APF Paper 8 Technical Supplement — The Admissibility-Capacity Ledger

Technical Supplement to Paper 8 of the Admissibility Physics Framework (APF), The Admissibility-Capacity Ledger. Paper 8 is a formal framework paper: the unifying record (K, deff) is read through six regime projections πT (thermodynamic), πG (gauge), πQ (quantum), πF (fermion), πC (cosmological), πA (action), with four consistency identities I1, I2, I3, I4 encoding holographic, gauge–cosmological, thermo-quantum, and action–thermo coherence respectively. Hierarchy of claims. Three of the four identities are intentionally modest: I1 is a horizon-convention identity under the Bekenstein cell-count convention Khorizon = A/(4ℓP2), reproducing SBH = A/(4ℓP2) by construction; I3 is the standard finite-dimensional entropy closure SvN(ρmax) = ln N; I4 is the standard high-temperature limit limβ → 0 ln Z(β) = ln N. The paper's only non-trivial structural claim is the gauge–cosmological bridge I2: Theorem 1.1 (Formal Kernel; Gauge–Cosmological Bridge). There exists a unique GSM-invariant 42-dimensional subspace VΛ ⊂ V61 under the Standard-Model gauge action GSM = SU(3) × SU(2) × U(1) satisfying the T12 partition constraint. This subspace induces the residual partition 3 + 16 + 42 = 61 and the cosmological fraction ΩΛ = 42/61 ≈ 0.6885. Proof: Maschke's complete-reducibility theorem (Hall 2015) applied to the GSM-representation on V61, combined with dimension-counting under the T12 partition. What the supplement provides. The self-contained proof of Theorem 1.1 via Maschke semisimplicity. A minimal working example at toy interface K = 3, deff = 4 auditable in ≤10 lines of numpy. A full-covariance Planck 2018 multivariate Bayes factor across (Ωb, Ωc, ΩΛ) yielding χ2 = 1.18 for 3 d.o.f., p ≈ 0.76. A finite-volume C*-algebra formulation with Type III1 factor classification at infinite volume (Araki 1968 / Connes 1973 / Bratteli–Robinson). Two-tier carrier construction with the single-scale impossibility theorem (Theorem 5.5 gauge-uniqueness strengthening). Sensitivity theorem quantifying the brittleness of downstream consequences under unit shifts in upstream integer counts. A self-contained APF-internal derivation of the Standard-Model ACC record (KSM, deffSM) = (61, 102) from Papers 2 / 4 / 6 upstream identifications. Centerpiece §O ("The Gauge–Cosmological Bridge: Integrated Nontrivial Content") gathers Theorem 1.1, the forcing chain, two-tier uniqueness, and brittle consequences into a single focal reading. What this paper does not claim. A competitive joint likelihood fit to Planck / BAO / SN under full nuisance parameters; a derivation of log-A corrections to SBH that microscopic frameworks (LQG, string, SUSY-localisation) supply; a first-principles microscopic derivation of the integer counts (61, 102) beyond APF's internal stack; a resolution of the observer-dependence question (formalised with explicit transformation laws, not resolved). Four Phase 16–20 reviewer-response passes culminated in this v2.5 supplement + v2.9 main paper. 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