APF Paper 5 — Quantum Structure from Finite Enforceability (Technical Supplement)

Abstract

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 derives what canonical quantum-reconstruction programs (Hardy 2001; Chiribella–D'Ariano–Perinotti 2011; Masanes–Müller 2011; Barnum–Wilce 2014) postulate. 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. 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 = ℝ[x]/(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/quantum_admissibility.py (branch-taxonomy inclusions, κBool minimum, capacity lower-bound certificate, QAC, ℂ-selection uniform-defect form, Born trace rule — 6 checks); apf/closed_world_completeness.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 verify_all checks load cleanly; 467 PASS. 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 reposito