Technical Supplement to Paper 4 of the Admissibility Physics Framework (APF), Admissibility Constraints and the Standard Model Field Content. Paper 4 specifies the complete Standard-Model field content — 45 chiral fermions, 4 electroweak gauge bosons, 12 QCD gauge bosons, totalling Ctot = 61 — as the unique admissible solution to the seven structural filters of Paper 2's 1,680-template scan, and derives the fermion mass theorem TM and the PMNS mixing matrix. Contents. (i) Full assumption inventory including paper-specific structural hypotheses layered on top of PLEC: chirality closure, anomaly cancellation by construction, and the hypercharge-uniqueness constraint. (ii) Formal proofs for the main theorems (compressed in the main paper to ≤20-line sketches): Mass theorem TM. The chiral fermion mass spectrum derived from admissibility saturation at the electroweak interface. The charm-quark mass mc is irreducible at the 2.6% Schur structural limit; the top-quark mass mt and bottom-quark mass mb are Type-B anchors (open, requiring absolute-scale σ-derivation). Across the currently tested mass predictions, APF achieves mean error 3.83% and median error 0.37%. PMNS derivation. Neutrino mixing matrix elements derived from admissibility constraints on the leptonic sector, including the CP-violating Dirac phase δPMNS. Type-I seesaw closure for right-handed neutrino masses, with the seesaw scale set by capacity saturation. (iii) Countermodels explicitly excluding the main alternative field-content families — four-generation models, extended gauge groups, non-chiral matter, and vector-like fermion augmentations — by constraint violation. (iv) Red-team section analysing the paper-specific structural hypotheses under the v6.9 Phase-13 PLEC anchor analysis, flagging which filters are P internally derived versus P+lattice requiring lattice-QCD input. (v) Theorem index and dependency diagram. (vi) Changelog from the February 2026 PDF-only release through the April 2026 v2.0 rebuild which restored the supplement to full dual-document architecture and added the Type-I seesaw formalisation. The supplement is the canonical source of truth for Paper 4's formal content. Where the main paper and the supplement appear to conflict, the supplement is correct by convention. 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
Ethan Brooke (Fri,) studied this question.