APF Paper 7 — A Minimal Quantum of Action from Finite Admissibility (Technical Supplement)

Abstract

Technical Supplement to Paper 7 of the Admissibility Physics Framework (APF), A Minimal Quantum of Action from Finite Admissibility. Paper 7's central structural identity: the Connes–Chamseddine spectral action coincides with the APF partition function under the Boltzmann cutoff. The Seeley–DeWitt expansion of the resulting heat trace then reproduces the cosmological constant Λ, the Einstein–Hilbert action ∫ R √(−g) d4x, and the Yang–Mills + Higgs sector Tr(Fμν Fμν) + |Dμφ|2 + V(φ) — from a single variational principle. Six formal sections plus a red-team audit appendix. All derivations use only A1, MD, A2, BW (PLEC) plus constitutive semantics SP, K3, plus the capacity quantization Ctot = 61, deff = 102 imported from Papers 1–5. §S.1 Minimum quantum of action. Zero action is inadmissible for record-forming paths from MD's positive cost floor. Multi-record bookkeeping produces the k! permutation factor underlying Planck-type quantization ℏ. §S.2 Non-interacting partition function. Z0(β) = (1 + deff e−βε*)Ctot, derived from the PLEC saturation form. Gapped-ground-state regime: kT ≪ ε*. §S.3 Saturation factorisation. Zsat(β, η) = Nmicro e−β Esat(η) with two structural independences isolated. Zsat is independent of β through Nmicro, and independent of interaction couplings η through the saturation counting. §S.4 The spectral-action identity (the central result). The Connes spectral action Tr(f(D/Λcut)) coincides with the APF partition function under the Boltzmann cutoff. The Seeley–DeWitt expansion Tr(e−tD²) ∼ Σk ≥ 0 ak(D²) tk−n/2 yields a0 → Λ, a2 → R, a4 → Fμν Fμν + |Dμφ|2 + V(φ). Uniqueness follows from spectral invariance, gauge invariance, and positivity (check_spectral_action_internal in internalization.py). §S.5 Geometric corollaries. Kolmogorov-type continuum limit. Lipschitz chartability. Lovelock uniqueness of the Einstein equations in d = 4. Coleman–Mandula internalisation. Hawking–King–McCarthy causal-order uniqueness. Malament full-metric recovery from causal structure. §S.6 Standard-Model extensions. One-loop β-function coefficients from APF field content. Type-I seesaw for right-handed neutrino masses. Pauli–Jordan spin–statistics from commutator symmetry. McKean–Singer index identity. Tannaka–Krein reconstruction of the gauge group. §S.7 Red-team audit flagging additional hypotheses at point of use: spectral-triple ansatz, Lipschitz regularity, causal-order topology. Code and reproducibility. GitHub repository Colab walkthrough notebook (one-click) Interactive dependency DAG Codebase backing. Every theorem traces to a named check function in the canonical apf/ codebase (v7.5; 454 bank-registered theorems, 471 verify_all checks, 39 modules + standalone). Paper 7's spectral-action / partition-function identity, Seeley–DeWitt expansion, and Standard-Model Lagrangian extensions live in apf/internalization.py, apf/internalization_geo.py, and apf/extensions.py. About the APF series. The Admissibility Physics Framework derives Standard Model and cosmological structure from a single primitive — finite enforcement capacity — through a chain of papers plus a machine-verifiable engine. Each paper has a main text and, where applicable, a Technical Supplement, deposited on Zenodo and collected in the admissibility_physics community; the canonical engine (APF-Engine v24.3.427, 3,918 bank-registered checks) backs every quantitative claim. The full series index is at admissibilityphysics.com. DOIs below are concept DOIs (they always resolve to the latest version). Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — 10.5281/zenodo.18439200 · supplement Paper 2 — Finite Admissibility and the Failure of Global Description — 10.5281/zenodo.18439274 · supplement Paper 3 — Entropy, Time, and Accumulated Cost — 10.5281/zenodo.18439363 · supplement Paper 4 — Admissibility Constraints and Structural Saturation — 10.5281/zenodo.18439397 · supplement Paper 5 — Quantum Structure from Finite Enforceability — 10.5281/zenodo.18439433 · supplement Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — 10.5281/zenodo.18439445 · supplement Paper 7 — A Minimal Quantum of Action from Finite Admissibility — 10.5281/zenodo.18439513 · supplement Paper 8 — The Admissibility-Capacity Ledger — 10.5281/zenodo.19721384 · supplement Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — 10.5281/zenodo.20041675 · supplement Paper 10 — The Calculus of Finite Continuability — 10.5281/zenodo.20041680 · supplement Paper 11 — Forced Universality from Capacity-Bounded Admissibility — 10.5281/zenodo.20684198 · supplement Paper 12 — The Gauge Sector of Admissibility Physics — 10.5281/zenodo.21212521 · supplement Paper 13 — The Minimal Admissibility Core — 10.5281/zenodo.18361446 Paper 14 — The Quantum Story — 10.5281/zenodo.21212188 · supplement Paper 16 — Markov Breakdown and the Hard Problems — 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — 10.5281/zenodo.20684209 Paper 19 — Large-Angle CMB Suppression as a Finite-Continuability Constraint — 10.5281/zenodo.21199218 Paper 20 — The Enforcement Crystal — 10.5281/zenodo.18531732 · supplement Engine (Paper 21) — Admissibility Physics: An Executable Constraint Framework and Constants Map — 10.5281/zenodo.18529115 Paper 23 — Magnetically Dark Winding-Class Quantum Processors — 10.5281/zenodo.21199220 Paper 24 — The Recruitment-Radius Extension — Foundations — 10.5281/zenodo.20684211 · supplement Paper 25 — The Recruitment-Radius Extension: Applications — 10.5281/zenodo.21199222 · supplement Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — 10.5281/zenodo.20684222 Paper 32 — Quark Anchors as APF Trace Masses — 10.5281/zenodo.21199226 Paper 33 — Trace-to-Scheme Export Architecture — 10.5281/zenodo.20684224 · supplement Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — 10.5281/zenodo.20684228 · supplement Paper 36 — The Missing Floor — 10.5281/zenodo.21199228 Paper 37 — Collapse as Realignment — 10.5281/zenodo.21199231 · supplement Paper 38 — Unification as Ledger — 10.5281/zenodo.21199233 · supplement Paper 39 — Coherent Phase Regimes in the APF Interface Engine — 10.5281/zenodo.21199235 · supplement Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — 10.5281/zenodo.20684235 · supplement Paper 41 — The Horizon as a Continuation Ledger — 10.5281/zenodo.20684241 · supplement Paper 42 — The Weak Mixing Angle Is Not Free — 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — 10.5281/zenodo.21195874 · supplement Paper 45 — What the Vacuum Can Keep — 10.5281/zenodo.21194195 · supplement Paper 46 — The Strong CP Angle Is Not Free — 10.5281/zenodo.21194197 · supplement Paper 47 — The Confinement Antecedent Is Not Forced — 10.5281/zenodo.21194199 · supplement Paper 48 — The Light-Bending Weight Is Not Free — 10.5281/zenodo.21199627 · supplement Technical Supplements cross-link to their main paper via isSupplementTo and to the companion GitHub repository (github.com/Ethan-Brooke) via isDocumentedBy. Author. Ethan Brooke, Independent Researcher, San Anselmo, California, USA. ORCID: 0009-0001-2261-4682 LinkedIn: linkedin.com/in/ethanbrooke GitHub: github.com/Ethan-Brooke Contact: brooke.ethan@gmail.com