.T96 Central Theorem --- physical form theorem IV.T97 Boundary Triad Theorem theorem IV.T98 Uniqueness of Phi theorem IV.T99 Temporal Complement theorem