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

Abstract

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 gives 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 structural filters (Tfield), admitting 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 of Paper 6. Saturation independence and uncertainty analysis. The structural partition is proved here. Its identification with the Friedmann density fractions (Ωb, Ωc, ΩΛ) is taken up in Paper 6. The one-loop β-function sign is verified using an imported formula. Its derivation from capacity arguments is taken up in Paper 4A. Appendix C gives complete, self-contained proofs of all ten results imported from the Paper 1 Technical Supplement. 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 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