Admissibility Physics

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Published May 5, 2026 | Version v6.4
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APF Paper 5 Technical Supplement: Quantum Structure from Finite Enforceability

Authors/Creators

  • 1. Independent Researcher, San Anselmo, California, USA

Description

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 ?op = Jac(??) 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 ? structurally because Mn(?) ⊗? Mm(?) ≅ M2nm(?), not quaternionic), and (ii) finite tomographic locality (rules out ? via Wootters–Hardy local-marginal parameter count: 2×2 joint dimension 10 < locally-reconstructible 15). ? 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(?) 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 → ? 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 ?? = ?[x]/(x3), giving a quotient ??/(x) ≅ ? that is semisimple and eligible for matrix-sector classification.
  • Tsplit-closed-world-complex-selection derives ?-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.

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, ?-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 ?-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 treats finite enforcement seriously as substance. Three primitive commitments about physical content, plus a finite-physical-regime hypothesis, generate the Standard Model and cosmological structure with zero free parameters via the Principle of Least Enforcement Cost. Papers 0–10 form the core derivational sequence; Paper 13 is the master reference. Each paper’s main text and Technical Supplement is deposited separately on Zenodo; each paper has a companion GitHub repository carrying the vendored apf/ codebase, a one-click Colab notebook, and an interactive D3.js dependency DAG.

Author. Ethan Brooke, Independent Researcher, San Anselmo, California, USA.

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