APF Paper 8 — The Admissibility-Capacity Ledger (Technical Supplement)

Abstract

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. 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). Code and reproducibility. GitHub repository Colab walkthrough notebook (one-click) Interactive dependency DAG Codebase backing. Every theorem traces to a named check function in the canonical apf/ codebase (v7.5; 454 bank-registered theorems, 471 verify_all checks, 39 modules + standalone). Paper 8's I1–I4 consistency identities and Theorem 1.1 (gauge–cosmological bridge) live in apf/unification.py; the three-level integer/scalar/subspace witnesses live in apf/unification_three_levels.py; the I1-bridge construction lives in apf/horizon_joint_bridge.py; the VΛ uniqueness witness lives in apf/formal_kernel.py. 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