APF Paper 5 Technical Supplement — Quantum Structure as Least-Cost Admissible Bookkeeping

Technical Supplement to Paper 5 of the Admissibility Physics Framework (APF), Quantum Structure as Least-Cost Admissible Bookkeeping. Paper 5 derives the complete quantum formalism — linearity, complex Hilbert space, tensor products, Born's rule, unitary and CPTP dynamics, measurement as class change, spectral-action internalisation, and UV-completion closure — from PLEC (A1, MD, A2, BW) plus the capacity quantization Ctot = 61, deff = 102 imported from Papers 1–4. The supplement provides self-contained proofs of the theorems compressed to sketch form in the main paper. Linearity from coherent-distinction closure. Tlinearity: if sector states |ψ1⟩, |ψ2⟩ are admissible and coherent-distinction-closed, then their normalised linear combination α|ψ1⟩ + β|ψ2⟩ with |α|2 + |β|2 = 1 is admissible. Complex Hilbert space. THilbert: the state space is a complex Hilbert space ℋ with inner product induced by the cost functional, and the field selection 𝔽 = ℂ is inherited from Paper 1's field-selection tier. Tensor product. Ttensor: composite systems factorise as ℋAB = ℋA ⊗ ℋB under admissibility of joint distinction. Born's rule. TBorn: measurement outcome probabilities are Pr (k | ρ) = Tr (Pk ρ) with Pk the projective decomposition operators corresponding to admissible partition classes — derived, not postulated. Unitary and CPTP dynamics. THermitian and TCPTP: admissible state evolution is Hermitian-generated for closed systems and CPTP for open systems, both from minimum-cost bookkeeping. Measurement as class change. Tdecoherence: measurement is the transition from an admissible coherent superposition to an admissible classical register, formalised as a class change on the equivalence structure of admissible states. Spectral-action internalisation. Lspectralₐctionᵢnternal: the Connes spectral action coincides with the APF partition function under the Boltzmann cutoff. Full proof in Paper 7 Technical Supplement §S. 4. UV-completion closure. LQGP1closure: the admissibility framework remains consistent under Paper 1's structural extension to UV scales. Explicit countermodels exclude the main published quantum-reconstruction alternatives (Hardy 2001, Masanes–Müller 2011, d'Ariano–Chiribella–Perinotti 2010, Rovelli relational). Red-team section analyses three paper-specific hypotheses (H1 coherent-distinction closure, H2 tensor-product admissibility, H3 spectral-action compatibility). Theorem index, dependency diagram, 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 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