Physics Regime Extension II

Joshua K. Cliff, 2026 · 165 pages · 47 chapters · 6 appendices · 89 formal results · 85 proofs · CC BY 4. 0 Overview This volume is the discrete-to-continuum carrier construction and conditional structural-unification volume for Principal Dynamics in the physics regime. Where Physics Regime Extension I (10. 5281/zenodo. 19334815) closes the baseline bridge from a stipulated continuum carrier, this volume constructs the carrier from a discrete physics-regime datum and re-derives the sector equations on the produced carrier under one generator stack. The volume is positioned as a Layer 2 regime-extension volume under the Program Guide's claim taxonomy and EC1–EC8 charter. It does not alter axioms A1–A5 or any proved foundation theorem; it adds a discrete-to-continuum carrier construction, the conditional low-order structural-unification theorem on a declared target-neutral sieve, the relativistic matter-sector reductions on eligible bundle branches, and the branch-evaluation architecture. Numerical execution, full Standard Model parameter synthesis, frontier closures (quantum gravity, cosmology, dark sector, horizons), benchmark concordance, and empirical adequacy are routed outward to future work covering the relevant physics domains. Volume-II results (closed within the volume on declared classes) The following targets are closed on declared admissible classes: Discrete-to-continuum carrier construction: kernel path metric on the projected informational degrees of freedom, metric-measure compactness in the Gromov–Hausdorff–Prokhorov topology, Dirichlet-form / semigroup / Sobolev-scale convergence under Mosco hypotheses, graph-to-CTE convergence, smooth-upgrade theorem with Fisher-information metric, and effective Lorentz-branch realization. Sector reductions on the produced carrier: Schrödinger reduction in the Hamiltonian window with Born trace-form readout and no-signaling; Einstein variational reduction with cosmological term, plus geodesic and weak-field recoveries; bundle-and-connection realization from frame transport, Yang–Mills reduction with matter current, Maxwell as the abelian specialization; corrected-domain coercivity and branch-qualified spectral closure. Single-generator consistency audit: graph, continuum, smooth, quantum, gravity, gauge, and Yang–Mills sectors generated from one stack (F, G, Π, C) ; projection-limit commutation across sectors; non-conflict with the foundation and with Physics Regime Extension I. Conditional low-order structural unification (the load-bearing theorem): target-neutral low-order sieve with non-circular admissibility predicates; allowed generator-basis theorem proved before any sector equation is named; classification lemmas for the surviving low-order representatives; equation-forcing theorem; non-circularity audit; branch-constant canonicity and convergence under spectral and Gamma-convergence hypotheses. Matter-sector branch reductions: gauge-covariant Dirac equation on spin/spinᶜ branches under Clifford square-root compatibility (Theorem 23. 1) ; gauge-covariant Klein–Gordon equation on scalar-bundle branches under scalar-principal-coefficient admissibility (Theorem 24. 1) ; admissible Yukawa slot form classified on chiral spinor and scalar order-parameter branches (Theorem 26. 1). Branch-evaluation architecture: concrete branch-family format with two illustrative carriers, admissibility-certificate gate, computable estimator package for ℏ*, c*, G*, Λ*, gauge couplings, and persistence masses, unit-lock theorem for dimensionless ratios, refinement and coarsening invariance, observable pipeline, benchmark suite specification, failure taxonomy, and calibration discipline. Routed outward The volume explicitly does not claim, and routes to future work covering the relevant physics domains: Empirical confirmation, measured-number concordance, executed benchmark rows, external audit verdicts Unrestricted Yang–Mills mass gap and unrestricted confinement Standard Model spectrum, family count, mass hierarchy, mixing matrices, numerical Yukawa matrices, flavor hierarchy Chiral anomaly cancellation on declared content, neutrino sector, electroweak vacuum closure, Higgs-sector identification beyond the Klein–Gordon kernel Scalar-field quantization (CCR algebra, microcausality), fermionic statistics (CAR algebra), spin-statistics theorem Quantum gravity, cosmology and early-universe physics, dark-sector closure, horizon and black-hole sectors Cross-regime closure to chemistry, materials, and engineered systems Identification of emergent constants (ℏ*, c*, G*, Λ*) with measured values Structure Part I (Reference Architecture and Scope): position within the corpus, source-chain audit, claim taxonomy with Program Guide D/E/F/G crosswalk, dependency graph, theorem ledger. Part II (Discrete-to-Continuum Carrier): discrete physics-regime datum and discrete CTE; projected informational degrees of freedom; kernel metric, measure, dimension, locality; Dirichlet-form / Sobolev-scale / graph-to-CTE convergence; smooth physics carrier with effective Lorentz branch. Part III (Sector Reductions on the Carrier): quantum sector (Hilbert, Schrödinger, Born readout, no-signaling) ; gravity sector (curvature variation, Einstein, geodesic, weak-field) ; gauge sector (bundle realization, Yang–Mills, Maxwell, matter current) ; corrected domains. Part IV (Conditional Structural Unification): single-generator consistency audit; conditional structural-unification theorem with target-neutral sieve, generator-basis theorem, classification lemmas, equation forcing, branch-constant canonicity, non-circularity audit, falsification interface. Part V (Theorem-to-Evaluation Boundary): explicit boundary between the conditional theorem chain and the branch-evaluation pipeline. Part VI (Quantitative Branch Evaluation): concrete branch data with two illustrative branches; admissibility certificate; constant-functional evaluation; dimensionless ratios with unit lock; scaling and renormalization behavior. Part VII (Matter-Sector Closure): matter-sector taxonomy; spin and fermion emergence; Dirac branch reduction (Chapter 23) ; Klein–Gordon branch reduction (Chapter 24) ; gauge-representation eligibility; symmetry breaking and Yukawa slot (Chapter 26) ; family count and flavor structure. Part VIII (Quantitative Gauge and Gravity Closure): gauge-coupling evaluation; confinement and mass-gap branch; gravity constant and cosmological term; semiclassical coupling consistency. Part IX (Empirical Interface and Benchmarks): observable pipeline; benchmark suite; failure taxonomy; calibration discipline. Part X (Computational Realization): numerical representation; algorithms; verification and error bounds. Part XI (Frontier Sectors): quantum-gravity, cosmology, dark-sector, horizon, and cross-regime routing chapters. Part XII (Final Synthesis): master theorem map; integrated synthesis claim; what would complete empirical unification; open-problems register. Appendices: A. Why this volume is not circular. B. Standard results register. C. Assumption registry. D. Worked example: flat-torus scalar-gauge branch through the full pipeline. E. Worked example: flat-torus spinor-gauge branch and the Dirac reduction. F. Symbol table. Corpus context This volume is part of the Principal Dynamics public corpus governed by the Program Guide. Related volumes: Foundation Monograph: 10. 5281/zenodo. 19334802 Foundation Paper: 10. 5281/zenodo. 19432514 Physics Regime Extension I — Monograph-to-Physics Bridge: 10. 5281/zenodo. 19334815 Intelligence Regime Extension I — Monograph-to-Intelligence Bridge: 10. 5281/zenodo. 19334838 Keywords: Principal Dynamics, Physics Regime Extension II, discrete-to-continuum carrier construction, regime closure, conditional structural unification, target-neutral sieve, least-generator-cost selection, equation forcing, kernel metric, metric-measure convergence, Gromov–Hausdorff–Prokhorov, Dirichlet-form convergence, Mosco convergence, Sobolev-scale convergence, graph-to-CTE convergence, smooth-upgrade theorem, Fisher-information metric, Lorentz emergence, single-generator consistency, projection-limit commutation, Schrödinger reduction, Einstein variational reduction, Yang–Mills reduction, gauge-covariant Dirac equation, gauge-covariant Klein–Gordon equation, scalar-principal-coefficient admissibility, Schur's lemma, Yukawa slot admissibility classification, branch functionals, branch certificate, dimensionless ratios, unit-lock theorem, observable pipeline, benchmark suite, failure taxonomy, calibration discipline, claim discipline, non-circularity audit