Physics Regime Companion — From Axioms to Quantitative Closure Joshua K. Cliff, 2026 323 pages · 31 chapters · 14 appendices · 276 formal results · 259 proofs · CC BY 4.0 Overview This volume is the physics regime companion to the Principal Dynamics foundation monograph (10.5281/zenodo.19334803). It specializes the domain-neutral PD foundation to the physics regime and proves three tiers of results: direct reductions, structural closure, and quantitative closure on the canonical completion branch. Every theorem is either a direct specialization of a monograph result or a consequence of such specializations composed with explicitly stated regime closure laws satisfying the Extension Charter (EC1–EC8). The volume does not alter axioms A1–A5 or any proved foundation theorem. Direct Reductions (Part I) The Coherence Transformation Equation, specialized to the physics regime window, reproduces the equations and structures of quantum mechanics and relativistic field theory as direct parameter-limit reductions: the Schrödinger equation, Hilbert-space structure, canonical commutation relations, unitarity, the Born rule, no-signaling, the Einstein field equations, the geodesic equation, the Dirac equation, Yang–Mills gauge equations, a corrected-domain mass gap, exponential clustering, reflection positivity, ghost decoupling, and the scalar/Yukawa sector. These reductions use only the monograph foundation and standard mathematics, without regime closure laws. Structural Closure (Part II) Six structural closure theorems are proved from the foundation plus explicitly stated regime closure laws: universality of the low-order infrared sector, native Lorentz emergence, observable-functor uniqueness, gauge-group selection (recovering SU(3) × SU(2) × U(1)), a constructive continuum gauge bridge (conditional on witness-family existence in the declared bridge class), and three-generation family count. Quantitative Closure (Part III) On the canonical completion branch and declared admissible class, all charged-sector quantitative data (masses, mixing angles, coupling constants) are uniquely determined as branch functionals of a single normalized datum, with no free continuous moduli. The neutrino sector is fully fixed, with the discrete mass ordering resolved on the canonical positive MX branch. The existence, uniqueness, and finiteness of these functionals are proved; this part does not carry out the full numerical evaluation of every functional against experiment except where explicitly stated. Deeper Sectors and Predictions (Parts IV–V) The closed branch is extended to cosmological perturbations, baryogenesis (branch-limited, source-normalized), dark-sector classification, and black-hole information. Part V states the benchmark protocol with explicit claim boundary: a subset of non-anchor quantitative rows has been publicly executed and independently audited; the remaining blocked rows have not yet been numerically evaluated. What Is Not Claimed Full empirical adequacy is not established. Agreement of the complete benchmark sheet with measurement has not been verified. Unrestricted formal universality is not claimed. Universality is proved across the tested enlargement ladder A₀ ⊂ · · · ⊂ A₅; universality across all conceivable admissible classes is not asserted. A universal unconditional 4D constructive Yang–Mills theorem is not claimed. The continuum bridge is conditional on witness-family existence in the declared constructive bridge class. Full non-perturbative quantum gravity on a larger class is not claimed. The mixed quantum-gravity completion is a branch-level candidate on the declared class. All quantitative results are branch-qualified: they hold on the declared admissible class and canonical branch. Related Volumes Foundation Monograph: 10.5281/zenodo.19334803 Intelligence and Computation Regime Companion: 10.5281/zenodo.19334839 Keywords: Principal Dynamics, physics regime, quantum mechanics, gauge theory, Yang–Mills, Lorentz emergence, structural closure, quantitative closure, branch functional, benchmark, Born rule, Standard Model