APF Paper 8 — The Admissibility-Capacity Ledger

The Admissibility-Capacity Ledger formalises a two-scalar record ACC (K, deff) = (K, K·ln deff) read through six regime projections πT, πG, πQ, πF, πC, πA (thermo / gravitational / quantum / gauge-field / cosmological / action), tied together by four consistency identities I1, I2, I3, I4. Paper 8’s only nontrivial structural claim is the gauge-cosmological bridge I2 (Theorem 1. 1 of the Technical Supplement Formal Kernel): there exists a unique 𝐺SM-invariant 42-dimensional subspace VΛ ⊂ V61 under the Standard Model gauge action and the T12 partition constraint, forcing the residual partition 3 + 16 + 42 = 61 and the cosmological fraction ΩΛ = 42/61 ≈ 0. 6885 (Planck 2018: 0. 6847 ± 0. 0073, agreement 0. 5σ). Hierarchy of claims (read this before anything else) Paper 8 contains four regime-consistency identities. Three are intentionally modest: I1 — horizon convention identity. Definitional under the cell-count convention Khorizon = A/ (4ℓP2). I3 — thermo-quantum closure SvN (ρmax) = ln dim ℋ = ACC. Standard finite-dimensional identity. I4 — action-thermo closure limβ→0 ln Z (β) = K ln deff. Standard high-temperature partition-function limit. Only I2 is nontrivial. The paper’s weight is there, in a single conventionally-statable theorem backed by a self-contained Maschke-semisimplicity proof (Hall 2015 as the only external dependency). Core theorems Theorem 1. 1 (Gauge-cosmological bridge). Given V61 with explicit ordered basis (12 gauge + 4 Higgs + 45 fermion slots), the GSM = SU (3) × SU (2) × U (1) representation ρSM on V61, and the T12 partition V61 = Vlocal ⊕ VΛ with dim Vlocal = 19: there exists a unique GSM-invariant 42-dim complement VΛ; the decomposition V61 = Vb ⊕ Vc ⊕ VΛ satisfies 3 + 16 + 42 = 61; under the Fractional Reading Equivalence, ΩΛ = dim VΛ / dim V61 = 42/61. Theorem 5. 6 (SM ACC record uniqueness). Under four stated upstream identifications (U1) – (U4) from Papers 2/4/6, the SM ACC record (KSM, deffSM) = (61, 102) is unique. Theorem 7. 14′ (Type III1 factor at infinite volume). Composition of Araki 1968 extremal-KMS factor property + Kastler–Haag + Connes 1973 classification on the UHF algebra indexed by ledger slots; assumptions (A1 non-degenerate local H, A2 translation action, A3 limit state) stated explicitly. Theorem C. 5 (Single-scale impossibility). No single-scale carrier supports a non-trivial πF + πQ + πC simultaneously. Two-tier architecture is structurally mandatory. §R Minimal Working Example. K = 3, deff = 4 toy interface with all six projections in closed form, one bridge identity verified by hand, one operator expectation verified by hand, 10-line numpy reproduction with committed expected output. Key predictions ΩΛ = 42/61 ≈ 0. 6885 (Planck 2018: 0. 6847 ± 0. 0073, 0. 5σ). ρΛ / MPl4 = 42/10262 ≈ 10−122. 91 (observed: 10−122. 90, agreement 3% = 0. 012 decades, zero free parameters). H0 = 70. 03 km/s/Mpc via GR ρcrit inversion (midpoint of the Planck–SH0ES tension; 7. 09σ vs H0DN 2026 is stated as a framework falsifier in Paper 6 §11. 4). (TCMB / MPl) 4 = 48/10264 matches FIRAS to 0. 33%. Multivariate Planck 2018 Bayes factor over (Ωb, Ωc, ΩΛ) with ρcΛ ≈ −0. 95: χ2 = 1. 18 for 3 d. o. f. (p = 0. 76, Mahalanobis 1. 09σ). Phase 22 anti-smuggling discipline The v2. 9 / v2. 5 release includes a code-level anti-smuggling layer: accSM () now reads the residual partition (3, 16, 42) from upstream DAG keys populated by Lₑquip instead of hardcoding. apf/testₙoₛmuggling. py contains five adversarial mutation tests proving (i) accSM honours DAG mutations; (ii) I2 fails loudly on sum-violating partitions (non-tautological) ; (iii) CANON constants agree with Lequip derivation at v6. 9 (no drift) ; (iv) a toy rep-theory witness showing dimension-only complements are not G-invariant. apf/formalₖernel. py provides an executable witness for Theorem 1. 1: constructs V61 with explicit irrep structure, builds a representative GSM action via independent phase labels per irrep, verifies VΛ existence + uniqueness, and confirms adversarially that a random 42-dim subspace is NOT G-invariant. Code and reproducibility GitHub repository: APF-Paper-8-Admissibility-Capacity-Ledger Colab walkthrough (one-click): Open APFReviewerWalkthrough. ipynb — typeset theorem statements, prose walkthroughs, colour-coded epistemic badges, live anti-smuggling test run. Interactive DAG: GitHub Pages (dependency graph of 420 bank-registered theorems + 437 verifyₐll checks). Audit-native bundle in the repo’s aicontext/: ARGUMENTFLOW. md (one-page spine), LOCALVSIMPORTED. md (five-category partition), CLAIMSLEDGER. md (row-by-row attack surface), DONOTCLAIM. md (20 anti-hallucination guards). Minimal working example: minimalworkingₑxample/toyᵢnterfaceₙumpy. py — run with python3 minimalworkingₑxample/toyᵢnterfaceₙumpy. py, compare to committed expected output, verify the ledger machinery at K = 3, deff = 4 in under 30 seconds. Accompanies the Technical Supplement (107 pp, separate Zenodo deposit) and the v6. 9 canonical codebase. Paper 8 sits in position 8 of the APF series Core Theoretical Spine, downstream of the field-content derivation (Paper 4) and the spacetime/gravity formalism (Paper 6), upstream of no other paper (it is the architectural synthesis endpoint of the spine). APF paper series (cross-reference): Engine — Admissibility Physics: Unified Theorem Bank & Verification Engine — 10. 5281/zenodo. 18604548 — GitHub Derivation — The Weak Mixing Angle as a Capacity Equilibrium — 10. 5281/zenodo. 18603209 Paper 0 — What Physics Permits — 10. 5281/zenodo. 18605692 Paper 1 — The Enforceability of Distinction — 10. 5281/zenodo. 18604678 Paper 2 — Finite Admissibility and the Failure of Global Description — 10. 5281/zenodo. 18604839 Paper 3 — Entropy, Time, and Accumulated Cost — 10. 5281/zenodo. 18604844 Paper 4 — Admissibility Constraints and Structural Saturation — 10. 5281/zenodo. 18604845 Paper 5 — Quantum Structure from Finite Enforceability — 10. 5281/zenodo. 18604861 Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — 10. 5281/zenodo. 18604874 Paper 7 — A Minimal Quantum of Action from Finite Admissibility — 10. 5281/zenodo. 18604875 Paper 8 — The Admissibility-Capacity Ledger (this deposit) Paper 13 — The Minimal Admissibility Core — 10. 5281/zenodo. 18614663 Paper 14 — The Enforcement Crystal — 10. 5281/zenodo. 18615555 Author: Ethan S. Brooke — brooke. ethan@gmail. com — ORCID 0009-0001-2261-4682 — LinkedIn — GitHub