APF Paper 2 Technical Supplement — Finite Admissibility and the Failure of Global Description

Technical Supplement to Paper 2 of the Admissibility Physics Framework (APF), Finite Admissibility and the Failure of Global Description. Paper 2 carries the structural chain from PLEC through to the Standard Model gauge group SU(3) × SU(2) × U(1), the complete fermion content (45 chiral fermions from a 1,680-template scan), the cosmological capacity partition 42 + 3 + 16 = 61 with total effective dimension deff = 102, and the derived mixing ratio sin2θW = 3/13 at tree level with ratio constant γ = 17/4. The supplement provides self-contained proofs for every theorem whose derivation depends on companion-paper material, organised in three parts. Part I — Lorentzian spacetime from the cost kernel. Cost-function uniqueness lemma LCauchy (forcing F(d) = d and hence γ = 17/4); the algebraic bridge LΔ → Talg → gauge identification from superadditivity; three carrier prerequisites T7B, Lirr,uniform, B1′ closing Theorem R's dependency on formerly external lemmas; causal ordering from Lirr; spacetime dimension d = 4 from Lovelock uniqueness (T8) plus degrees-of-freedom minimality; Einstein field equations via Δsig, T9, T9,grav. Part II — Gauge template and matter content. The exhaustive Lie-algebra classification Lgauge,unique selecting 𝔰𝔲(3) ⊕ 𝔰𝔲(2) ⊕ 𝔲(1) as the unique admissible template; the 1,680-template fermion scan with seven formally specified filters (Tfield), admitting exactly 45 Standard-Model fermions out of 4,680 candidates; anomaly cancellation by construction; hypercharge uniqueness from capacity arguments. Part III — Capacity counting and cosmology. The MECE capacity partition Ctot = 45 + 4 + 12 = 61; horizon equipartition lemma Lequip; the bridge to the Friedmann equations conditional on the horizon identification established in Paper 6; saturation independence and uncertainty analysis. The structural partition is proved here; its identification with the Friedmann density fractions (Ωb, Ωc, ΩΛ) is deferred to Paper 6. The one-loop β-function sign is verified using an imported formula; its derivation from capacity arguments is deferred to Paper 4A. Appendix C provides complete, self-contained proofs of all ten results imported from the Paper 1 Technical Supplement, so no external document is needed to verify any theorem in this supplement. Appendix A compares the framework with existing reconstruction programs and "Standard Model from principles" approaches. Code and reproducibility. Every theorem traces to a named check in the apf/ codebase. 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