APF Paper 5 Technical Supplement — Quantum Structure as Least-Cost Admissible Bookkeeping (v5.97)

Technical Supplement to Paper 5 of the Admissibility Physics Framework (APF), Quantum Structure as Least-Cost Admissible Bookkeeping. v5. 97 (April 2026), 157pp, 36 sections, 235 theorem-class environments, 59 bibitems. The supplement gives a finite-regime reconstruction of the Hilbert–Born formalism from APF's enforcement-substance primitives: complex Hilbert space, the Born trace rule, tensor products, CPTP dynamics, and measurement as record-locking, all derived rather than postulated. Headline result. The Hilbert–Born endpoint is licensed by a chain of finite, certifiable gates traceable to a single APF primitive — closed-world ledger conservation paired with no-phantom-records. The framework is in the spirit of canonical quantum-reconstruction programs (Hardy 2001; Chiribella–D'Ariano–Perinotti 2011; Masanes–Müller 2011; Barnum–Wilce 2014). Core spine (sections in order): Quantum Admissibility Condition (QAC) — produced at branch- (IJC) record-complete coherent interfaces; coherent continuations whose Boolean record-locking incurs strictly positive preservation distortion. Branch taxonomy Sepstr / Sepadm / IJCstr / IJCadm / IJCpres / Seppres with forced inclusions Sepadm ⇒ Sepstr, IJCstr ⇒ IJCadm, and explicit anti-conflation: capacity-only failure (Sepstr + IJCadm) is structurally classical and does NOT trigger QAC. Operational radical via no-phantom-record quotient — every element of the operational radical does no record-distinguishing work and is quotient-eliminable; a Wedderburn bridge identifies ϱop = Jac (𝒜) under stable-simple completeness. Finite matrix-sector representation — the faithful semisimple quotient enters a finite ⊕-of-matrix-blocks structure via Artin–Wedderburn applied to the closed-world quotient. Field selection by SPLIT closed-world composite gates — APF-complete composite closure decomposes into two independently-derivable conditions: (i) finite tensor closure (rules out ℍ structurally because Mn (ℍ) ⊗ℝ Mm (ℍ) ≅ M2nm (ℝ), not quaternionic), and (ii) finite tomographic locality (rules out ℝ via Wootters–Hardy local-marginal parameter count: 2×2 joint dimension 10 < locally-reconstructible 15). ℂ is the unique field passing both legs; the selection is a conjunction of independently-derivable conditions, not a single black-box axiom. Born trace rule ⟨E, ρ⟩ = Tr (ρE) — every positive linear functional on Mn (ℂ) is a density-matrix-and-effect pairing; trace-rule positivity is licensed by upstream cone-preservation. Gate-certified Hilbert–Born pipeline — composite meta-theorem: four gates (positivity preservation under quotient + closed-ledger reciprocity + operational-radical-equals-Jacobson + split closed-world complex selection) jointly license the H–B endpoint. Each gate is independently necessary; no single gate is redundant. If any gate fails, the framework stops at the corresponding fallback rather than silently importing quantum formalism. The derivation chain made explicit. The supplement makes structurally visible that the standard "regime gates" — reciprocal calibration → self-duality + adjoint, stable simple-record completeness, APF-complete finite composite closure → ℂ selection — are not independent postulates. They are joint consequences of closed-world ledger conservation + no-phantom-records: Tclosed-ledger reciprocity derives reciprocal calibration → adjoint from no-hidden-debt symmetric pairing B (p, m) = ∑ pi mi on a finite ledger; the adjoint is the swap partner under B, not a postulated involution. Tno-phantom-record quotient + Toperational-radical-equals-Jacobson derive stable simple-record completeness on 𝒜 = ℝx/ (x3), giving a quotient 𝒜/ (x) ≅ ℝ that is semisimple and eligible for matrix-sector classification. Tsplit-closed-world-complex-selection derives ℂ-selection through the explicit split into tensor closure + tomographic locality. Six countermodels and red-team. Explicit countermodels for Weinberg non-linear quantum mechanics, real-amplitude QM, Bohmian mechanics, Kochen–Specker hidden variables, GRW/CSL collapse, quaternionic QM. Red-team analyses for the three paper-specific hypotheses (H1 coherent-distinction closure, H2 tensor-product admissibility, H3 spectral-action compatibility). Theorem index, dependency diagram, regime-gate certification audit table, and changelog close the document. Readable without prior exposure to the APF series. Code and reproducibility. GitHub repository Colab walkthrough notebook (one-click) Interactive dependency DAG Codebase backing (v7. 5). 23 bank-registered checks span the supplement's quantum-derivation chain across three modules: apf/aps. py (Admissible Possibility Space primitive + continuation preorder, 3 checks) ; apf/quantumₐdmissibility. py (branch-taxonomy inclusions, κBool minimum, capacity lower-bound certificate, QAC, ℂ-selection uniform-defect form, Born trace rule — 6 checks) ; apf/closedworldcompleteness. py (the closed-world derivation chain itself, 14 checks: closed-ledger reciprocity, no-phantom-record quotient, operational-radical-equals-Jacobson, positive-cone quotient compatibility, split tensor closure, split tomographic locality, split closed-world ℂ-selection, preservation-IJC obstruction, constructive commuting realization, closed read/write self-duality, capacity-only-distinct-from-structural-IJC, gate-certified Hilbert–Born pipeline, closed-world-completeness-derives-three-gates, adjoint closure of stable simple sectors). All 471 verifyₐll checks load cleanly; 467 PASS. 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