APF Paper 3 Technical Supplement — Entropy, Time, and Accumulated Cost

Technical Supplement to Paper 3 of the Admissibility Physics Framework (APF), Ledgers: Admissible Dynamics, Entropy, and the Arrow of Time. Paper 3 promotes complete positivity from postulate to theorem, derives the directed-enforcement multiplier κ = 2 from irreversibility plus locality, identifies von Neumann entropy as committed capacity, and elevates the Admissibility-Capacity Ledger to a first-class object with six regime projections and four consistency identities. Eight formal sections. §1 Complete positivity as theorem. TCPTP derives the Kraus-form evolution ℰ(ρ) = Σk Kk ρ Kk† with Σk Kk† Kk = 𝟙 directly from PLEC, in explicit comparison to the Stinespring–Choi–GKSL infrastructure (Stinespring 1955, Choi 1975, Lindblad 1976, Gorini–Kossakowski–Sudarshan 1976, Kraus 1983). §2 Directed-enforcement multiplier. Tκ proves κ = 2 from irreversibility + locality. §3 Von Neumann entropy as committed capacity. Tentropy identifies S(ρ) = −Tr(ρ log ρ) with the committed-capacity functional, anchored to the strong-subadditivity backbone (Lieb–Ruskai 1973, Wehrl 1978). §4 Second law and arrow of time as structural consequences of Lirr, not phenomenological postulates. §5 ACC and horizon entropy. SBH = A/4 with the formal proof of the ½ × ½ bookkeeping. §6 The Admissibility-Capacity Ledger. TACC unification elevates (K, deff) to a single ledger with six regime projections πT (thermodynamic), πG (gauge), πQ (quantum), πF (fermion), πC (cosmological), πA (action), and four consistency identities I1, I2, I3, I4 linking holographic, gauge–cosmological, thermo-quantum, and action–thermo readings. §7 Interface-sector bridge. Tinterface sector bridge is the subspace-level refinement: the two recurring "42"s in the codebase — the T11 cosmological partition and the Lself exclusion commitment decomposition — co-witness on the same 42-dimensional subspace Vglobal ⊂ V61. §8 Fisher-information metric. The Fisher metric on admissible states; RG flow expressed as Fisher gradient descent (Braunstein–Caves 1994, Petz 1996, Amari–Nagaoka 2000). Red-team appendix flags four additional hypotheses at point of use: H1 (Markovian-composition closure), H2 (Hamiltonian-constraint redundancy), H3 (constraint-functional monotonicity), H4 (subspace-level uniqueness of Vglobal). Theorem index and dependency-diagram appendices close the document. 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