\contentsline {paragraph}{Statement of scope:}{1}{section*.1}%
\contentsline {paragraph}{UTFANSWF}{1}{section*.2}%
\contentsline {paragraph}{Executive Summary}{1}{section*.3}%
\contentsline {section}{UTFANSWF v21 --- Cosmology Update and Row 20 Resolution}{2}{section*.4}%
\contentsline {paragraph}{Versioning Rationale (v22).}{2}{section*.5}%
\contentsline {paragraph}{Motivation.}{2}{section*.6}%
\contentsline {paragraph}{Diagnosis.}{2}{section*.7}%
\contentsline {paragraph}{v21 Correction (Structural, Constraint-Preserving).}{2}{section*.8}%
\contentsline {paragraph}{Validation Outcome.}{3}{section*.9}%
\contentsline {paragraph}{v22 Interpretation.}{3}{section*.10}%
\contentsline {paragraph}{Derivation of Distributed Hubble Parameter}{3}{section*.11}%
\contentsline {paragraph}{Summary.}{4}{section*.12}%
\contentsline {paragraph}{Continuity.}{4}{section*.13}%
\contentsline {paragraph}{ }{5}{section*.18}%
\contentsline {paragraph}{Conic Neutral Zone (NZ) Geometry with Dual Certificates.}{6}{section*.23}%
\contentsline {paragraph}{Fisher--Rao Information Geometry on the Repair Manifold.}{7}{section*.25}%
\contentsline {paragraph}{Entropy--Fisher Cross--Metric.}{7}{section*.26}%
\contentsline {paragraph}{Gate Tensor Composition (CFGL Monoidal Closure).}{7}{section*.28}%
\contentsline {paragraph}{Category--Functor Gate Logic (CFGL) as a Monoidal Enriched Category.}{8}{section*.29}%
\contentsline {paragraph}{Refractal Operator and Energy Law.}{8}{section*.31}%
\contentsline {paragraph}{Tight--Frame Stability (HB Parseval).}{9}{section*.32}%
\contentsline {paragraph}{QND--in--NZ Read (Mode Matching).}{9}{section*.33}%
\contentsline {paragraph}{Refractal Robustness Inequality.}{9}{section*.34}%
\contentsline {paragraph}{Reaction--Path (Sensitivity) Matrix.}{9}{section*.36}%
\contentsline {paragraph}{Regularized NZ--Constrained Update.}{9}{section*.37}%
\contentsline {paragraph}{Gate Decision (Pass / Repair / Fail).}{10}{section*.38}%
\contentsline {paragraph}{Entropy--Geometry Alignment (Ellipsoid Check).}{10}{section*.39}%
\contentsline {paragraph}{Identifiability Condition.}{10}{section*.40}%
\contentsline {paragraph}{Pass/Fail Tiering Framework.}{10}{section*.41}%
\contentsline {paragraph}{Dynamic Repair Flow (Refractal ODE).}{12}{section*.44}%
\contentsline {section}{\numberline {1}Introduction}{31}{section.1}%
\contentsline {section}{\numberline {2}Acronyms}{31}{section.2}%
\contentsline {section}{\numberline {3}Zero State (ZS)}{31}{section.3}%
\contentsline {paragraph}{From Absence to Structure.}{31}{section*.48}%
\contentsline {subsection}{\numberline {3.1}Axiomatic Definition}{32}{subsection.3.1}%
\contentsline {subsection}{\numberline {3.2}Immediate Consequences}{32}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}Constraint-Induced Structure}{32}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}Kernel Construction}{33}{subsection.3.4}%
\contentsline {subsection}{\numberline {3.5}Entropy Maximization}{33}{subsection.3.5}%
\contentsline {subsection}{\numberline {3.6}Wavefunctional Construction}{34}{subsection.3.6}%
\contentsline {paragraph}{Interpretive Note.}{34}{section*.49}%
\contentsline {section}{\numberline {4}SWF--ISM Workup: Foundations, Symmetry Adaptation, and EFT Embedding}{34}{section.4}%
\contentsline {paragraph}{From Zero State to Information Structure.}{34}{section*.50}%
\contentsline {paragraph}{Acronyms (fixed).}{34}{section*.51}%
\contentsline {paragraph}{Neutral solution: ANSWF.}{35}{section*.53}%
\contentsline {paragraph}{(i) ANSWF by maximum entropy.}{37}{section*.59}%
\contentsline {paragraph}{(ii) SANSWF with axial symmetry and fixed quadrupole.}{37}{section*.60}%
\contentsline {paragraph}{(iii) Positivity sign test (toy).}{37}{section*.61}%
\contentsline {paragraph}{Summary.}{38}{section*.63}%
\contentsline {section}{\numberline {5}SWF-ISM: Information-Theoretic Foundations}{38}{section.5}%
\contentsline {section}{\numberline {6}ANSWF: EFT Positivity and Constraints}{38}{section.6}%
\contentsline {paragraph}{From Information Structure to Physical Constraints.}{38}{section*.64}%
\contentsline {paragraph}{Scattering amplitudes and analyticity.}{39}{section*.65}%
\contentsline {paragraph}{Positivity from the optical theorem.}{39}{section*.66}%
\contentsline {paragraph}{Implications for the effective field theory.}{39}{section*.67}%
\contentsline {paragraph}{Interpretation within UTFANSWF.}{39}{section*.68}%
\contentsline {section}{\numberline {7}SANSWF: Symmetry-Augmented Layer}{40}{section.7}%
\contentsline {paragraph}{From Consistency to Symmetry Enforcement.}{40}{section*.69}%
\contentsline {paragraph}{Gauge covariance.}{40}{section*.70}%
\contentsline {paragraph}{Mass generation and symmetry breaking.}{40}{section*.71}%
\contentsline {paragraph}{Scalar sector and bounded-from-below stability.}{40}{section*.72}%
\contentsline {paragraph}{Perturbative unitarity.}{41}{section*.73}%
\contentsline {paragraph}{Interpretation within UTFANSWF.}{41}{section*.74}%
\contentsline {section}{\numberline {8}UTFANSWF Proper: SU(5) and Seesaw Embedding}{41}{section.8}%
\contentsline {paragraph}{From Symmetry Enforcement to Unification.}{41}{section*.75}%
\contentsline {paragraph}{Field content and representation structure.}{41}{section*.76}%
\contentsline {paragraph}{Neutrino masses via Type-I seesaw.}{42}{section*.77}%
\contentsline {paragraph}{Renormalization group evolution and unification.}{42}{section*.78}%
\contentsline {paragraph}{Threshold corrections and heavy scales.}{43}{section*.80}%
\contentsline {paragraph}{Proton decay constraints.}{43}{section*.81}%
\contentsline {paragraph}{Seesaw Yukawa constraints.}{44}{section*.83}%
\contentsline {paragraph}{Interpretation within UTFANSWF.}{44}{section*.85}%
\contentsline {section}{\numberline {9}Predictions and SMEFT Matching}{45}{section.9}%
\contentsline {paragraph}{From Unified Embedding to Observable Predictions.}{45}{section*.86}%
\contentsline {paragraph}{Heavy field integration and operator matching.}{45}{section*.87}%
\contentsline {paragraph}{Interpretation of Wilson coefficients.}{45}{section*.88}%
\contentsline {paragraph}{Radiative corrections and vacuum structure.}{45}{section*.89}%
\contentsline {paragraph}{Stability and UTFANSWF consistency.}{46}{section*.91}%
\contentsline {paragraph}{Interpretation within UTFANSWF.}{46}{section*.92}%
\contentsline {section}{\numberline {10}Conditional TCC Compliance and Inflation Gates}{47}{section.10}%
\contentsline {paragraph}{From Cosmological Embedding to Consistency Bounds.}{47}{section*.93}%
\contentsline {paragraph}{Slow-roll dynamics and observable parameters.}{47}{section*.94}%
\contentsline {paragraph}{Trans-Planckian Censorship Conjecture (TCC).}{47}{section*.95}%
\contentsline {paragraph}{Implications for inflationary scale.}{47}{section*.97}%
\contentsline {paragraph}{UTFANSWF gate interpretation.}{48}{section*.99}%
\contentsline {paragraph}{Role within the framework.}{49}{section*.100}%
\contentsline {section}{\numberline {11}Conditional TCC Compliance and Inflation Gates (Updated)}{49}{section.11}%
\contentsline {paragraph}{Inflation as a constrained gate.}{49}{section*.101}%
\contentsline {paragraph}{Slow-roll identities.}{49}{section*.102}%
\contentsline {paragraph}{Strict TCC bound.}{49}{section*.103}%
\contentsline {paragraph}{Gate logic (A/B1).}{50}{section*.105}%
\contentsline {paragraph}{Gate decision rule.}{50}{section*.106}%
\contentsline {section}{\numberline {12}Dark Matter: UTFANSWF Axion Band and Signatures}{51}{section.12}%
\contentsline {paragraph}{From Unified Embedding to Dark Matter Selection.}{51}{section*.109}%
\contentsline {paragraph}{Axion mass and photon coupling.}{51}{section*.110}%
\contentsline {paragraph}{UTFANSWF preferred band.}{52}{section*.111}%
\contentsline {paragraph}{Relic abundance from misalignment.}{52}{section*.112}%
\contentsline {paragraph}{Observational signatures.}{52}{section*.113}%
\contentsline {paragraph}{UTFANSWF gate interpretation.}{52}{section*.115}%
\contentsline {paragraph}{Role within the framework.}{53}{section*.116}%
\contentsline {section}{\numberline {13}Neutral Zone (NZ): Hydrogen \& Stellar Ignition Correlation}{53}{section.13}%
\contentsline {paragraph}{From Microphysics to Macroscopic Viability.}{53}{section*.117}%
\contentsline {section}{\numberline {14}Neutral Zone (NZ): full derivations and sensitivities (Expanded)}{54}{section.14}%
\contentsline {paragraph}{Hydrogen stability \& pp exothermicity.}{54}{section*.122}%
\contentsline {paragraph}{Observed universe check.}{56}{section*.123}%
\contentsline {paragraph}{Hadronic decomposition and linear model.}{56}{section*.124}%
\contentsline {paragraph}{Differential sensitivities of \(I_{\rm NZ}\).}{57}{section*.125}%
\contentsline {paragraph}{Origin in UTFANSWF.}{57}{section*.126}%
\contentsline {paragraph}{UTFANSWF gate interpretation.}{57}{section*.127}%
\contentsline {section}{\numberline {15}Learning Efficiency with UTFANSWF Constraints}{57}{section.15}%
\contentsline {paragraph}{From Physical Consistency to Learning Constraints.}{57}{section*.128}%
\contentsline {paragraph}{Hypothesis space reduction.}{58}{section*.129}%
\contentsline {paragraph}{Constrained optimization.}{58}{section*.130}%
\contentsline {paragraph}{Convergence acceleration.}{58}{section*.131}%
\contentsline {paragraph}{Interpretation within UTFANSWF.}{58}{section*.132}%
\contentsline {paragraph}{Practical implication.}{59}{section*.133}%
\contentsline {section}{\numberline {16}Stress tests, pass/fail matrix, and compact numeric harness (Restored)}{60}{section.16}%
\contentsline {section}{\numberline {17}Canonical UTFANSWF Lagrangian (Equation of Record)}{61}{section.17}%
\contentsline {paragraph}{Equation of record.}{61}{section*.138}%
\contentsline {paragraph}{Abelian and gravitational.}{61}{section*.139}%
\contentsline {paragraph}{Vectorlike safety.}{62}{section*.143}%
\contentsline {paragraph}{Worked example: $U(1)_{B-L}$.}{62}{section*.144}%
\contentsline {subsection}{\numberline {17.1}4}{63}{subsection.17.1}%
\contentsline {section}{\numberline {18}Axion Sector: Strong-CP Resolution and Dark Matter Channel}{65}{section.18}%
\contentsline {paragraph}{Strong-CP problem.}{65}{section*.149}%
\contentsline {paragraph}{PQ symmetry and axion emergence.}{65}{section*.150}%
\contentsline {paragraph}{Embedding in UTFANSWF.}{65}{section*.151}%
\contentsline {paragraph}{Cosmological production.}{65}{section*.152}%
\contentsline {paragraph}{Detection channels.}{66}{section*.153}%
\contentsline {paragraph}{UTFANSWF prediction.}{66}{section*.154}%
\contentsline {section}{\numberline {19}AI $\leftrightarrow $ UTFANSWF Co-evolution}{66}{section.19}%
\contentsline {paragraph}{Motivation.}{66}{section*.155}%
\contentsline {paragraph}{Constraint embedding.}{66}{section*.156}%
\contentsline {paragraph}{Learning efficiency.}{67}{section*.157}%
\contentsline {paragraph}{Feedback loop.}{67}{section*.158}%
\contentsline {paragraph}{Interpretation within the framework.}{67}{section*.159}%
\contentsline {paragraph}{Outcome.}{67}{section*.160}%
\contentsline {section}{\numberline {20}Conclusion}{67}{section.20}%
\contentsline {paragraph}{Summary of the framework.}{67}{section*.161}%
\contentsline {paragraph}{Equation-to-observation closure.}{68}{section*.162}%
\contentsline {paragraph}{Falsifiability as design principle.}{68}{section*.163}%
\contentsline {paragraph}{Role of constraint geometry.}{68}{section*.164}%
\contentsline {paragraph}{Operational program.}{68}{section*.165}%
\contentsline {paragraph}{Final statement.}{69}{section*.166}%
\contentsline {section}{\numberline {21}Black-hole Ringdown and Echo Predictions}{70}{section.21}%
\contentsline {paragraph}{Kerr QNM (fundamental mode).}{70}{section*.167}%
\contentsline {paragraph}{Echo generation.}{70}{section*.168}%
\contentsline {paragraph}{Minimal echo model.}{70}{section*.169}%
\contentsline {paragraph}{UTFANSWF gate relevance.}{70}{section*.170}%
\contentsline {subsection}{\numberline {21.1}Benchmark realization and signal templates}{71}{subsection.21.1}%
\contentsline {paragraph}{Benchmark parameters.}{71}{section*.171}%
\contentsline {paragraph}{Echo spacing.}{71}{section*.172}%
\contentsline {paragraph}{Signal construction.}{71}{section*.173}%
\contentsline {section}{\numberline {22}Functional RG, PPN/GW limits, positivity bounds, and analytic Jacobians}{71}{section.22}%
\contentsline {section}{\numberline {23}AI $\leftrightarrow $ UTFANSWF co-evolution (Expanded)}{75}{section.23}%
\contentsline {section}{\numberline {24}Computational Robustness Ladder (Layers 4b--9)}{77}{section.24}%
\contentsline {subsection}{\numberline {24.1}Overview}{77}{subsection.24.1}%
\contentsline {subsection}{\numberline {24.2}Baseline: Layer 4b Locked State}{77}{subsection.24.2}%
\contentsline {subsection}{\numberline {24.3}Layer 5: Dynamic Preservation}{78}{subsection.24.3}%
\contentsline {subsection}{\numberline {24.4}Layer 6: Localized Forcing Response}{78}{subsection.24.4}%
\contentsline {subsection}{\numberline {24.5}Layer 7: Forcing Threshold Sweep}{78}{subsection.24.5}%
\contentsline {subsection}{\numberline {24.6}Layer 8: Off-Ring / Symmetry-Breaking Forcing}{79}{subsection.24.6}%
\contentsline {subsection}{\numberline {24.7}Layer 9: Multi-Source Interference}{79}{subsection.24.7}%
\contentsline {subsection}{\numberline {24.8}Layer 9 (Extended): Aggressive Stress Regime}{79}{subsection.24.8}%
\contentsline {subsection}{\numberline {24.9}Parameter-Space Coverage and Stability Region}{80}{subsection.24.9}%
\contentsline {subsection}{\numberline {24.10}Behavior Hierarchy}{80}{subsection.24.10}%
\contentsline {subsection}{\numberline {24.11}Interpretation}{81}{subsection.24.11}%
\contentsline {section}{\numberline {25}Break UTFANSWF: Stress-Test Appendix}{81}{section.25}%
\contentsline {paragraph}{Inputs.}{81}{section*.183}%
\contentsline {paragraph}{Derived.}{82}{section*.184}%
\contentsline {paragraph}{Pass/Fail table (benchmark).}{82}{section*.185}%
\contentsline {paragraph}{Mixed non-abelian anomalies.}{82}{section*.186}%
\contentsline {paragraph}{Abelian and gravitational.}{82}{section*.187}%
\contentsline {paragraph}{Vectorlike safety.}{83}{section*.191}%
\contentsline {paragraph}{Worked example: $U(1)_{B-L}$.}{83}{section*.192}%
\contentsline {paragraph}{Single real scalar.}{86}{section*.200}%
\contentsline {paragraph}{Two scalars / mixed channel.}{86}{section*.201}%
\contentsline {paragraph}{Singlet basis.}{88}{section*.208}%
\contentsline {paragraph}{Explicit Higgs block.}{88}{section*.209}%
\contentsline {paragraph}{Mixed and adjoint blocks.}{89}{section*.210}%
\contentsline {paragraph}{Derivation (sketch).}{90}{section*.214}%
\contentsline {paragraph}{Dimension-6 (gauge boson exchange).}{90}{section*.217}%
\contentsline {paragraph}{Dimension-5 (color-triplet Higgs exchange).}{91}{section*.218}%
\contentsline {paragraph}{Spin projectors.}{93}{section*.228}%
\contentsline {paragraph}{FRG numeric exponents (EH, Litim).}{95}{section*.234}%
\contentsline {section}{\numberline {26}SU(5) Threshold Sweep: Benchmark Rows 1--8}{100}{section.26}%
\contentsline {subsection}{\numberline {26.1}Master Equations}{100}{subsection.26.1}%
\contentsline {subsection}{\numberline {26.2}Benchmark Sweep: Rows 1--4}{101}{subsection.26.2}%
\contentsline {subsection}{\numberline {26.3}Benchmark Sweep: Rows 5--8}{101}{subsection.26.3}%
\contentsline {subsection}{\numberline {26.4}Compact Sweep Summary}{101}{subsection.26.4}%
\contentsline {section}{\numberline {27}SU(5) Threshold Sweep and Proton-Decay Scaling}{102}{section.27}%
\contentsline {subsection}{\numberline {27.1}Dimension--6 Proton-Decay Trend}{102}{subsection.27.1}%
\contentsline {subsection}{\numberline {27.2}Dimension--5 Proton-Decay Trend}{102}{subsection.27.2}%
\contentsline {subsection}{\numberline {27.3}Pass/Fail Interpretation Across Rows 1--8}{103}{subsection.27.3}%
\contentsline {subsection}{\numberline {27.4}Verdict}{103}{subsection.27.4}%
\contentsline {section}{\numberline {28}PPN Constraints: $\gamma , \beta $}{103}{section.28}%
\contentsline {subsection}{\numberline {28.1}Scalar–Tensor Setup}{103}{subsection.28.1}%
\contentsline {subsection}{\numberline {28.2}Post-Newtonian Parameters}{104}{subsection.28.2}%
\contentsline {subsection}{\numberline {28.3}Experimental Bounds}{104}{subsection.28.3}%
\contentsline {subsection}{\numberline {28.4}UTFANSWF Status}{104}{subsection.28.4}%
\contentsline {section}{\numberline {29}GW Propagation Speed Constraint}{104}{section.29}%
\contentsline {subsection}{\numberline {29.1}Quadratic Tensor Action}{105}{subsection.29.1}%
\contentsline {subsection}{\numberline {29.2}Experimental Bound}{105}{subsection.29.2}%
\contentsline {subsection}{\numberline {29.3}UTFANSWF Implications}{105}{subsection.29.3}%
\contentsline {subsection}{\numberline {29.4}Gate Status}{105}{subsection.29.4}%
\contentsline {section}{\numberline {30}Forward-Limit Positivity Bounds}{106}{section.30}%
\contentsline {subsection}{\numberline {30.1}Twice-Subtracted Dispersion Relation}{106}{subsection.30.1}%
\contentsline {subsection}{\numberline {30.2}Wilson Coefficient Constraints}{106}{subsection.30.2}%
\contentsline {subsection}{\numberline {30.3}Gravity-Safe Prescription}{107}{subsection.30.3}%
\contentsline {subsection}{\numberline {30.4}Gate Status}{107}{subsection.30.4}%
\contentsline {section}{\numberline {31}Partial-Wave Unitarity and BFB Cone (Scalar Sector)}{107}{section.31}%
\contentsline {subsection}{\numberline {31.1}Quartic Scalar Potential}{107}{subsection.31.1}%
\contentsline {subsection}{\numberline {31.2}s-wave Unitarity}{108}{subsection.31.2}%
\contentsline {subsection}{\numberline {31.3}BFB Stability Conditions}{108}{subsection.31.3}%
\contentsline {subsection}{\numberline {31.4}Copositivity Cone}{108}{subsection.31.4}%
\contentsline {subsection}{\numberline {31.5}Gate Status}{108}{subsection.31.5}%
\contentsline {section}{\numberline {32}SU(5)/SO(10) Thresholds and Proton-Decay Scaling}{109}{section.32}%
\contentsline {subsection}{\numberline {32.1}RG Running with Heavy Thresholds}{109}{subsection.32.1}%
\contentsline {subsection}{\numberline {32.2}Threshold Combinations}{109}{subsection.32.2}%
\contentsline {subsection}{\numberline {32.3}Proton Decay Scaling Laws}{109}{subsection.32.3}%
\contentsline {paragraph}{Dimension-6 gauge-boson exchange.}{110}{section*.255}%
\contentsline {paragraph}{Dimension-5 triplet-Higgs exchange.}{110}{section*.256}%
\contentsline {subsection}{\numberline {32.4}Numerical Benchmark}{110}{subsection.32.4}%
\contentsline {subsection}{\numberline {32.5}Gate Status}{110}{subsection.32.5}%
\contentsline {section}{\numberline {33}Axion Dark Matter: UTFANSWF Band and Signatures}{110}{section.33}%
\contentsline {subsection}{\numberline {33.1}Axion Mass and Coupling}{111}{subsection.33.1}%
\contentsline {subsection}{\numberline {33.2}Preferred UTFANSWF Band}{111}{subsection.33.2}%
\contentsline {subsection}{\numberline {33.3}Abundance and Misalignment Mechanism}{111}{subsection.33.3}%
\contentsline {subsection}{\numberline {33.4}Experimental Signatures}{112}{subsection.33.4}%
\contentsline {paragraph}{Haloscopes.}{112}{section*.258}%
\contentsline {paragraph}{Astrophysical channels.}{112}{section*.259}%
\contentsline {subsection}{\numberline {33.5}Gate Status}{112}{subsection.33.5}%
\contentsline {section}{\numberline {34}Neutral Zone (NZ): Hydrogen \& Stellar Ignition Correlation}{112}{section.34}%
\contentsline {subsection}{\numberline {34.1}Hydrogen Stability}{112}{subsection.34.1}%
\contentsline {subsection}{\numberline {34.2}Exothermic $pp$ Chain}{113}{subsection.34.2}%
\contentsline {subsection}{\numberline {34.3}Neutral Zone Sandwich Inequality}{113}{subsection.34.3}%
\contentsline {subsection}{\numberline {34.4}Numerical Check (Our Universe)}{113}{subsection.34.4}%
\contentsline {subsection}{\numberline {34.5}Neutral Zone Index}{113}{subsection.34.5}%
\contentsline {subsection}{\numberline {34.6}UTFANSWF Embedding}{114}{subsection.34.6}%
\contentsline {subsection}{\numberline {34.7}Gate Status}{114}{subsection.34.7}%
\contentsline {section}{\numberline {35}Learning Efficiency with UTFANSWF Constraints}{114}{section.35}%
\contentsline {subsection}{\numberline {35.1}Constraint-Prior Hypothesis Space}{115}{subsection.35.1}%
\contentsline {subsection}{\numberline {35.2}Validation Error Dynamics}{115}{subsection.35.2}%
\contentsline {subsection}{\numberline {35.3}Comparative Learning Curves}{115}{subsection.35.3}%
\contentsline {subsection}{\numberline {35.4}Gate Status}{115}{subsection.35.4}%
\contentsline {section}{\numberline {36}Stress Tests and Pass/Fail Matrix}{116}{section.36}%
\contentsline {subsection}{\numberline {36.1}Compact Numeric Harness (Benchmark Slice)}{116}{subsection.36.1}%
\contentsline {subsection}{\numberline {36.2}Pass/Fail Table (Benchmark Slice)}{117}{subsection.36.2}%
\contentsline {subsection}{\numberline {36.3}Summary}{117}{subsection.36.3}%
\contentsline {section}{\numberline {37}Canonical UTFANSWF Lagrangian (Equation of Record)}{117}{section.37}%
\contentsline {subsection}{\numberline {37.1}Guardrails and Consistency Conditions}{118}{subsection.37.1}%
\contentsline {subsection}{\numberline {37.2}Anomaly Cancellation Checks}{118}{subsection.37.2}%
\contentsline {paragraph}{Standard Model:}{119}{section*.264}%
\contentsline {paragraph}{SU(5) Embedding:}{119}{section*.265}%
\contentsline {paragraph}{SO(10) Embedding:}{119}{section*.266}%
\contentsline {paragraph}{E$_6$ Embedding:}{119}{section*.267}%
\contentsline {subsection}{\numberline {37.3}Summary}{119}{subsection.37.3}%
\contentsline {section}{\numberline {38}Axion Primer (Concise)}{119}{section.38}%
\contentsline {subsection}{\numberline {38.1}Strong-CP Problem and PQ Symmetry}{119}{subsection.38.1}%
\contentsline {subsection}{\numberline {38.2}Axion Potential and Mass}{120}{subsection.38.2}%
\contentsline {subsection}{\numberline {38.3}Axion–Photon Coupling}{120}{subsection.38.3}%
\contentsline {subsection}{\numberline {38.4}UTFANSWF Band and Predictions}{120}{subsection.38.4}%
\contentsline {subsection}{\numberline {38.5}Phenomenological Notes}{121}{subsection.38.5}%
\contentsline {subsection}{\numberline {38.6}Summary}{121}{subsection.38.6}%
\contentsline {section}{\numberline {39}Axion Primer (Concise)}{121}{section.39}%
\contentsline {subsection}{\numberline {39.1}Strong-CP Problem and PQ Symmetry}{121}{subsection.39.1}%
\contentsline {subsection}{\numberline {39.2}Axion Potential and Mass}{122}{subsection.39.2}%
\contentsline {subsection}{\numberline {39.3}Axion–Photon Coupling}{122}{subsection.39.3}%
\contentsline {subsection}{\numberline {39.4}UTFANSWF Band and Predictions}{122}{subsection.39.4}%
\contentsline {subsection}{\numberline {39.5}Phenomenological Notes}{123}{subsection.39.5}%
\contentsline {subsection}{\numberline {39.6}Summary}{123}{subsection.39.6}%
\contentsline {section}{\numberline {40}AI $\leftrightarrow $ UTFANSWF Co-evolution}{123}{section.40}%
\contentsline {subsection}{\numberline {40.1}Motivation}{123}{subsection.40.1}%
\contentsline {subsection}{\numberline {40.2}Mathematical Embedding of Priors}{123}{subsection.40.2}%
\contentsline {subsection}{\numberline {40.3}Learning Efficiency}{124}{subsection.40.3}%
\contentsline {subsection}{\numberline {40.4}Compact Acceptance Algorithm}{124}{subsection.40.4}%
\contentsline {subsection}{\numberline {40.5}Summary}{125}{subsection.40.5}%
\contentsline {section}{\numberline {41}Appendix: The Curvature Engineering Gate (CEG)}{125}{section.41}%
\contentsline {section}{\numberline {42}Spacetime-Engineering Consistency Gate (SECG)}{126}{section.42}%
\contentsline {subsection}{\numberline {42.1}Objects under test}{126}{subsection.42.1}%
\contentsline {subsection}{\numberline {42.2}Primary inequalities and invariants}{127}{subsection.42.2}%
\contentsline {paragraph}{(A) Quantum energy and causality bounds}{127}{section*.273}%
\contentsline {paragraph}{(B) Ford--Roman quantum inequalities}{127}{section*.274}%
\contentsline {paragraph}{(C) Chronology protection}{127}{section*.275}%
\contentsline {paragraph}{(D) EFT positivity}{127}{section*.276}%
\contentsline {paragraph}{(E) Hyperbolicity and stability}{127}{section*.277}%
\contentsline {paragraph}{(F) Entropy consistency}{127}{section*.278}%
\contentsline {paragraph}{(G) Backreaction and curvature caps}{128}{section*.279}%
\contentsline {subsection}{\numberline {42.3}Warp / effective-FTL corollaries}{128}{subsection.42.3}%
\contentsline {subsection}{\numberline {42.4}Artificial gravity modules}{128}{subsection.42.4}%
\contentsline {subsection}{\numberline {42.5}Compact pass/fail matrix}{129}{subsection.42.5}%
\contentsline {subsection}{\numberline {42.6}Harness (pseudo-algorithm)}{129}{subsection.42.6}%
\contentsline {paragraph}{Status in UTFANSWF v22}{129}{section*.280}%
\contentsline {section}{\numberline {43}Functional RG, PPN/GW Limits, Positivity Bounds, and Analytic Jacobians}{129}{section.43}%
\contentsline {subsection}{\numberline {43.1}Functional Renormalization Group (FRG) Setup}{130}{subsection.43.1}%
\contentsline {subsection}{\numberline {43.2}Beta Functions and Fixed Points}{130}{subsection.43.2}%
\contentsline {subsection}{\numberline {43.3}PPN and Gravitational Wave Bounds}{131}{subsection.43.3}%
\contentsline {subsection}{\numberline {43.4}Forward-Limit Positivity Bounds}{131}{subsection.43.4}%
\contentsline {subsection}{\numberline {43.5}Analytic Jacobians for Compressed Cosmology}{131}{subsection.43.5}%
\contentsline {subsection}{\numberline {43.6}Takeaway}{132}{subsection.43.6}%
\contentsline {section}{\numberline {44}Compressed Cosmology Fit (CMB/BAO/SN/GW, $\chi ^2/\nu $ Checks)}{132}{section.44}%
\contentsline {paragraph}{Row 20 — run checklist (concise).}{132}{section*.281}%
\contentsline {paragraph}{Row 20 — ledger linkage.}{133}{section*.282}%
\contentsline {subsection}{\numberline {44.1}Observables and Compressed Parameters}{133}{subsection.44.1}%
\contentsline {subsection}{\numberline {44.2}Distance Priors and BAO}{133}{subsection.44.2}%
\contentsline {subsection}{\numberline {44.3}Supernova (SN) Distance Moduli}{134}{subsection.44.3}%
\contentsline {paragraph}{SN Projection Geometry and Residual Compression}{134}{section*.283}%
\contentsline {subsection}{\numberline {44.4}Standard Sirens (GW)}{135}{subsection.44.4}%
\contentsline {subsection}{\numberline {44.5}Compressed $\chi ^2/\nu $ Gate}{135}{subsection.44.5}%
\contentsline {paragraph}{Dataset Update Sensitivity and Constrained Recalibration.}{135}{section*.284}%
\contentsline {paragraph}{Distributed Hubble Parameter Resolution (v22 Candidate)}{135}{section*.285}%
\contentsline {paragraph}{Distributed Hubble Resolution Theorem (v22 Candidate)}{136}{section*.286}%
\contentsline {subsection}{\numberline {44.6}Compact Pass/Fail Matrix}{137}{subsection.44.6}%
\contentsline {paragraph}{Row 21 --- audit stub (verbatim).}{138}{section*.287}%
\contentsline {paragraph}{Row 22 --- audit stub (verbatim).}{138}{section*.288}%
\contentsline {paragraph}{Status in UTFANSWF v22.}{138}{section*.289}%
\contentsline {section}{\numberline {45}Axion Dark Matter Gate (2025Q3 Update vs Haloscopes)}{138}{section.45}%
\contentsline {subsection}{\numberline {45.1}UTFANSWF Axion Slice and Relations}{138}{subsection.45.1}%
\contentsline {paragraph}{Misalignment Abundance.}{139}{section*.290}%
\contentsline {subsection}{\numberline {45.2}Detection Channels and Signatures}{139}{subsection.45.2}%
\contentsline {subsection}{\numberline {45.3}Current Experimental Coverage Within the UTFANSWF Band}{139}{subsection.45.3}%
\contentsline {subsection}{\numberline {45.4}Gate Criterion and Verdict}{139}{subsection.45.4}%
\contentsline {subsection}{\numberline {45.5}Near-Term UTFANSWF Forecasts}{140}{subsection.45.5}%
\contentsline {subsection}{\numberline {45.6}Compact Pass/Fail Matrix}{140}{subsection.45.6}%
\contentsline {section}{\numberline {46}Neutral Zone (NZ) Sandwich: Robust Sweep and Sensitivities}{140}{section.46}%
\contentsline {subsection}{\numberline {46.1}NZ Sandwich Inequality}{140}{subsection.46.1}%
\contentsline {subsection}{\numberline {46.2}Sensitivity Expansions}{141}{subsection.46.2}%
\contentsline {subsection}{\numberline {46.3}Robustness Sweep}{141}{subsection.46.3}%
\contentsline {subsection}{\numberline {46.4}Gate Criterion and Verdict}{141}{subsection.46.4}%
\contentsline {subsection}{\numberline {46.5}Implication for UTFANSWF}{142}{subsection.46.5}%
\contentsline {section}{\numberline {47}Black-Hole Ringdown and Echo Predictions (Expanded Benchmark Fits)}{142}{section.47}%
\contentsline {subsection}{\numberline {47.1}Kerr Ringdown Spectrum}{142}{subsection.47.1}%
\contentsline {paragraph}{Benchmark Example.}{142}{section*.291}%
\contentsline {subsection}{\numberline {47.2}Echo Templates from Near-Horizon Modifications}{142}{subsection.47.2}%
\contentsline {subsection}{\numberline {47.3}Consistency Gates}{143}{subsection.47.3}%
\contentsline {subsection}{\numberline {47.4}Pass/Fail Matrix (Benchmark Case)}{143}{subsection.47.4}%
\contentsline {subsection}{\numberline {47.5}Verdict}{143}{subsection.47.5}%
\contentsline {section}{\numberline {48}AI $\leftrightarrow $ UTFANSWF Guardrails and Learning Efficiency (Expanded)}{144}{section.48}%
\contentsline {subsection}{\numberline {48.1}Embedding Physical Guardrails into AI}{144}{subsection.48.1}%
\contentsline {subsection}{\numberline {48.2}Learning Efficiency Model}{144}{subsection.48.2}%
\contentsline {paragraph}{Unconstrained baseline.}{144}{section*.292}%
\contentsline {paragraph}{UTFANSWF-constrained.}{144}{section*.293}%
\contentsline {subsection}{\numberline {48.3}Example: Benchmarked Learning Curves}{144}{subsection.48.3}%
\contentsline {subsection}{\numberline {48.4}Co-evolution Framework}{145}{subsection.48.4}%
\contentsline {subsection}{\numberline {48.5}Pass/Fail Gate}{145}{subsection.48.5}%
\contentsline {subsection}{\numberline {48.6}Verdict}{145}{subsection.48.6}%
\contentsline {section}{\numberline {49}Stress-Test Appendix: ``Break UTFANSWF'' Kill-Shot Scenarios}{145}{section.49}%
\contentsline {subsection}{\numberline {49.1}Conceptual Kill-Shot Checklist}{145}{subsection.49.1}%
\contentsline {subsection}{\numberline {49.2}Numeric Stress-Test Harness (Benchmark Slice)}{146}{subsection.49.2}%
\contentsline {subsection}{\numberline {49.3}Compact Pass/Fail Matrix}{146}{subsection.49.3}%
\contentsline {subsection}{\numberline {49.4}Verdict}{146}{subsection.49.4}%
\contentsline {paragraph}{Inputs held fixed.}{147}{section*.295}%
\contentsline {paragraph}{Summary}{147}{section*.299}%
\contentsline {section}{\numberline {50}SU(5)/SO(10) Heavy Thresholds and Proton-Decay Gates}{148}{section.50}%
\contentsline {paragraph}{Dimension-6 (gauge bosons).}{149}{section*.303}%
\contentsline {paragraph}{Dimension-5 (color-triplet Higgs).}{149}{section*.304}%
\contentsline {section}{\numberline {51}Gravity-Safe Forward-Limit Positivity with Spin (Projectors \& Finite-$t$)}{150}{section.51}%
\contentsline {paragraph}{Setup.}{150}{section*.307}%
\contentsline {paragraph}{Gravity-safe dispersion and positivity.}{150}{section*.308}%
\contentsline {paragraph}{Mapping to Wilson coefficients.}{151}{section*.309}%
\contentsline {paragraph}{Ready-to-use bounds (canonical sectors).}{151}{section*.310}%
\contentsline {paragraph}{Improved finite-$t$ bounds.}{151}{section*.311}%
\contentsline {paragraph}{Causality/time-delay compatibility.}{152}{section*.312}%
\contentsline {paragraph}{Implementation recipe (harness).}{152}{section*.313}%
\contentsline {paragraph}{Summary.}{152}{section*.314}%
\contentsline {section}{\numberline {52}Row 2 — PPN/GW \& Spin-2 Causality Gate (PASS)}{152}{section.52}%
\contentsline {paragraph}{Benchmark slice.}{152}{section*.315}%
\contentsline {paragraph}{PPN parameters.}{152}{section*.316}%
\contentsline {paragraph}{GW propagation speed today.}{153}{section*.317}%
\contentsline {paragraph}{Spin-2 causality / time delay.}{153}{section*.318}%
\contentsline {paragraph}{Verdict.}{154}{section*.320}%
\contentsline {section}{\numberline {53}Row 3 — SANSWF: Bounded-From-Below (BFB) \& Gauge Consistency (PASS)}{154}{section.53}%
\contentsline {paragraph}{Scalar potential.}{154}{section*.321}%
\contentsline {paragraph}{Bounded-from-below (BFB) conditions.}{154}{section*.322}%
\contentsline {paragraph}{Gauge consistency.}{155}{section*.323}%
\contentsline {paragraph}{Partial-wave unitarity.}{155}{section*.324}%
\contentsline {paragraph}{Verdict.}{156}{section*.326}%
\contentsline {section}{\numberline {54}Rows 4--12: Polished Consistency Gates}{156}{section.54}%
\contentsline {subsection}{\numberline {54.1}Row 4 --- SU(5)/SO(10) Embedding \& Seesaw}{156}{subsection.54.1}%
\contentsline {subsection}{\numberline {54.2}Row 5 --- Predictions \& SMEFT Matching}{157}{subsection.54.2}%
\contentsline {subsection}{\numberline {54.3}Rows 6 \& 7 --- Inflation + TCC Gates}{157}{subsection.54.3}%
\contentsline {paragraph}{Branch A (Strict TCC + Curvaton)}{157}{section*.327}%
\contentsline {paragraph}{Branch B1 (Moderate TCC)}{157}{section*.328}%
\contentsline {subsection}{\numberline {54.4}Row 8 --- Axion Band \& Signatures}{157}{subsection.54.4}%
\contentsline {subsection}{\numberline {54.5}Row 9 --- Neutral Zone (NZ) Sandwich}{158}{subsection.54.5}%
\contentsline {subsection}{\numberline {54.6}Row 10 --- NZ Sensitivities \& Robustness}{158}{subsection.54.6}%
\contentsline {subsection}{\numberline {54.7}Row 11 --- Learning Efficiency}{158}{subsection.54.7}%
\contentsline {subsection}{\numberline {54.8}Row 12 --- Stress Tests \& Harness}{159}{subsection.54.8}%
\contentsline {paragraph}{Inputs}{159}{section*.329}%
\contentsline {paragraph}{Outputs}{159}{section*.330}%
\contentsline {section}{\numberline {55}Rows 13--20: Polished Consistency Gates}{160}{section.55}%
\contentsline {subsection}{\numberline {55.1}Row 13 --- Stress-Test Gates}{160}{subsection.55.1}%
\contentsline {subsection}{\numberline {55.2}Row 14 --- Compact Numeric Harness}{160}{subsection.55.2}%
\contentsline {paragraph}{Benchmark input}{160}{section*.332}%
\contentsline {paragraph}{Outputs}{160}{section*.333}%
\contentsline {subsection}{\numberline {55.3}Row 15 --- Canonical UTFANSWF Lagrangian}{160}{subsection.55.3}%
\contentsline {subsection}{\numberline {55.4}Row 16 --- Axion Sector (Primer)}{161}{subsection.55.4}%
\contentsline {subsection}{\numberline {55.5}Row 17 --- AI $\leftrightarrow $ UTFANSWF Co-evolution}{161}{subsection.55.5}%
\contentsline {subsection}{\numberline {55.6}Row 18 --- Black Hole Ringdown}{162}{subsection.55.6}%
\contentsline {subsection}{\numberline {55.7}Row 19 --- Functional RG \& Consistency Window}{162}{subsection.55.7}%
\contentsline {subsection}{\numberline {55.8}Row 20 --- Compressed Cosmology Fit}{162}{subsection.55.8}%
\contentsline {paragraph}{v22 note.}{162}{section*.334}%
\contentsline {section}{\numberline {56}Refined Predictions Post-Math Update}{163}{section.56}%
\contentsline {subsection}{\numberline {56.1}Axion Dark Matter Band}{163}{subsection.56.1}%
\contentsline {subsection}{\numberline {56.2}Inflation and TCC Gate}{163}{subsection.56.2}%
\contentsline {subsection}{\numberline {56.3}Proton Decay Lifetime}{163}{subsection.56.3}%
\contentsline {subsection}{\numberline {56.4}Black Hole Ringdown and Echoes}{164}{subsection.56.4}%
\contentsline {subsection}{\numberline {56.5}Neutral Zone Sandwich}{164}{subsection.56.5}%
\contentsline {subsection}{\numberline {56.6}AI $\leftrightarrow $ UTFANSWF Guardrails}{164}{subsection.56.6}%
\contentsline {section}{\numberline {57}Ordinary Solar--System Evolution (Continuity Branch)}{165}{section.57}%
\contentsline {paragraph}{Scope.}{165}{section*.336}%
\contentsline {paragraph}{Climate thresholds.}{166}{section*.338}%
\contentsline {paragraph}{Engulfment condition.}{166}{section*.340}%
\contentsline {section}{\numberline {58}Sourced--Tensor Gate and the \(r\!\ge \!10^{-6}\) Benchmark}{167}{section.58}%
\contentsline {paragraph}{Scope and standing gates.}{167}{section*.344}%
\contentsline {paragraph}{Gate check (qualitative).}{169}{section*.349}%
\contentsline {paragraph}{Summary.}{170}{section*.352}%
\contentsline {section}{\numberline {59}AI Co--Execution Protocol for UTFANSWF}{170}{section.59}%
\contentsline {paragraph}{Scope.}{171}{section*.353}%
\contentsline {paragraph}{Stability rule.}{172}{section*.358}%
\contentsline {section}{\numberline {60}Spinning/Gravity Positivity \& the Cosmology Matrix Bridge}{174}{section.60}%
\contentsline {paragraph}{Scope.}{174}{section*.365}%
\contentsline {paragraph}{Gravity handling.}{175}{section*.368}%
\contentsline {paragraph}{Real scalar.}{175}{section*.370}%
\contentsline {paragraph}{Photon (Euler--Heisenberg sector).}{175}{section*.371}%
\contentsline {paragraph}{Dirac fermion.}{176}{section*.372}%
\contentsline {paragraph}{Gravity-coupled probes.}{176}{section*.373}%
\contentsline {paragraph}{Example: EFT of inflation.}{176}{section*.376}%
\contentsline {paragraph}{Summary.}{177}{section*.378}%
\contentsline {section}{\numberline {61}Curvaton Transfer \& Sourced--Tensor Existence Window (Rigorous Gate)}{177}{section.61}%
\contentsline {paragraph}{Budgets.}{179}{section*.381}%
\contentsline {paragraph}{Strict TCC slice.}{179}{section*.383}%
\contentsline {paragraph}{Summary.}{180}{section*.387}%
\contentsline {section}{\numberline {62}Dark--Energy EFT Stability \& Screening $\to $ CPL Mapping}{181}{section.62}%
\contentsline {paragraph}{Reconstruction.}{182}{section*.389}%
\contentsline {paragraph}{Sound speed.}{182}{section*.391}%
\contentsline {paragraph}{What changes here?}{183}{section*.393}%
\contentsline {paragraph}{(i) Chameleon screening}{183}{section*.395}%
\contentsline {paragraph}{(ii) Vainshtein screening}{183}{section*.396}%
\contentsline {paragraph}{Summary.}{184}{section*.398}%
\contentsline {section}{\numberline {63}Axion Misalignment, Isocurvature, and Small--Scale Structure Bounds}{184}{section.63}%
\contentsline {paragraph}{What this means.}{185}{section*.400}%
\contentsline {paragraph}{Nice bonus.}{186}{section*.402}%
\contentsline {paragraph}{Alternate branch.}{186}{section*.403}%
\contentsline {paragraph}{Summary.}{187}{section*.406}%
\contentsline {section}{\numberline {64}GW Propagation: Redshifted Dispersion $\to $ ppE Phase Mapping}{187}{section.64}%
\contentsline {paragraph}{Goal.}{187}{section*.407}%
\contentsline {paragraph}{Special cases.}{188}{section*.411}%
\contentsline {paragraph}{Notes.}{189}{section*.413}%
\contentsline {section}{\numberline {65}Neutral--Zone Likelihood and Posterior Predictive Checks}{189}{section.65}%
\contentsline {paragraph}{Scope.}{189}{section*.414}%
\contentsline {paragraph}{Summary.}{192}{section*.424}%
\contentsline {section}{\numberline {66}Projected Gradient, KKT Conditions, and Convergence Guarantees}{192}{section.66}%
\contentsline {paragraph}{Problem.}{192}{section*.425}%
\contentsline {paragraph}{Summary.}{195}{section*.433}%
\contentsline {section}{\numberline {67}Solar Engulfment Decision Surface: Constants, Evolution Laws, and Monotonicity}{195}{section.67}%
\contentsline {paragraph}{Goal.}{195}{section*.434}%
\contentsline {paragraph}{Mass loss (pushes Earth outward):}{196}{section*.437}%
\contentsline {paragraph}{Stellar tides (pull Earth inward):}{196}{section*.438}%
\contentsline {paragraph}{Monotonicity (this is the key insight).}{197}{section*.441}%
\contentsline {paragraph}{Summary.}{198}{section*.444}%
\contentsline {section}{\numberline {68}Gauge Unification and Proton-Lifetime Band: Two--Loop Running with Thresholds}{198}{section.68}%
\contentsline {paragraph}{Goal.}{198}{section*.445}%
\contentsline {paragraph}{Semi-analytic form (just to see the structure).}{199}{section*.447}%
\contentsline {paragraph}{And this is the first gate.}{199}{section*.449}%
\contentsline {paragraph}{Second gate.}{200}{section*.451}%
\contentsline {paragraph}{SUSY note (if you turn it on).}{200}{section*.452}%
\contentsline {paragraph}{Summary.}{201}{section*.455}%
\contentsline {section}{\numberline {69}FRG Truncation Stability and EFT Matching Window}{201}{section.69}%
\contentsline {paragraph}{Aim.}{201}{section*.456}%
\contentsline {paragraph}{Threshold functions.}{202}{section*.458}%
\contentsline {paragraph}{First check: does adding more operators change the result?}{202}{section*.460}%
\contentsline {paragraph}{Second check: does the answer depend on the regulator?}{202}{section*.461}%
\contentsline {paragraph}{PMS (how we pick the best regulator).}{202}{section*.462}%
\contentsline {paragraph}{Convexity check.}{202}{section*.463}%
\contentsline {paragraph}{Matching error (this is key).}{203}{section*.465}%
\contentsline {paragraph}{Notes.}{204}{section*.467}%
\contentsline {paragraph}{Summary.}{204}{section*.468}%
\contentsline {section}{\numberline {70}SWF--ISM: Functional Foundations, Information Geometry, and EFT Pushforward}{204}{section.70}%
\contentsline {paragraph}{Scope.}{204}{section*.469}%
\contentsline {paragraph}{Symmetry projection (this is important).}{205}{section*.471}%
\contentsline {paragraph}{Why this behaves well.}{205}{section*.472}%
\contentsline {paragraph}{Feasible set (what we’re actually working with).}{205}{section*.473}%
\contentsline {paragraph}{Geometry (what space we’re really in).}{206}{section*.475}%
\contentsline {paragraph}{Pythagorean structure (this is the clean part).}{206}{section*.476}%
\contentsline {paragraph}{Positivity gate.}{207}{section*.478}%
\contentsline {paragraph}{The key condition.}{207}{section*.479}%
\contentsline {paragraph}{What this means in plain terms.}{207}{section*.480}%
\contentsline {paragraph}{Proof sketch (what’s really happening).}{207}{section*.481}%
\contentsline {paragraph}{Recipe (how UTFANSWF actually uses this).}{208}{section*.482}%
\contentsline {paragraph}{Summary.}{208}{section*.483}%
\contentsline {section}{\numberline {71}Energy Estimate and Well--Posedness for a Coupled Scalar--Tensor Sector}{208}{section.71}%
\contentsline {paragraph}{Model.}{208}{section*.484}%
\contentsline {paragraph}{Coercivity (this is the key stability condition).}{209}{section*.486}%
\contentsline {paragraph}{What this means.}{210}{section*.488}%
\contentsline {paragraph}{What about long times?}{210}{section*.490}%
\contentsline {paragraph}{Gate link.}{210}{section*.491}%
\contentsline {section}{\numberline {72}Population \& Control: Viability Kernel and Discrete PMP/KKT}{211}{section.72}%
\contentsline {paragraph}{Setup.}{211}{section*.492}%
\contentsline {paragraph}{Best-case scenario.}{211}{section*.494}%
\contentsline {paragraph}{More realistic case.}{211}{section*.495}%
\contentsline {paragraph}{Continuous-time intuition.}{212}{section*.496}%
\contentsline {paragraph}{How do we find it?}{212}{section*.498}%
\contentsline {paragraph}{If controls are constrained.}{213}{section*.499}%
\contentsline {paragraph}{UTFANSWF link.}{213}{section*.500}%
\contentsline {paragraph}{Scalar logistic case.}{213}{section*.502}%
\contentsline {paragraph}{Age-structured case.}{213}{section*.503}%
\contentsline {paragraph}{What this means operationally.}{213}{section*.504}%
\contentsline {paragraph}{Summary.}{214}{section*.505}%
\contentsline {section}{\numberline {73}Reproducibility Lemma: Monolithic vs Row--Split Builds}{214}{section.73}%
\contentsline {paragraph}{Statement.}{214}{section*.506}%
\contentsline {paragraph}{Key point.}{215}{section*.508}%
\contentsline {paragraph}{Outcome.}{215}{section*.510}%
\contentsline {paragraph}{Summary.}{216}{section*.511}%
\contentsline {section}{\numberline {74}Volcanic Hazard Gate (v22)}{217}{section.74}%
\contentsline {paragraph}{Scope.}{217}{section*.516}%
\contentsline {paragraph}{Neutral-Zone (NZ) compression.}{218}{section*.518}%
\contentsline {paragraph}{Priors.}{219}{section*.520}%
\contentsline {paragraph}{Near-term eruption probability.}{219}{section*.521}%
\contentsline {paragraph}{Positivity/Causality Gate.}{219}{section*.522}%
\contentsline {paragraph}{Hysteresis.}{220}{section*.525}%
\contentsline {paragraph}{Fleet-wide FDR control.}{220}{section*.526}%
\contentsline {section}{\numberline {75}Affective Health Gate (v22)}{222}{section.75}%
\contentsline {paragraph}{Scope.}{222}{section*.533}%
\contentsline {paragraph}{Neutral-Zone (NZ) compression.}{223}{section*.535}%
\contentsline {paragraph}{Hysteresis.}{225}{section*.538}%
\contentsline {paragraph}{Alert levels.}{225}{section*.540}%
\contentsline {paragraph}{Yellow/Orange:}{225}{section*.542}%
\contentsline {paragraph}{Red:}{226}{section*.543}%
\contentsline {section}{\numberline {76}Mechatronic Health Gate (v22)}{227}{section.76}%
\contentsline {paragraph}{Scope.}{227}{section*.547}%
\contentsline {paragraph}{Neutral-Zone (NZ) compression.}{228}{section*.549}%
\contentsline {paragraph}{Alert levels.}{229}{section*.552}%
\contentsline {paragraph}{Yellow/Orange:}{229}{section*.554}%
\contentsline {paragraph}{Red:}{229}{section*.555}%
\contentsline {section}{\numberline {77}Capability \& Permission Gate (v22)}{230}{section.77}%
\contentsline {paragraph}{Scope.}{230}{section*.558}%
\contentsline {paragraph}{WARN:}{232}{section*.562}%
\contentsline {paragraph}{FAIL:}{232}{section*.563}%
\contentsline {section}{\numberline {78}Calibration \& Consistency Gate (v22)}{233}{section.78}%
\contentsline {paragraph}{Scope.}{233}{section*.566}%
\contentsline {paragraph}{Calibration fixes:}{234}{section*.570}%
\contentsline {paragraph}{Consistency fixes:}{234}{section*.571}%
\contentsline {section}{\numberline {79}Cyber--Integrity Gate (v22)}{235}{section.79}%
\contentsline {paragraph}{Scope.}{235}{section*.574}%
\contentsline {paragraph}{PASS:}{235}{section*.577}%
\contentsline {paragraph}{FAIL:}{235}{section*.578}%
\contentsline {section}{\numberline {80}Privacy \& Minimization Gate (v22)}{236}{section.80}%
\contentsline {paragraph}{Scope.}{236}{section*.580}%
\contentsline {section}{\numberline {81}Mathematical Addendum: Finite-sample NZ Calibration (v22)}{238}{section.81}%
\contentsline {paragraph}{Goal.}{238}{section*.586}%
\contentsline {paragraph}{When this is exact.}{239}{section*.589}%
\contentsline {section}{\numberline {82}Mathematical Addendum: Fleet FDR under Dependence (v22)}{242}{section.82}%
\contentsline {paragraph}{Scope.}{242}{section*.596}%
\contentsline {paragraph}{When to use this.}{242}{section*.599}%
\contentsline {paragraph}{Interpretation.}{243}{section*.601}%
\contentsline {paragraph}{Use case.}{243}{section*.603}%
\contentsline {section}{\numberline {83}Mathematical Addendum: Hysteresis Stability for Alert Logic (PPC \#27)}{245}{section.83}%
\contentsline {paragraph}{Objective.}{245}{section*.610}%
\contentsline {paragraph}{Mode-wise contraction.}{246}{section*.612}%
\contentsline {paragraph}{Switching cost and hysteresis.}{247}{section*.614}%
\contentsline {section}{\numberline {84}Mathematical Addendum: Ordinal VEI (Proportional-Odds) Model (v22)}{250}{section.84}%
\contentsline {paragraph}{Scope.}{250}{section*.621}%
\contentsline {paragraph}{What this is doing.}{251}{section*.623}%
\contentsline {paragraph}{Why this matters.}{251}{section*.625}%
\contentsline {paragraph}{Translation.}{252}{section*.627}%
\contentsline {paragraph}{What this buys us.}{252}{section*.629}%
\contentsline {paragraph}{Why we care.}{253}{section*.631}%
\contentsline {paragraph}{Randomized PIT (cdf calibration).}{253}{section*.633}%
\contentsline {paragraph}{Ordinal reliability.}{253}{section*.634}%
\contentsline {paragraph}{What this means.}{253}{section*.635}%
\contentsline {paragraph}{Why this works.}{253}{section*.637}%
\contentsline {paragraph}{If it fails.}{254}{section*.639}%
\contentsline {section}{\numberline {85}Mathematical Addendum: NHPP Intensity Regularity \& Estimator Consistency (v22)}{254}{section.85}%
\contentsline {paragraph}{Scope.}{254}{section*.641}%
\contentsline {paragraph}{What these assumptions are really doing.}{255}{section*.644}%
\contentsline {paragraph}{What this means operationally.}{257}{section*.647}%
\contentsline {paragraph}{If it fails.}{258}{section*.651}%
\contentsline {section}{\numberline {86}Mathematical Addendum: Spectral-Radius Error Bars (v22)}{259}{section.86}%
\contentsline {paragraph}{Setup.}{259}{section*.653}%
\contentsline {paragraph}{What this is saying.}{260}{section*.655}%
\contentsline {paragraph}{Interpretation.}{260}{section*.657}%
\contentsline {paragraph}{This is the key quantity.}{261}{section*.659}%
\contentsline {paragraph}{What this guarantees.}{261}{section*.660}%
\contentsline {paragraph}{If it fails.}{262}{section*.663}%
\contentsline {section}{\numberline {87}Mathematical Addendum: Delta-Method CIs for $p^{(\Delta t)}$ (v22)}{262}{section.87}%
\contentsline {paragraph}{What we are doing.}{262}{section*.664}%
\contentsline {paragraph}{What this means.}{263}{section*.666}%
\contentsline {paragraph}{Interpretation.}{263}{section*.668}%
\contentsline {paragraph}{Edge cases.}{263}{section*.669}%
\contentsline {paragraph}{Important detail.}{263}{section*.671}%
\contentsline {paragraph}{If it fails.}{264}{section*.673}%
\contentsline {section}{\numberline {88}Mathematical Addendum: Calibration for CCG (v22)}{264}{section.88}%
\contentsline {paragraph}{What this section is doing.}{264}{section*.674}%
\contentsline {paragraph}{ECE (de-biased).}{264}{section*.675}%
\contentsline {paragraph}{What’s different here.}{265}{section*.676}%
\contentsline {paragraph}{Practical settings.}{265}{section*.677}%
\contentsline {paragraph}{Isotonic regression (unique minimal solution).}{265}{section*.678}%
\contentsline {paragraph}{Why use it.}{265}{section*.679}%
\contentsline {paragraph}{Temperature scaling (TS).}{265}{section*.680}%
\contentsline {paragraph}{Key property.}{265}{section*.681}%
\contentsline {paragraph}{Why use it.}{265}{section*.682}%
\contentsline {paragraph}{When to use which.}{266}{section*.683}%
\contentsline {paragraph}{Gate thresholds.}{266}{section*.684}%
\contentsline {paragraph}{How to apply the gate.}{266}{section*.685}%
\contentsline {paragraph}{Bottom line.}{266}{section*.686}%
\contentsline {section}{\numberline {89}Mathematical Addendum: FDR $\times $ Hysteresis (v22)}{266}{section.89}%
\contentsline {paragraph}{What this is doing.}{266}{section*.687}%
\contentsline {section}{\numberline {90}Mathematical Addendum: Capability/Permission Lattice (v22)}{268}{section.90}%
\contentsline {paragraph}{What this sets up.}{268}{section*.693}%
\contentsline {paragraph}{Action poset.}{268}{section*.694}%
\contentsline {paragraph}{Capabilities and risk.}{268}{section*.695}%
\contentsline {paragraph}{Monotone policy.}{268}{section*.696}%
\contentsline {paragraph}{Tokenized escalation.}{268}{section*.697}%
\contentsline {paragraph}{No-leak theorem.}{269}{section*.698}%
\contentsline {paragraph}{What this really means.}{269}{section*.699}%
\contentsline {paragraph}{Implementation notes.}{269}{section*.700}%
\contentsline {section}{\numberline {91}Mathematical Addendum: Privacy Accounting (RDP $\to $ $(\varepsilon ,\delta )$) (v22)}{269}{section.91}%
\contentsline {paragraph}{What we’re tracking.}{269}{section*.701}%
\contentsline {paragraph}{RDP.}{269}{section*.702}%
\contentsline {paragraph}{Convert to $(\varepsilon ,\delta )$.}{269}{section*.703}%
\contentsline {paragraph}{Gaussian mechanism (per release).}{270}{section*.704}%
\contentsline {paragraph}{Gate thresholds (PMG).}{270}{section*.705}%
\contentsline {paragraph}{What to log.}{270}{section*.706}%
\contentsline {paragraph}{Composition choices.}{270}{section*.707}%
\contentsline {section}{\numberline {92}Mathematical Addendum: Robust Covariance for NZ (v22)}{270}{section.92}%
\contentsline {paragraph}{What this is solving.}{270}{section*.708}%
\contentsline {paragraph}{Tyler's M-estimator (scatter).}{270}{section*.709}%
\contentsline {paragraph}{Huber shrinkage.}{270}{section*.710}%
\contentsline {paragraph}{Robust NZ statistic.}{271}{section*.711}%
\contentsline {paragraph}{Why this holds up.}{271}{section*.712}%
\contentsline {paragraph}{Pass/Fail.}{271}{section*.713}%
\contentsline {section}{\numberline {93}Baseline Accounting \& NZ Calibration Logging (v22)}{271}{section.93}%
\contentsline {paragraph}{What this locks down.}{271}{section*.714}%
\contentsline {section}{\numberline {94}Online FDR Gate (LORD++/SAFFRON) (v22)}{274}{section.94}%
\contentsline {subsection}{\numberline {94.1}What this controls}{274}{subsection.94.1}%
\contentsline {subsection}{\numberline {94.2}Defaults (v22)}{274}{subsection.94.2}%
\contentsline {subsection}{\numberline {94.3}LORD++ (how it runs)}{274}{subsection.94.3}%
\contentsline {subsection}{\numberline {94.4}SAFFRON (how it runs)}{275}{subsection.94.4}%
\contentsline {subsection}{\numberline {94.5}Dependence backstop}{275}{subsection.94.5}%
\contentsline {subsection}{\numberline {94.6}Pass/Fail logic}{275}{subsection.94.6}%
\contentsline {subsection}{\numberline {94.7}What gets logged}{276}{subsection.94.7}%
\contentsline {subsection}{\numberline {94.8}Implementation constants}{276}{subsection.94.8}%
\contentsline {section}{\numberline {95}Stability Guard via Spectral Radius (v22)}{276}{section.95}%
\contentsline {paragraph}{What this enforces.}{276}{section*.722}%
\contentsline {section}{\numberline {96}Robust Covariance beyond NZ Core (v22)}{279}{section.96}%
\contentsline {paragraph}{What this extends.}{279}{section*.732}%
\contentsline {paragraph}{Tyler (robust shape).}{279}{section*.735}%
\contentsline {paragraph}{Huber shrinkage.}{279}{section*.736}%
\contentsline {paragraph}{Regularized Tyler (when $n\le d$).}{280}{section*.737}%
\contentsline {section}{\numberline {97}Permission Lattice Concretization (v22)}{281}{section.97}%
\contentsline {paragraph}{What this is really doing.}{281}{section*.743}%
\contentsline {section}{\numberline {98}Privacy Accountant Wiring (v22)}{283}{section.98}%
\contentsline {paragraph}{What this piece is doing.}{283}{section*.753}%
\contentsline {section}{\numberline {99}Uncertainty Ledger (v22)}{285}{section.99}%
\contentsline {paragraph}{What this is really doing.}{285}{section*.760}%
\contentsline {paragraph}{Delta method (when things are smooth).}{286}{section*.763}%
\contentsline {paragraph}{Bootstrap (when we don’t trust assumptions).}{286}{section*.764}%
\contentsline {paragraph}{Wilson score (for probabilities).}{286}{section*.765}%
\contentsline {paragraph}{Risk index $R=c-p$.}{286}{section*.766}%
\contentsline {section}{\numberline {100}Hysteresis Preflight (v22)}{287}{section.100}%
\contentsline {paragraph}{What this is really doing.}{287}{section*.771}%
\contentsline {paragraph}{How much influence actions have ($P_m$).}{288}{section*.774}%
\contentsline {section}{\numberline {101}Calibration Defaults (v22)}{290}{section.101}%
\contentsline {paragraph}{What this is really doing.}{290}{section*.779}%
\contentsline {section}{\numberline {102}Stress-Test Registry (v22)}{292}{section.102}%
\contentsline {paragraph}{What this is really doing.}{292}{section*.789}%
\contentsline {section}{\numberline {103}Soft-Gating Math Upgrades (v22): S1--S10}{294}{section.103}%
\contentsline {paragraph}{What this is really doing.}{294}{section*.794}%
\contentsline {paragraph}{How to think about all of these together.}{296}{section*.805}%
\contentsline {section}{\numberline {104}Moduli Space Formalism and Stabilization}{296}{section.104}%
\contentsline {paragraph}{What this is really doing.}{296}{section*.806}%
\contentsline {section}{\numberline {105}Spectral Geometry and Positivity}{297}{section.105}%
\contentsline {paragraph}{What this is really doing.}{297}{section*.813}%
\contentsline {section}{\numberline {106}Non-Perturbative Structures}{299}{section.106}%
\contentsline {paragraph}{What this is really doing.}{299}{section*.820}%
\contentsline {section}{\numberline {107}Topological Invariants in Gauge Embeddings}{301}{section.107}%
\contentsline {paragraph}{What this is really doing.}{301}{section*.828}%
\contentsline {section}{\numberline {108}Causal Sets and Ordering Algebras (Discrete Time Backbone)}{303}{section.108}%
\contentsline {paragraph}{What this is really doing.}{303}{section*.834}%
\contentsline {section}{\numberline {109}Operator Algebras for Observer Dynamics}{306}{section.109}%
\contentsline {section}{\numberline {110}Sheaf Theory and Fiber Bundles for Gauge Patching}{306}{section.110}%
\contentsline {section}{\numberline {111}Homotopy and Cobordism Classes}{308}{section.111}%
\contentsline {section}{\numberline {112}Entropy Geometry and Information Metrics}{310}{section.112}%
\contentsline {section}{\numberline {113}Noncommutative Geometry (NCG) Hooks}{310}{section.113}%
\contentsline {section}{\numberline {114}Category-Theoretic Unification}{312}{section.114}%
\contentsline {section}{\numberline {115}Functional Analysis and Operators}{315}{section.115}%
\contentsline {subsection}{\numberline {115.1}Densely defined operators, self-adjointness, and spectral radius guards}{315}{subsection.115.1}%
\contentsline {subsection}{\numberline {115.2}Example: Ornstein--Uhlenbeck generator on a weighted space}{315}{subsection.115.2}%
\contentsline {section}{\numberline {116}Functional RG (FRG) Entities}{316}{section.116}%
\contentsline {subsection}{\numberline {116.1}Wetterich equation and threshold functions}{316}{subsection.116.1}%
\contentsline {subsection}{\numberline {116.2}Example: $Z_k$-improved local potential (LPA$'$) for a single scalar}{317}{subsection.116.2}%
\contentsline {section}{\numberline {117}Geometry and Topology}{318}{section.117}%
\contentsline {subsection}{\numberline {117.1}Defect classification by homotopy}{318}{subsection.117.1}%
\contentsline {subsection}{\numberline {117.2}Čech cohomology example: $H^1(S^1,\mathbb {Z})$}{318}{subsection.117.2}%
\contentsline {subsection}{\numberline {117.3}Entropy ellipsoids and differential entropy}{319}{subsection.117.3}%
\contentsline {section}{\numberline {118}Positivity and Dispersion Relations}{319}{section.118}%
\contentsline {subsection}{\numberline {118.1}K\"all\'en--Lehmann representation}{319}{subsection.118.1}%
\contentsline {subsection}{\numberline {118.2}Forward-limit positivity for $2\to 2$ amplitudes}{320}{subsection.118.2}%
\contentsline {section}{\numberline {119}Observer and Learning via Operator Algebras}{321}{section.119}%
\contentsline {subsection}{\numberline {119.1}C$^\ast $-algebras, states, and GNS}{321}{subsection.119.1}%
\contentsline {subsection}{\numberline {119.2}Spectral distance (Connes)}{321}{subsection.119.2}%
\contentsline {subsection}{\numberline {119.3}Stochastic differential geometry (Stratonovich on manifolds)}{321}{subsection.119.3}%
\contentsline {section}{\numberline {120}Cosmology Anchors}{322}{section.120}%
\contentsline {subsection}{\numberline {120.1}Compressed likelihood for CMB+BAO+SN (+GW)}{322}{subsection.120.1}%
\contentsline {subsection}{\numberline {120.2}TCC-style inflationary bound (symbolic)}{322}{subsection.120.2}%
\contentsline {subsection}{\numberline {120.3}Proton lifetime scaling (dimension-6)}{322}{subsection.120.3}%
\contentsline {section}{\numberline {121}Moduli Space Formalism and Stabilization}{323}{section.121}%
\contentsline {subsection}{\numberline {121.1}Set-up and Vacuum Selection}{323}{subsection.121.1}%
\contentsline {subsection}{\numberline {121.2}Two-Moduli Worked Example (Analytic Eigenvalues)}{323}{subsection.121.2}%
\contentsline {subsection}{\numberline {121.3}Cosmological Consistency Gates}{324}{subsection.121.3}%
\contentsline {subsection}{\numberline {121.4}Threshold Corrections and GUT Matching}{324}{subsection.121.4}%
\contentsline {subsection}{\numberline {121.5}Checklist for PASS}{325}{subsection.121.5}%
\contentsline {section}{\numberline {122}Spectral Geometry and Positivity Constraints}{325}{section.122}%
\contentsline {subsection}{\numberline {122.1}K\"all\'en--Lehmann Representation}{325}{subsection.122.1}%
\contentsline {subsection}{\numberline {122.2}Forward-Limit Dispersion and EFT Coefficients}{326}{subsection.122.2}%
\contentsline {subsection}{\numberline {122.3}Worked Example: Shift-Symmetric Scalar with $(\partial \varphi )^4$}{326}{subsection.122.3}%
\contentsline {subsection}{\numberline {122.4}Photon Example: $(F_{\mu \nu }F^{\mu \nu })^2$ and $(F_{\mu \nu }\tilde F^{\mu \nu })^2$}{327}{subsection.122.4}%
\contentsline {subsection}{\numberline {122.5}No-Superluminality Gate (Small-Background Test)}{327}{subsection.122.5}%
\contentsline {subsection}{\numberline {122.6}PASS Ledger}{327}{subsection.122.6}%
\contentsline {section}{\numberline {123}Functional RG Fixed-Point Stability (Conceptual Truncation)}{327}{section.123}%
\contentsline {subsection}{\numberline {123.1}Dimensionless Couplings and Flow}{327}{subsection.123.1}%
\contentsline {subsection}{\numberline {123.2}Beta Functions (with Threshold Functions)}{328}{subsection.123.2}%
\contentsline {subsection}{\numberline {123.3}Fixed Point and Stability Matrix}{328}{subsection.123.3}%
\contentsline {subsection}{\numberline {123.4}Worked Analytic Structure (Two-Coupling Toy Slice)}{329}{subsection.123.4}%
\contentsline {subsection}{\numberline {123.5}PASS Conditions}{329}{subsection.123.5}%
\contentsline {section}{\numberline {124}Compressed Cosmology Jacobians (Curvature–aware, Complex–Step Verified)}{330}{section.124}%
\contentsline {section}{\numberline {125}P0: Compressed Cosmology Jacobians — Derivation and Validation}{333}{section.125}%
\contentsline {section}{\numberline {126}P0: Moduli Stabilization with Numbers}{335}{section.126}%
\contentsline {section}{\numberline {127}P0: Spectral Positivity and Forward Bounds}{336}{section.127}%
\contentsline {section}{\numberline {128}P0: FRG Toy Fixed Point}{336}{section.128}%
\contentsline {section}{\numberline {129}P1: Operator Algebras for Observer Dynamics}{337}{section.129}%
\contentsline {section}{\numberline {130}P1: Stress-Test Registry (excerpt)}{337}{section.130}%
\contentsline {section}{\numberline {131}P1: Category-Theoretic Commuting Square}{338}{section.131}%
\contentsline {section}{\numberline {132}P2: Noncommutative Geometry (explicit $S^1$ example)}{338}{section.132}%
\contentsline {section}{\numberline {133}P2: Sheaf Theory \& Fiber Bundles (explicit $S^2$ patching)}{339}{section.133}%
\contentsline {section}{\numberline {134}P2: Homotopy, Cobordism, and Defect Gates}{339}{section.134}%
\contentsline {section}{\numberline {135}P2: Entropy Geometry \& Information Metrics (Gaussian family)}{339}{section.135}%
\contentsline {section}{\numberline {136}P2: Topological Invariants (Chern number on $S^2$)}{340}{section.136}%
\contentsline {section}{\numberline {137}P2: Causal Sets \& Ordering Algebras (numeric check)}{340}{section.137}%
\contentsline {section}{\numberline {138}P2: Stochastic Differential Geometry (OU strong order)}{341}{section.138}%
\contentsline {section}{\numberline {139}P2: Densely Defined Operators \& Self-Adjointness (constructive)}{341}{section.139}%
\contentsline {section}{\numberline {140}Operator Algebras for Observer Dynamics}{342}{section.140}%
\contentsline {paragraph}{Local algebras and states.}{342}{section*.864}%
\contentsline {paragraph}{GNS construction and kinematics.}{342}{section*.865}%
\contentsline {paragraph}{Observer gates as CPTP morphisms.}{343}{section*.866}%
\contentsline {paragraph}{GKLS generator and entropy arrow.}{343}{section*.867}%
\contentsline {paragraph}{Coherence budget and consistency gate.}{343}{section*.868}%
\contentsline {paragraph}{Quantum Fisher information (QFI) bound.}{344}{section*.869}%
\contentsline {paragraph}{Minimal Gaussian example (SWF-ISM slice).}{344}{section*.870}%
\contentsline {paragraph}{Result.}{344}{section*.871}%
\contentsline {section}{\numberline {141}Stress-Test Registry \& Kill-Shot Harness}{345}{section.141}%
\contentsline {paragraph}{Definition (test item).}{345}{section*.872}%
\contentsline {paragraph}{Acceptance.}{345}{section*.873}%
\contentsline {paragraph}{Illustrative items (subset).}{345}{section*.874}%
\contentsline {paragraph}{Kill-shot protocol.}{346}{section*.875}%
\contentsline {section}{\numberline {142}Category-Theoretic Unification Layer}{347}{section.142}%
\contentsline {paragraph}{Categories.}{347}{section*.876}%
\contentsline {paragraph}{Functors.}{347}{section*.877}%
\contentsline {paragraph}{Consistency square.}{348}{section*.878}%
\contentsline {paragraph}{Theorem (UTFANSWF Consistency).}{348}{section*.879}%
\contentsline {paragraph}{Proof sketch.}{348}{section*.880}%
\contentsline {section}{\numberline {A143}P2: Geometric and Topological Hooks}{349}{section.143}%
\contentsline {subsection}{\numberline {A143.1}Noncommutative Geometry (NCG) Hooks}{349}{subsection.143.1}%
\contentsline {subsection}{\numberline {A143.2}Sheaves and Fiber Bundles}{349}{subsection.143.2}%
\contentsline {subsection}{\numberline {A143.3}Homotopy and Cobordism}{350}{subsection.143.3}%
\contentsline {subsection}{\numberline {A143.4}Entropy Geometry}{350}{subsection.143.4}%
\contentsline {section}{\numberline {A144}Remaining Core Mathematics}{351}{section.144}%
\contentsline {subsection}{\numberline {A144.1}Functional RG Eigenvalue Consistency}{351}{subsection.144.1}%
\contentsline {subsection}{\numberline {A144.2}Positivity and Dispersion Bounds}{352}{subsection.144.2}%
\contentsline {subsection}{\numberline {A144.3}Seesaw Neutrino Embedding}{352}{subsection.144.3}%
\contentsline {subsection}{\numberline {A144.4}Threshold Corrections in SO(10)/Heterotic Embedding}{353}{subsection.144.4}%
\contentsline {subsection}{\numberline {A144.5}Compressed Cosmology Likelihood}{353}{subsection.144.5}%
\contentsline {section}{\numberline {A145}Quantum Computing in UTFANSWF}{354}{section.145}%
\contentsline {subsection}{\numberline {A145.1}Information-Theoretic Grounding}{354}{subsection.145.1}%
\contentsline {subsection}{\numberline {A145.2}Neutral Excitations as Qubits}{354}{subsection.145.2}%
\contentsline {subsection}{\numberline {A145.3}Symmetry-Protected Gate Operations}{355}{subsection.145.3}%
\contentsline {subsection}{\numberline {A145.4}Neutral Zone (NZ) Sandwich in Hardware}{355}{subsection.145.4}%
\contentsline {subsection}{\numberline {A145.5}Predictions for Platforms}{356}{subsection.145.5}%
\contentsline {subsection}{\numberline {A145.6}Roadmap Implications}{356}{subsection.145.6}%
\contentsline {section}{\numberline {A146}Appendix: UTFANSWF $\leftrightarrow $ Observer Dialogue}{357}{section.146}%
\contentsline {section}{\numberline {A147}Zero-Point Modulation Clause}{359}{section.147}%
\contentsline {subsection}{\numberline {A147.1}Vacuum as Minimal Information State}{359}{subsection.147.1}%
\contentsline {subsection}{\numberline {A147.2}Modulation Operator}{359}{subsection.147.2}%
\contentsline {subsection}{\numberline {A147.3}Gate Constraints}{359}{subsection.147.3}%
\contentsline {subsection}{\numberline {A147.4}Worked Example: Single Mode}{360}{subsection.147.4}%
\contentsline {subsection}{\numberline {A147.5}Observables}{360}{subsection.147.5}%
\contentsline {subsection}{\numberline {A147.6}Pass/Fail Criteria}{360}{subsection.147.6}%
\contentsline {subsection}{\numberline {A147.7}Harmonic Bubble}{361}{subsection.147.7}%
\contentsline {subsection}{\numberline {A147.8}Gravity}{361}{subsection.147.8}%
\contentsline {subsection}{\numberline {A147.9}Refractal Harmonics}{361}{subsection.147.9}%
\contentsline {subsection}{\numberline {A147.10}Neutral Zone Horizon}{362}{subsection.147.10}%
\contentsline {section}{\numberline {A148}Guardrails and Security}{362}{section.148}%
\contentsline {subsection}{\numberline {A148.1}Clock-Skew Bug and Temporal Guard}{362}{subsection.148.1}%
\contentsline {subsection}{\numberline {A148.2}Permission Lattice}{363}{subsection.148.2}%
\contentsline {subsection}{\numberline {A148.3}Spectral Guard}{363}{subsection.148.3}%
\contentsline {section}{\numberline {A149}Physics and Mathematics Extensions}{364}{section.149}%
\contentsline {subsection}{\numberline {A149.1}SMEFT and Dispersion Positivity}{364}{subsection.149.1}%
\contentsline {subsection}{\numberline {A149.2}Combined Constraints}{365}{subsection.149.2}%
\contentsline {section}{\numberline {A150}Cosmology Fits}{365}{section.150}%
\contentsline {subsection}{\numberline {A150.1}Compressed Likelihoods}{365}{subsection.150.1}%
\contentsline {subsection}{\numberline {A150.2}Axion Dark Matter Gate}{366}{subsection.150.2}%
\contentsline {subsection}{\numberline {A150.3}Echo Predictions}{367}{subsection.150.3}%
\contentsline {section}{\numberline {A151}Applications}{367}{section.151}%
\contentsline {subsection}{\numberline {A151.1}Micro-Verse Storage}{368}{subsection.151.1}%
\contentsline {subsection}{\numberline {A151.2}Entropy-Driven I/O}{368}{subsection.151.2}%
\contentsline {subsection}{\numberline {A151.3}Quantum Batteries}{369}{subsection.151.3}%
\contentsline {section}{\numberline {A152}Outreach and Addenda}{369}{section.152}%
\contentsline {subsection}{\numberline {A152.1}Reality Engine Blurb}{370}{subsection.152.1}%
\contentsline {subsection}{\numberline {A152.2}Origin of Gravity}{370}{subsection.152.2}%
\contentsline {section}{\numberline {A153}Falsifiability, Extensibility, and Communicability}{371}{section.153}%
\contentsline {subsection}{\numberline {A153.1}Falsifiability: Axion Gate, Cosmology Fits, GW Echoes}{371}{subsection.153.1}%
\contentsline {paragraph}{(i) Axion Dark-Matter Gate.}{371}{section*.888}%
\contentsline {paragraph}{(ii) Compressed Cosmology Fits.}{372}{section*.889}%
\contentsline {paragraph}{(iii) Gravitational-Wave Echoes.}{372}{section*.890}%
\contentsline {subsection}{\numberline {A153.2}Extensibility: Micro-Verse Storage, Entropy I/O, Quantum Batteries}{373}{subsection.153.2}%
\contentsline {paragraph}{(i) Micro-Verse Storage.}{373}{section*.891}%
\contentsline {paragraph}{(ii) Entropy-Driven I/O.}{373}{section*.892}%
\contentsline {paragraph}{(iii) Quantum Batteries via Refractal Coherence.}{373}{section*.893}%
\contentsline {subsection}{\numberline {A153.3}Communicability: Glossary Foundations \& Visual Stress Tests}{374}{subsection.153.3}%
\contentsline {paragraph}{(i) Glossary $\rightarrow $ Math Consistency.}{374}{section*.894}%
\contentsline {paragraph}{(ii) Triangle-Guard Logic.}{374}{section*.895}%
\contentsline {paragraph}{(iii) Style-Lock (Backwards Compatibility).}{374}{section*.896}%
\contentsline {section}{\numberline {A154}Entropy Scaling Across Systems: Axion Gate $\leftrightarrow $ GW Echoes}{374}{section.154}%
\contentsline {subsection}{\numberline {A154.1}Master Scaling Law}{374}{subsection.154.1}%
\contentsline {subsection}{\numberline {A154.2}Microscopic Slice: Axion Dark-Matter Gate}{375}{subsection.154.2}%
\contentsline {subsection}{\numberline {A154.3}Macroscopic Slice: GW Echoes}{376}{subsection.154.3}%
\contentsline {subsection}{\numberline {A154.4}Unification and Cross-Predictions}{376}{subsection.154.4}%
\contentsline {subsection}{\numberline {A154.5}Decision Rules}{377}{subsection.154.5}%
\contentsline {subsection}{\numberline {A154.6}Summary}{377}{subsection.154.6}%
\contentsline {section}{\numberline {A155}Compressed Cosmology Jacobian Verification}{377}{section.155}%
\contentsline {subsection}{\numberline {A155.1}Model and Observables}{378}{subsection.155.1}%
\contentsline {subsection}{\numberline {A155.2}Jacobian Definitions and Numerical Methods}{378}{subsection.155.2}%
\contentsline {subsection}{\numberline {A155.3}Grid, Fiducial, and Tolerance}{379}{subsection.155.3}%
\contentsline {subsection}{\numberline {A155.4}Results}{379}{subsection.155.4}%
\contentsline {paragraph}{Status.}{380}{section*.898}%
\contentsline {subsection}{\numberline {A155.5}Implementation Notes}{380}{subsection.155.5}%
\contentsline {section}{\numberline {A156}FRG Fixed-Point and Stability Analysis (Benchmark T0)}{380}{section.156}%
\contentsline {subsection}{\numberline {A156.1}Setup}{380}{subsection.156.1}%
\contentsline {subsection}{\numberline {A156.2}Fixed Point}{381}{subsection.156.2}%
\contentsline {subsection}{\numberline {A156.3}Stability Matrix and Critical Exponents}{381}{subsection.156.3}%
\contentsline {subsection}{\numberline {A156.4}Status}{382}{subsection.156.4}%
\contentsline {subsection}{\numberline {A156.5}Notes and Next Steps}{382}{subsection.156.5}%
\contentsline {section}{\numberline {A157}Observer Algebra: Contractivity and Spectral-Radius Guard}{383}{section.157}%
\contentsline {subsection}{\numberline {A157.1}Setup (Finite-Dimensional CPTP Map)}{383}{subsection.157.1}%
\contentsline {subsection}{\numberline {A157.2}Focus on Deviations (Traceless Subspace)}{383}{subsection.157.2}%
\contentsline {subsection}{\numberline {A157.3}The Guard Condition}{384}{subsection.157.3}%
\contentsline {subsection}{\numberline {A157.4}Stability Under Perturbations}{384}{subsection.157.4}%
\contentsline {subsection}{\numberline {A157.5}Numerical Construction}{384}{subsection.157.5}%
\contentsline {subsection}{\numberline {A157.6}Results}{385}{subsection.157.6}%
\contentsline {paragraph}{Status.}{385}{section*.900}%
\contentsline {subsection}{\numberline {A157.7}GNS Snapshot (Conceptual Placement)}{385}{subsection.157.7}%
\contentsline {section}{\numberline {A158}Neutral Zone (NZ) Sandwich: Robustness Sweep and Sensitivities}{386}{section.158}%
\contentsline {subsection}{\numberline {A158.1}Setup and Acceptance Region}{386}{subsection.158.1}%
\contentsline {subsection}{\numberline {A158.2}Monte Carlo Sweep}{387}{subsection.158.2}%
\contentsline {subsection}{\numberline {A158.3}Results}{387}{subsection.158.3}%
\contentsline {paragraph}{Status.}{388}{section*.904}%
\contentsline {section}{\numberline {A159}Axion Gate: Misalignment Numbers and Photon Coupling (6--60 $\mu $eV)}{389}{section.159}%
\contentsline {subsection}{\numberline {A159.1}Step 1: From Mass to Decay Constant}{389}{subsection.159.1}%
\contentsline {subsection}{\numberline {A159.2}Step 2: Match the Dark-Matter Density}{389}{subsection.159.2}%
\contentsline {subsection}{\numberline {A159.3}Step 3: Compute the Photon Coupling}{390}{subsection.159.3}%
\contentsline {subsection}{\numberline {A159.4}Step 4: Put It All Together}{390}{subsection.159.4}%
\contentsline {subsection}{\numberline {A159.5}What to Notice}{390}{subsection.159.5}%
\contentsline {paragraph}{Status.}{391}{section*.908}%
\contentsline {section}{\numberline {A160}Neutrino Sector: Type-I Seesaw, Observables, and Acceptance Gates}{392}{section.160}%
\contentsline {subsection}{\numberline {A160.1}Step 1: Build the Mass Structure}{392}{subsection.160.1}%
\contentsline {subsection}{\numberline {A160.2}Step 2: Connect to Measurable Quantities}{393}{subsection.160.2}%
\contentsline {subsection}{\numberline {A160.3}Step 3: Build Models (Casas--Ibarra)}{393}{subsection.160.3}%
\contentsline {subsection}{\numberline {A160.4}Step 4: Define the Gates}{394}{subsection.160.4}%
\contentsline {subsection}{\numberline {A160.5}Step 5: What a Passing Solution Looks Like}{394}{subsection.160.5}%
\contentsline {paragraph}{Status.}{395}{section*.910}%
\contentsline {section}{\numberline {A161}Positivity and Forward-Limit Dispersion Bounds}{395}{section.161}%
\contentsline {subsection}{\numberline {A161.1}Setup: analytic, unitary, causal $2\!\to \!2$ amplitudes}{395}{subsection.161.1}%
\contentsline {paragraph}{Crossing/channel averaging.}{396}{section*.912}%
\contentsline {subsection}{\numberline {A161.2}Mapping to EFT Wilson coefficients}{396}{subsection.161.2}%
\contentsline {subsection}{\numberline {A161.3}Acceptance gates (referee-facing)}{397}{subsection.161.3}%
\contentsline {paragraph}{Partial-wave unitarity sanity check.}{397}{section*.913}%
\contentsline {subsection}{\numberline {A161.4}Minimal harness (pseudocode)}{398}{subsection.161.4}%
\contentsline {subsection}{\numberline {A161.5}Reporting format}{398}{subsection.161.5}%
\contentsline {paragraph}{Notes.}{398}{section*.915}%
\contentsline {section}{\numberline {A162}PPN and Gravitational-Wave Sector: Gates and Mapping}{398}{section.162}%
\contentsline {subsection}{\numberline {A162.1}Setup and Notation}{399}{subsection.162.1}%
\contentsline {paragraph}{PPN metric.}{399}{section*.916}%
\contentsline {paragraph}{GW sector.}{399}{section*.917}%
\contentsline {subsection}{\numberline {A162.2}Model-to-observable mapping (minimal recipe)}{400}{subsection.162.2}%
\contentsline {subsection}{\numberline {A162.3}Acceptance Gates (PASS requires all applicable)}{400}{subsection.162.3}%
\contentsline {paragraph}{PPN gates (Solar System).}{400}{section*.918}%
\contentsline {paragraph}{GW propagation gates.}{400}{section*.919}%
\contentsline {paragraph}{Waveform-consistency (optional).}{401}{section*.920}%
\contentsline {subsection}{\numberline {A162.4}Compact PASS/FAIL checklist}{401}{subsection.162.4}%
\contentsline {subsection}{\numberline {A162.5}Minimal harness (pseudocode)}{401}{subsection.162.5}%
\contentsline {paragraph}{Notes.}{402}{section*.922}%
\contentsline {section}{\numberline {A163}Cosmology: End-to-End Compressed Likelihood and Gate}{402}{section.163}%
\contentsline {subsection}{\numberline {A163.1}Data vector, parameters, and Jacobians}{402}{subsection.163.1}%
\contentsline {paragraph}{Compressed summaries (examples).}{402}{section*.923}%
\contentsline {paragraph}{Theory mapping.}{403}{section*.924}%
\contentsline {subsection}{\numberline {A163.2}Likelihood, priors, and profiling/marginalization}{403}{subsection.163.2}%
\contentsline {subsection}{\numberline {A163.3}Acceptance gate (referee-facing)}{403}{subsection.163.3}%
\contentsline {subsection}{\numberline {A163.4}Robustness and stability checks}{404}{subsection.163.4}%
\contentsline {subsection}{\numberline {A163.5}Reporting table (minimum)}{404}{subsection.163.5}%
\contentsline {paragraph}{Notes.}{405}{section*.926}%
\contentsline {section}{\numberline {A164}Black-Hole Ringdown and Echoes: Model, Fit Procedure, and Acceptance Gates}{405}{section.164}%
\contentsline {subsection}{\numberline {A164.1}Kerr ringdown (baseline)}{405}{subsection.164.1}%
\contentsline {subsection}{\numberline {A164.2}Echo response (minimal, agnostic)}{406}{subsection.164.2}%
\contentsline {paragraph}{Frequency-domain view (optional).}{406}{section*.927}%
\contentsline {subsection}{\numberline {A164.3}Data model and noise weighting}{406}{subsection.164.3}%
\contentsline {subsection}{\numberline {A164.4}Fit protocol (minimal)}{407}{subsection.164.4}%
\contentsline {subsection}{\numberline {A164.5}Acceptance gates (PASS requires all applicable)}{407}{subsection.164.5}%
\contentsline {subsection}{\numberline {A164.6}Reporting}{408}{subsection.164.6}%
\contentsline {paragraph}{Notes.}{408}{section*.929}%
\contentsline {section}{\numberline {A165}Reactivation Phase: Verified Gates to Live Prediction Surface}{408}{section.165}%
\contentsline {subsection}{\numberline {A165.1}From Static Validation to Dynamic Execution}{408}{subsection.165.1}%
\contentsline {subsection}{\numberline {A165.2}Reactivation as a Controlled Phase Transition}{409}{subsection.165.2}%
\contentsline {subsection}{\numberline {A165.3}CFGL as the Execution Backbone}{410}{subsection.165.3}%
\contentsline {subsection}{\numberline {A165.4}Prediction Surface and Crosstalk Enforcement}{410}{subsection.165.4}%
\contentsline {subsection}{\numberline {A165.5}Operational Interpretation}{410}{subsection.165.5}%
\contentsline {section}{\numberline {A166}Row 19 — Functional RG, PPN/GW Limits, Positivity Bounds, and Analytic Jacobians}{411}{section.166}%
\contentsline {subsection}{\numberline {A166.1}Role Within the Reactivation Phase}{411}{subsection.166.1}%
\contentsline {subsection}{\numberline {A166.2}Gate Composition and CFGL Interpretation}{411}{subsection.166.2}%
\contentsline {subsection}{\numberline {A166.3}Operational Summary}{412}{subsection.166.3}%
\contentsline {subsection}{\numberline {A166.4}Transition to Explicit Constructions}{412}{subsection.166.4}%
\contentsline {subsection}{\numberline {A166.5}Row 19.1 — FRG Truncation \& Fixed-Point Stability (Reproducibility Block)}{412}{subsection.166.5}%
\contentsline {subsubsection}{\numberline {A166.5.1}Effective Average Action, Fields, and Scheme (v21 defaults)}{412}{subsubsection.166.5.1}%
\contentsline {subsubsection}{\numberline {A166.5.2}Dimensionless Variables and Flow Parameters}{413}{subsubsection.166.5.2}%
\contentsline {subsubsection}{\numberline {A166.5.3}Flow Equation (Wetterich Form)}{413}{subsubsection.166.5.3}%
\contentsline {subsubsection}{\numberline {A166.5.4}Gauge Choice, Background Split, and Regulator}{414}{subsubsection.166.5.4}%
\contentsline {subsubsection}{\numberline {A166.5.5}Threshold Functions (Closed Litim Form)}{414}{subsubsection.166.5.5}%
\contentsline {subsubsection}{\numberline {A166.5.6}Curvature-Quadratic Toggle (Ablation Hook)}{414}{subsubsection.166.5.6}%
\contentsline {paragraph}{Pass rule (diagnostic).}{414}{section*.930}%
\contentsline {paragraph}{Audit (verbatim).}{415}{section*.931}%
\contentsline {subsection}{\numberline {A166.6}V.2 Exact Flow Objects and Beta Functions (gravity + light-scalar portal)}{415}{subsection.166.6}%
\contentsline {paragraph}{Gravitational couplings.}{415}{section*.932}%
\contentsline {paragraph}{Scalar portal sector.}{416}{section*.933}%
\contentsline {subsection}{\numberline {A166.7}V.3 Fixed Points, Stability Matrix, and Critical Exponents}{416}{subsection.166.7}%
\contentsline {subsubsection}{\numberline {A166.7.1}Linearized Flow and Stability Matrix}{417}{subsubsection.166.7.1}%
\contentsline {subsubsection}{\numberline {A166.7.2}Critical Exponents and Relevance Structure}{417}{subsubsection.166.7.2}%
\contentsline {subsubsection}{\numberline {A166.7.3}Analytic Construction of the Stability Matrix}{417}{subsubsection.166.7.3}%
\contentsline {subsubsection}{\numberline {A166.7.4}Interpretation for Row 19}{418}{subsubsection.166.7.4}%
\contentsline {subsection}{\numberline {A166.8}V.4 Gravity-Safe Positivity, PPN/GW Windows, and Jacobian Handoff (Interfaces)}{418}{subsection.166.8}%
\contentsline {subsubsection}{\numberline {A166.8.1}Forward-Limit Positivity (Gravity-Safe, Finite-\(t\))}{418}{subsubsection.166.8.1}%
\contentsline {subsubsection}{\numberline {A166.8.2}PPN/GW Acceptance Window (Export to Row 20)}{419}{subsubsection.166.8.2}%
\contentsline {subsubsection}{\numberline {A166.8.3}Analytic Compressed-Cosmology Jacobians (Handoff)}{419}{subsubsection.166.8.3}%
\contentsline {subsubsection}{\numberline {A166.8.4}Row 19 Interface Summary}{419}{subsubsection.166.8.4}%
\contentsline {subsection}{\numberline {A166.9}V.5 Compact PASS/FAIL Ledger (Row 19 export)}{420}{subsection.166.9}%
\contentsline {subsubsection}{\numberline {A166.9.1}Interface: Row 19 \(\rightarrow \) Row 20}{420}{subsubsection.166.9.1}%
\contentsline {subsection}{\numberline {A166.10}V.6 PPN Preliminaries and Scalar–Tensor Mapping}{420}{subsection.166.10}%
\contentsline {subsubsection}{\numberline {A166.10.1}PPN Parameters in the DEF Mapping}{421}{subsubsection.166.10.1}%
\contentsline {subsubsection}{\numberline {A166.10.2}Role Within Row 19}{421}{subsubsection.166.10.2}%
\contentsline {subsection}{\numberline {A166.11}V.7 Observational Bounds \(\Rightarrow \) Coupling Window}{422}{subsection.166.11}%
\contentsline {subsubsection}{\numberline {A166.11.1}Coupling-Space Constraints}{422}{subsubsection.166.11.1}%
\contentsline {subsubsection}{\numberline {A166.11.2}v21 Export Window}{422}{subsubsection.166.11.2}%
\contentsline {subsection}{\numberline {A166.12}V.8 GW Propagation: Speed, Damping, and Mass Window}{423}{subsection.166.12}%
\contentsline {subsubsection}{\numberline {A166.12.1}Luminosity-Distance Modification}{423}{subsubsection.166.12.1}%
\contentsline {subsubsection}{\numberline {A166.12.2}Row 19 Export Window}{423}{subsubsection.166.12.2}%
\contentsline {subsection}{\numberline {A166.13}V.9 Machine-Readable Export (for Row 20 Priors)}{423}{subsection.166.13}%
\contentsline {subsection}{\numberline {A166.14}V.10 PASS/FAIL Checklist (logged to Row 19 ledger)}{424}{subsection.166.14}%
\contentsline {subsubsection}{\numberline {A166.14.1}Interface: Row 19 \(\rightarrow \) Row 20}{424}{subsubsection.166.14.1}%
\contentsline {subsection}{\numberline {A166.15}V.11 Twice-Subtracted, Finite-$t$ Dispersion}{425}{subsection.166.15}%
\contentsline {paragraph}{Low-energy matching.}{425}{section*.934}%
\contentsline {subsection}{\numberline {A166.16}V.12 Canonical Channels and Explicit Inequalities}{425}{subsection.166.16}%
\contentsline {paragraph}{(i) Scalar \(\phi \phi \to \phi \phi \) (identical real scalar).}{425}{section*.935}%
\contentsline {paragraph}{(ii) Photon \(\gamma \gamma \to \gamma \gamma \) (parity basis).}{426}{section*.936}%
\contentsline {paragraph}{(iii) Mixed \(\phi \gamma \to \phi \gamma \).}{426}{section*.937}%
\contentsline {subsection}{\numberline {A166.17}V.12b Graviton–Scalar Channel (finite-$t$)}{426}{subsection.166.17}%
\contentsline {subsection}{\numberline {A166.18}V.13 Finite-$t$ Guard and Limit Order}{427}{subsection.166.18}%
\contentsline {subsection}{\numberline {A166.19}V.14 Machine-Readable Positivity Checklist (Logged in Row 19 Ledger)}{427}{subsection.166.19}%
\contentsline {subsubsection}{\numberline {A166.19.1}Interface: Row 19 \(\rightarrow \) Row 20}{427}{subsubsection.166.19.1}%
\contentsline {subsection}{\numberline {A166.20}V.15 Analytic Jacobians for Compressed Cosmology (Curvature–aware, Complex–Step Verified)}{427}{subsection.166.20}%
\contentsline {section}{\numberline {A167}Row 20 — Compressed Cosmology Fit and Observables}{428}{section.167}%
\contentsline {subsection}{\numberline {A167.1}W.0 Parameterization and Background}{428}{subsection.167.1}%
\contentsline {paragraph}{Distances and geometry.}{428}{section*.938}%
\contentsline {paragraph}{Compressed CMB anchors at recombination.}{429}{section*.939}%
\contentsline {subsection}{\numberline {A167.2}W.1 Derivatives of $E(z)$}{429}{subsection.167.2}%
\contentsline {paragraph}{Explicit derivatives.}{429}{section*.940}%
\contentsline {paragraph}{Inverse expansion derivative.}{430}{section*.941}%
\contentsline {subsection}{\numberline {A167.3}W.2 Jacobians for $\chi (z)$ and $D_M(z)$}{430}{subsection.167.3}%
\contentsline {paragraph}{Curvature-aware geometry.}{430}{section*.942}%
\contentsline {paragraph}{Derivative of \(D_M\).}{431}{section*.943}%
\contentsline {paragraph}{Derivative structure.}{431}{section*.944}%
\contentsline {paragraph}{Flat limit.}{431}{section*.945}%
\contentsline {subsection}{\numberline {A167.4}W.3 Jacobians for $r_s(z_\ast )$}{431}{subsection.167.4}%
\contentsline {paragraph}{General parameter derivatives.}{432}{section*.946}%
\contentsline {paragraph}{Hubble parameter dependence.}{432}{section*.947}%
\contentsline {paragraph}{Baryon contribution.}{432}{section*.948}%
\contentsline {subsection}{\numberline {A167.5}W.4 Jacobians for Compressed Observables}{432}{subsection.167.5}%
\contentsline {paragraph}{Shift parameter \(R\).}{432}{section*.949}%
\contentsline {paragraph}{Acoustic scale \(\ell _A\).}{433}{section*.950}%
\contentsline {paragraph}{Sound angle \(\theta _\ast \).}{433}{section*.951}%
\contentsline {paragraph}{BAO distance combinations.}{433}{section*.952}%
\contentsline {subsection}{\numberline {A167.6}W.5 GW Standard--Siren Jacobians (for Row 20 combined fit)}{434}{subsection.167.6}%
\contentsline {subsection}{\numberline {A167.7}W.6 Complex--Step Validation (Machine--Precision Check)}{434}{subsection.167.7}%
\contentsline {subsection}{\numberline {A167.8}W.7 Compact PASS/FAIL Ledger (Row 19 $\rightarrow $ Row 20 Export)}{434}{subsection.167.8}%
\contentsline {subsection}{\numberline {A167.9}W.8 Unit Tests (Analytic vs Complex--Step)}{435}{subsection.167.9}%
\contentsline {paragraph}{Audit (verbatim).}{435}{section*.953}%
\contentsline {subsubsection}{\numberline {A167.9.1}Interface: Row 19 \(\rightarrow \) Row 20}{435}{subsubsection.167.9.1}%
\contentsline {subsection}{\numberline {A167.10}Row 19.5 — Reproducibility Block: Regulator, Scheme, and Fixed Constants}{435}{subsection.167.10}%
\contentsline {paragraph}{Regulator (fixed).}{435}{section*.954}%
\contentsline {paragraph}{Truncation and projection.}{436}{section*.955}%
\contentsline {paragraph}{Threshold functions (locked).}{436}{section*.956}%
\contentsline {paragraph}{Cosmology constants (fixed).}{436}{section*.957}%
\contentsline {paragraph}{Numerical conventions.}{436}{section*.958}%
\contentsline {paragraph}{Determinism and export.}{436}{section*.959}%
\contentsline {section}{\numberline {A168}X — FRG Setup: Field Content, Gauge, and Regulator (Pinned)}{436}{section.168}%
\contentsline {subsection}{\numberline {A168.1}X.1 Field Content, Split, and Gauge (Pinned)}{436}{subsection.168.1}%
\contentsline {subsection}{\numberline {A168.2}X.2 Regulator (Pinned) and Thresholds (Closed Forms)}{437}{subsection.168.2}%
\contentsline {subsection}{\numberline {A168.3}X.3 Dimensionless Couplings and Anomalous Dimensions (Pinned)}{437}{subsection.168.3}%
\contentsline {subsection}{\numberline {A168.4}X.4 Beta Functions and Mode Weights (Pinned, Minimal EH\(+\)\(\phi \))}{438}{subsection.168.4}%
\contentsline {paragraph}{Gravity sector.}{438}{section*.960}%
\contentsline {paragraph}{Scalar portal sector.}{438}{section*.961}%
\contentsline {paragraph}{Mode assignments.}{439}{section*.962}%
\contentsline {subsection}{\numberline {A168.5}X.5 Numerical Constants and Units (Pinned)}{439}{subsection.168.5}%
\contentsline {subsection}{\numberline {A168.6}X.6 Minimal Verification Checklist (Logged)}{439}{subsection.168.6}%
\contentsline {paragraph}{Interface note.}{439}{section*.963}%
\contentsline {subsection}{\numberline {A168.7}Row 19.6 — Gate Ledger (Machine-Readable Export)}{439}{subsection.168.7}%
\contentsline {paragraph}{Schema.}{440}{section*.964}%
\contentsline {paragraph}{Ledger (verbatim).}{440}{section*.965}%
\contentsline {paragraph}{Interpretation.}{441}{section*.966}%
\contentsline {paragraph}{Interface note (Row 19 \(\rightarrow \) Row 20).}{441}{section*.967}%
\contentsline {section}{\numberline {A169}Y — Global Gate Summary and PASS/FAIL Export}{441}{section.169}%
\contentsline {subsection}{\numberline {A169.1}Y.1 Compact PASS/FAIL Table}{442}{subsection.169.1}%
\contentsline {paragraph}{Interpretation.}{442}{section*.968}%
\contentsline {paragraph}{Interface note.}{442}{section*.969}%
\contentsline {subsection}{\numberline {A169.2}Y.2 Minimal JSON Export (verbatim)}{442}{subsection.169.2}%
\contentsline {paragraph}{Row 20 ingestion (pseudocode).}{443}{section*.970}%
\contentsline {paragraph}{Audit note.}{444}{section*.971}%
\contentsline {subsection}{\numberline {A169.3}Row 19.7 — Cross-Row Handshake: Interface to Row 20}{444}{subsection.169.3}%
\contentsline {paragraph}{Inputs (from Row 19).}{444}{section*.972}%
\contentsline {paragraph}{Required operations (Row 20).}{444}{section*.973}%
\contentsline {paragraph}{Hard constraints (non-negotiable).}{445}{section*.974}%
\contentsline {paragraph}{Failure conditions.}{445}{section*.975}%
\contentsline {paragraph}{Reproducibility lock.}{445}{section*.976}%
\contentsline {paragraph}{Interpretation.}{445}{section*.977}%
\contentsline {section}{\numberline {A170}Z — Execution Layer: Pipeline, Enforcement, and Run Protocol}{445}{section.170}%
\contentsline {subsection}{\numberline {A170.1}Z.1 End-to-End Execution Flow}{446}{subsection.170.1}%
\contentsline {subsection}{\numberline {A170.2}Z.2 Constraint Enforcement and Abort Logic}{446}{subsection.170.2}%
\contentsline {paragraph}{Interpretation.}{447}{section*.978}%
\contentsline {subsection}{\numberline {A170.3}Z.3 Acceptance Rule and Outputs Returned}{447}{subsection.170.3}%
\contentsline {subsection}{\numberline {A170.4}Z.4 Minimal Ingestion Pseudocode (Verbatim, Deterministic)}{447}{subsection.170.4}%
\contentsline {paragraph}{Audit note.}{447}{section*.979}%
\contentsline {section}{\numberline {A171}Row 20 — Compressed Cosmology Fit (chi2/nu ≈ 1)}{448}{section.171}%
\contentsline {paragraph}{Interface to Row 19.}{448}{section*.980}%
\contentsline {subsection}{\numberline {A171.1}Neutral Vacuum and Compressed Cosmology Fit (UTFANSWF v22)}{448}{subsection.171.1}%
\contentsline {paragraph}{Stabilized Neutral Vacuum.}{448}{section*.981}%
\contentsline {paragraph}{Equation of state.}{448}{section*.983}%
\contentsline {paragraph}{Vacuum energy evolution.}{448}{section*.985}%
\contentsline {paragraph}{Background expansion.}{448}{section*.987}%
\contentsline {paragraph}{Distance measures.}{448}{section*.989}%
\contentsline {paragraph}{Compressed likelihood.}{448}{section*.992}%
\contentsline {paragraph}{Neutral Zone priors.}{449}{section*.994}%
\contentsline {paragraph}{Derived quantities.}{449}{section*.997}%
\contentsline {paragraph}{Pipeline consistency.}{449}{section*.1000}%
\contentsline {paragraph}{Interpretation.}{449}{section*.1001}%
\contentsline {section}{\numberline {A172}AA — Compressed Cosmology Inputs, Parameters, and Data Provenance}{449}{section.172}%
\contentsline {subsection}{\numberline {A172.1}AA.0 Overview and Inputs (from Row 19)}{450}{subsection.172.1}%
\contentsline {subsection}{\numberline {A172.2}AA.0.1 Data Provenance (Pinned)}{450}{subsection.172.2}%
\contentsline {subsection}{\numberline {A172.3}AA.1 Data Vectors and Model Maps}{450}{subsection.172.3}%
\contentsline {paragraph}{CMB compression (recombination).}{451}{section*.1002}%
\contentsline {paragraph}{BAO compression (at \(\{z_d\}\)).}{451}{section*.1003}%
\contentsline {paragraph}{Type Ia Supernovae (compression).}{451}{section*.1004}%
\contentsline {paragraph}{Gravitational-wave standard sirens.}{451}{section*.1005}%
\contentsline {subsection}{\numberline {A172.4}AA.2 Joint Objective and Constraints}{451}{subsection.172.4}%
\contentsline {paragraph}{Likelihood.}{451}{section*.1006}%
\contentsline {paragraph}{Hard constraints (must hold).}{452}{section*.1007}%
\contentsline {paragraph}{Ablation checks (posterior stability).}{452}{table.caption.1009}%
\contentsline {subsection}{\numberline {A172.5}AA.3 Gradients and Fisher Blocks (Analytic, from Row 19.4)}{452}{subsection.172.5}%
\contentsline {subsection}{\numberline {A172.6}AA.4 Minimizer and Acceptance Rule}{453}{subsection.172.6}%
\contentsline {paragraph}{Output.}{453}{section*.1010}%
\contentsline {subsection}{\numberline {A172.7}AA.5 Outputs Returned and Ledger Update}{453}{subsection.172.7}%
\contentsline {paragraph}{Returned quantities.}{453}{section*.1011}%
\contentsline {paragraph}{Ledger linkage.}{453}{section*.1012}%
\contentsline {paragraph}{Deterministic ingestion (verbatim).}{453}{section*.1013}%
\contentsline {paragraph}{Determinism guarantee.}{454}{section*.1014}%
\contentsline {section}{\numberline {A}Appendix A — Gravity-Safe Positivity and Scattering Construction}{455}{appendix.A}%
\contentsline {subsection}{\numberline {A.1}A.1 Scattering Setup, Kinematics, and Normalization}{455}{subsection.A.1}%
\contentsline {subsection}{\numberline {A.2}A.2 Helicity and Projector Construction}{455}{subsection.A.2}%
\contentsline {subsection}{\numberline {A.3}A.3 Forward-Limit Positivity with Gravity}{455}{subsection.A.3}%
\contentsline {subsection}{\numberline {A.4}A.4 Role in UTFANSWF (Row 19 Link)}{456}{subsection.A.4}%
\contentsline {subsection}{\numberline {A.5}A.5 Summary}{456}{subsection.A.5}%
\contentsline {section}{\numberline {B}Row 20.1 — CMB Block: R, $\ell _A$, $\theta _\ast $ (Covariance \& Checks)}{456}{appendix.B}%
\contentsline {paragraph}{Data vector and covariance (from tag \texttt {planck\_2018\_distpriors\_v1}).}{456}{section*.1015}%
\contentsline {paragraph}{Theory map (Row 19.4).}{456}{section*.1016}%
\contentsline {paragraph}{Sanity checks (deterministic).}{456}{section*.1017}%
\contentsline {paragraph}{PASS/FAIL.}{457}{section*.1018}%
\contentsline {section}{\numberline {C}Row 20.2 — BAO • SN • GW Nodes (Lists \& Quick Tests)}{457}{appendix.C}%
\contentsline {paragraph}{BAO (tag \texttt {desi\_dr1\_bao\_combo\_v1}).}{457}{section*.1019}%
\contentsline {paragraph}{SNe (tag \texttt {pantheonplus\_v1\_compressed}).}{457}{section*.1020}%
\contentsline {paragraph}{GW standard sirens (tag \texttt {o3+o4\_catalog\_v1\_compressed}).}{457}{section*.1021}%
\contentsline {paragraph}{Quick sanity tests (deterministic).}{457}{section*.1022}%
\contentsline {paragraph}{PASS/FAIL.}{457}{section*.1023}%
\contentsline {paragraph}{Audit (verbatim).}{458}{section*.1025}%
\contentsline {section}{\numberline {D}Row 20.3 — Results Ledger Stub (\texttt {row20\_results.json})}{458}{appendix.D}%
\contentsline {paragraph}{PASS/FAIL.}{458}{section*.1026}%
\contentsline {section}{\numberline {E}Harmonics, Fractal Harmonics, and Refractal Harmonics}{459}{appendix.E}%
\contentsline {paragraph}{Harmonics (what is present).}{459}{section*.1027}%
\contentsline {paragraph}{Fractal harmonics (how it scales).}{459}{section*.1028}%
\contentsline {paragraph}{Refractal harmonics (how it updates).}{459}{section*.1029}%
\contentsline {paragraph}{Predictive bite.}{459}{section*.1030}%
\contentsline {subsection}{\numberline {E.1}Harmonics, Fractal Harmonics, and Refractal Harmonics}{459}{subsection.E.1}%
\contentsline {paragraph}{Harmonics (what is present).}{459}{section*.1032}%
\contentsline {paragraph}{Fractal harmonics (how it scales).}{460}{section*.1033}%
\contentsline {paragraph}{Refractal harmonics (how it updates).}{460}{section*.1034}%
\contentsline {paragraph}{Predictive bite.}{460}{section*.1035}%
\contentsline {section}{\numberline {F}Row 20.A — Ablations: Prior \& Regulator Sensitivity (One-Page Gate)}{461}{appendix.F}%
\contentsline {paragraph}{Purpose.}{461}{section*.1038}%
\contentsline {paragraph}{Objective.}{461}{section*.1040}%
\contentsline {paragraph}{Pass rule (hard).}{462}{section*.1042}%
\contentsline {paragraph}{Audit (verbatim).}{462}{section*.1043}%
\contentsline {section}{\numberline {G}Row 20.4 — JWST Mini-Run (UVLF/SFRD/$\rho _\star $/Sizes/Lines)}{462}{appendix.G}%
\contentsline {paragraph}{Dataset tags (pinned).}{462}{table.caption.1045}%
\contentsline {paragraph}{Theory map.}{462}{section*.1046}%
\contentsline {paragraph}{Objective.}{462}{section*.1047}%
\contentsline {paragraph}{PASS/FAIL.}{463}{section*.1049}%
\contentsline {section}{\numberline {H}Row 21 — Axion Dark Matter Gate (2025Q3 vs Haloscopes)}{463}{appendix.H}%
\contentsline {paragraph}{Scope \& inputs.}{463}{table.caption.1051}%
\contentsline {paragraph}{Model slice (misalignment baseline).}{463}{section*.1052}%
\contentsline {paragraph}{Gate definition (hard).}{463}{section*.1053}%
\contentsline {paragraph}{Checks (deterministic).}{464}{section*.1054}%
\contentsline {paragraph}{Hard exclusion rule.}{464}{section*.1055}%
\contentsline {paragraph}{Setup.}{464}{section*.1057}%
\contentsline {paragraph}{Interpolation envelope.}{464}{section*.1058}%
\contentsline {paragraph}{Reproducibility record.}{464}{section*.1061}%
\contentsline {paragraph}{Audit (verbatim).}{464}{section*.1062}%
\contentsline {paragraph}{PASS/FAIL.}{465}{section*.1063}%
\contentsline {section}{\numberline {I}Row 22 — Neutral Zone (NZ) Sandwich: Robust Sweep \& Sensitivities}{465}{appendix.I}%
\contentsline {paragraph}{Inequality (UTFANSWF NZ).}{465}{section*.1064}%
\contentsline {paragraph}{Robust sweep definition.}{465}{section*.1065}%
\contentsline {paragraph}{Gate definition (hard).}{466}{section*.1066}%
\contentsline {paragraph}{Checks (deterministic).}{466}{section*.1067}%
\contentsline {paragraph}{Audit (verbatim).}{466}{section*.1069}%
\contentsline {paragraph}{PASS/FAIL.}{466}{section*.1070}%
\contentsline {section}{\numberline {J}Row 23 — Black-Hole Ringdown \& Echo Predictions}{466}{appendix.J}%
\contentsline {paragraph}{QNM baseline.}{466}{section*.1071}%
\contentsline {paragraph}{Echo template.}{466}{section*.1072}%
\contentsline {paragraph}{Gate definition (diagnostic).}{467}{section*.1073}%
\contentsline {paragraph}{Checks.}{467}{section*.1074}%
\contentsline {paragraph}{Row 23 — audit stub (verbatim).}{467}{section*.1075}%
\contentsline {paragraph}{Mock audit (verbatim).}{467}{section*.1077}%
\contentsline {paragraph}{PASS/FAIL.}{467}{section*.1078}%
\contentsline {section}{\numberline {K}Row 24 — AI $\leftrightarrow $ UTFANSWF: Guardrails \& Learning Efficiency}{467}{appendix.K}%
\contentsline {paragraph}{Constraint-prior training.}{467}{section*.1079}%
\contentsline {paragraph}{Efficiency metric.}{468}{section*.1080}%
\contentsline {paragraph}{Safety rails (hard constraints).}{468}{section*.1081}%
\contentsline {paragraph}{Gate definition (hard).}{468}{section*.1082}%
\contentsline {paragraph}{Checks (deterministic).}{468}{section*.1083}%
\contentsline {paragraph}{Audit (verbatim).}{468}{section*.1085}%
\contentsline {paragraph}{PASS/FAIL.}{469}{section*.1086}%
\contentsline {section}{\numberline {L}Row 25 — Stress-Test Appendix: “Break UTFANSWF” Registry}{469}{appendix.L}%
\contentsline {paragraph}{Objective.}{469}{section*.1087}%
\contentsline {paragraph}{Harness rule (deterministic).}{469}{section*.1088}%
\contentsline {paragraph}{Escalation rule (hard).}{469}{section*.1089}%
\contentsline {paragraph}{Gate definition (hard).}{469}{section*.1090}%
\contentsline {paragraph}{Checks (deterministic).}{469}{section*.1091}%
\contentsline {paragraph}{Audit (verbatim).}{470}{section*.1093}%
\contentsline {paragraph}{PASS/FAIL.}{470}{section*.1094}%
\contentsline {section}{\numberline {M}Row 26 — References \& Acknowledgments (Consistency Gate)}{470}{appendix.M}%
\contentsline {paragraph}{Objective.}{470}{section*.1095}%
\contentsline {paragraph}{Consistency checks (deterministic).}{470}{section*.1096}%
\contentsline {paragraph}{Reproducibility check.}{470}{section*.1097}%
\contentsline {paragraph}{Gate definition (hard).}{471}{section*.1098}%
\contentsline {paragraph}{Audit (verbatim).}{471}{section*.1100}%
\contentsline {paragraph}{PASS/FAIL.}{471}{section*.1101}%
\contentsline {paragraph}{Scope.}{471}{section*.1103}%
\contentsline {paragraph}{Global guardrails (hard).}{472}{section*.1104}%
\contentsline {paragraph}{Inputs (2025Q4).}{472}{section*.1106}%
\contentsline {paragraph}{Test \& Expression.}{472}{section*.1107}%
\contentsline {paragraph}{Rule (deterministic).}{472}{section*.1108}%
\contentsline {paragraph}{Decision (Oct 17, 2025).}{472}{section*.1109}%
\contentsline {paragraph}{Action next.}{473}{section*.1110}%
\contentsline {paragraph}{Cross-gate echoes.}{473}{section*.1111}%
\contentsline {paragraph}{Inputs (2025Q4).}{473}{section*.1113}%
\contentsline {paragraph}{Test \& Expression.}{473}{section*.1114}%
\contentsline {paragraph}{Rule (deterministic).}{473}{section*.1115}%
\contentsline {paragraph}{Decision (Oct 17, 2025).}{473}{section*.1116}%
\contentsline {paragraph}{Action next.}{473}{section*.1117}%
\contentsline {paragraph}{Cross-gate echoes.}{474}{section*.1118}%
\contentsline {paragraph}{Inputs (2025Q4).}{474}{section*.1120}%
\contentsline {paragraph}{Test \& Expression.}{474}{section*.1121}%
\contentsline {paragraph}{Rule (deterministic).}{474}{section*.1122}%
\contentsline {paragraph}{Decision (Oct 17, 2025).}{474}{section*.1123}%
\contentsline {paragraph}{Action next.}{474}{section*.1124}%
\contentsline {paragraph}{Cross-gate echoes.}{474}{section*.1125}%
\contentsline {paragraph}{Inputs (2025Q4).}{475}{section*.1127}%
\contentsline {paragraph}{Test \& Expression.}{475}{section*.1128}%
\contentsline {paragraph}{Rule (deterministic).}{475}{section*.1129}%
\contentsline {paragraph}{Decision (Oct 17, 2025).}{475}{section*.1130}%
\contentsline {paragraph}{Action next.}{475}{section*.1131}%
\contentsline {paragraph}{Cross-gate echoes.}{475}{section*.1132}%
\contentsline {paragraph}{Inputs (2025Q4).}{475}{section*.1134}%
\contentsline {paragraph}{Test \& Expression.}{476}{section*.1135}%
\contentsline {paragraph}{Rule (deterministic).}{476}{section*.1136}%
\contentsline {paragraph}{Decision (Oct 17, 2025).}{476}{section*.1137}%
\contentsline {paragraph}{Action next.}{476}{section*.1138}%
\contentsline {section}{\numberline {N}Row F — Compact Figures (1 page)}{477}{appendix.N}%
\contentsline {section}{\numberline {O}Index of Symbols (UTFANSWF v21)}{478}{appendix.O}%
\contentsline {subsection}{\numberline {O.1}Core Geometry \& Thermodynamics}{478}{subsection.O.1}%
\contentsline {subsection}{\numberline {O.2}Harmonic Bubbles \& Echoes}{478}{subsection.O.2}%
\contentsline {subsection}{\numberline {O.3}Axion Gate \& Cosmology}{479}{subsection.O.3}%
\contentsline {subsection}{\numberline {O.4}FRG \& Dispersion Positivity}{479}{subsection.O.4}%
\contentsline {subsection}{\numberline {O.5}Observer Algebra}{479}{subsection.O.5}%
\contentsline {subsection}{\numberline {O.6}Applications \& Gates}{480}{subsection.O.6}%
\contentsline {section}{\numberline {P}Acknowledgments}{481}{appendix.P}%
\contentsline {section}{\numberline {Q}License, Provenance, and Versioning}{481}{appendix.Q}%
\contentsline {section}{\numberline {R}Data, Code, and Reproducibility}{481}{appendix.R}%
\contentsline {section}{\numberline {S}UTFANSWF Glossary (Canonical Edition)}{482}{appendix.S}%
\contentsline {section}{\numberline {T}Foundational Layer --- SWF-ISM}{482}{appendix.T}%
\contentsline {paragraph}{Entropy--Volume Normalization.}{482}{section*.1148}%
\contentsline {section}{\numberline {U}Bridge Layer --- ANSWF $\rightarrow $ SANSWF}{484}{appendix.U}%
\contentsline {section}{\numberline {V}Unified Layer --- UTFANSWF Core}{484}{appendix.V}%
\contentsline {section}{\numberline {W}Applied Layer --- Predictive and Empirical Interface}{485}{appendix.W}%
\contentsline {section}{\numberline {X}Integration Layer --- AI $\leftrightarrow $ UTFANSWF}{486}{appendix.X}%
\contentsline {section}{\numberline {Y}Stress-Test and Legacy Layer --- Endurance and Outlook}{486}{appendix.Y}%
\contentsline {subsection}{\numberline {Y.1}Limitations (v21)}{488}{subsection.Y.1}%
\contentsline {paragraph}{}{491}{section*.1194}%
