What question did this study set out to answer?

The paper aims to establish a minimal quantum of action from finite admissibility and reveal its connections to various fundamental principles.

April 25, 2026Open Access

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

Puntos clave

The paper aims to establish a minimal quantum of action from finite admissibility and reveal its connections to various fundamental principles.
Derived the non-interacting partition function Z0(β) from the PLEC saturation form.
Employs the Seeley–DeWitt expansion for heat trace derivations related to the cosmological constant and Einstein–Hilbert action.
Conducts a red-team audit to analyze additional hypotheses related to the spectral triple and causal-order topology.
Demonstrated the equivalency of the Connes–Chamseddine spectral action with the APF partition function under specific conditions.
Established how the Seeley–DeWitt expansion reproduces the cosmological constant, thus reaffirming the structure of the Einstein equations in four dimensions.
Highlighted extensions to the Standard Model through the interactions described by the derived one-loop β-function coefficients.

Resumen

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, and the Seeley–DeWitt expansion of the resulting heat trace 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) + constitutive semantics SP, K3 + 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 (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; the gapped-ground-state regime is 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 (checkₛpectralₐctionᵢnternal 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 About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, extending a single axiom (finite information capacity) through the Standard Model gauge group, fermion content, quantum formalism, Lorentzian spacetime, Einstein field equations, cosmological constant, and minimum quantum of action. Each paper's main text and Technical Supplement is deposited separately on Zenodo; each paper has a companion GitHub repository with the vendored apf/ codebase (v6. 9, 376 bank-registered theorems across 23 modules, 48 quantitative predictions), a one-click Colab notebook, and an interactive D3. js dependency DAG. Engine — Admissibility Physics Unified Theorem Bank & Verification Engine — DOI 10. 5281/zenodo. 18604548 · GitHub Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10. 5281/zenodo. 18605692 · GitHub Paper 1 — The Enforceability of Distinction — DOI 10. 5281/zenodo. 18604678 · GitHub Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10. 5281/zenodo. 18604839 · GitHub Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10. 5281/zenodo. 18604844 · GitHub Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10. 5281/zenodo. 18604845 · GitHub Paper 5 — Quantum Structure from Finite Enforceability — DOI 10. 5281/zenodo. 18604861 · GitHub Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10. 5281/zenodo. 18604874 · GitHub Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10. 5281/zenodo. 18604875 · GitHub Paper 8 — The Admissibility-Capacity Ledger — main paper DOI pending · GitHub Paper 13 — The Minimal Admissibility Core — DOI 10. 5281/zenodo. 18614663 · GitHub Companion derivation: The Weak Mixing Angle as a Capacity Equilibrium — DOI 10. 5281/zenodo. 18603209 Technical Supplement DOIs for Papers 1–8 (this series of deposits) cross-link to each main paper DOI via isSupplementTo and to each companion GitHub repository 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

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