APF Paper 1 Technical Supplement — The Enforceability of Distinction

Technical Supplement to Paper 1 of the Admissibility Physics Framework (APF), The Enforceability of Distinction. Paper 1's central formal claim is that a finite-dimensional complex Hilbert space, Born's rule, and a noncommutative operator *-algebra emerge from four structurally independent components of the Principle of Least Enforcement Cost (PLEC): A1 (finite enforcement capacity, upper bound on admissibility), MD (minimum distinction, uniform per-test cost floor), A2 (argmin selection, variational realisation), and BW (budget-window non-degeneracy of the cost spectrum). These four components are structurally separate and not merged. The supplement reconstructs the operator skeleton in three tiers. The core chain derives a finite-dimensional substrate from A1; a sector decomposition from the irreducibility lemma Lirr; a sector-orthogonal bilinear form ⟨·,·⟩ from A2's cost functional; self-adjoint projection operators; superadditivity from sector closure; a noncommutative *-algebra 𝒜 via Lnc with the Wedderburn–Artin decomposition 𝒜 ≅ ⨁i Mni(𝔽); and a GNS Hilbert space ℋ with faithful state ω. Field selection — the only tier requiring additional closure beyond PLEC — forces 𝔽 = ℂ: theorem T2c.1 excludes 𝔽 = ℝ via state-separation completeness at a single interface; theorem T2c.2 excludes 𝔽 = ℍ (quaternions) via composition closure across multi-interface junctions. External corollaries invoked but not rederived include Born's rule (Busch 2003), Tsirelson's bound (Cirel'son 1980), no-broadcasting (Barnum–Caves–Fuchs–Jozsa–Schumacher 1996), and the monogamy of entanglement. Constitutive semantics (SP simple-path, K3 closure, FD1–FD6 distinction-failure axioms) supply definitions but no empirical content; every auxiliary structure (vector space, inner product, adjoint, *-algebra, Hilbert space) is derived or explicitly constructed, with logical status flagged at point of use. The supplement is standalone — no companion-paper result is assumed — and is the canonical source of truth for Paper 1's formal content. Code and reproducibility. Every theorem cited in the main paper traces to a named check function in the bundled apf/ package. The companion repository contains the full codebase, manuscript .tex/.pdf, interactive D3.js dependency DAG, and a one-click Colab notebook that reproduces all Paper 1 results from first principles. GitHub repository Colab walkthrough notebook (one-click, no setup required) 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