A finite joint-query interface admits a single shared classical record if and only if its tested continuation-word algebra embeds faithfully into a commutative algebra of records — if and only if it admits a finite Boolean-record defender — if and only if a finite system of unit, product, recovery, positivity, and state–effect equations is solvable. The failure of every commuting completion is the precise definition of irreducible joint constraint. This is the Sep/IJC Representation Theorem, and it is the centerpiece of The Enforceability of Distinction. What this paper is A standalone formal-foundation paper for finite-interface physics. Self-contained: no prior exposure to the Admissibility Physics Framework is required. The mathematical core is a single biconditional theorem about finite joint-query interfaces, stated and proved in §11 using only finite linear-algebraic content over ℝ. The paper is built on a deliberately small primitive spine of three commitments: Physical identity is finite admissible continuation identity. A physical state is what it can still do. A physical distinction is a finite enforceable separator of continuation profiles. Distinctions are operational, not abstract. Physical distinctions carry positive enforcement cost. The positive cost floor is the only foundational scalar input. Everything else — the physical quotient, descent of distinctions and filters, the enforcement ledger, convex mixtures from finite classical control, and the continuation-word quotient algebra — is a forced construction. The finite-interface ingredients (finite query family Q, finite tested word set 𝒯, operational resolution, positive resolved floor) are not postulates: they are consequences of a finite act of interrogation plus the positive-cost spine. The Sep/IJC Representation Theorem Let (Γ, Q, 𝒯) be a finite joint-query interface with finite query family Q, finite tested word set 𝒯, finite operational resolution, and positive resolved floor. Let 𝒜Q,𝒯 be the tested continuation-word quotient algebra. Then the following are equivalent: (Γ, Q, 𝒯) admits a commuting-extension defender; 𝒜Q,𝒯 embeds faithfully into ℝΛ as a commutative record representation, for some finite labelling Λ; The associated finite Boolean-feasibility system over unit, product, recovery, positivity, and state–effect equations is solvable. The Sep branch is exactly the case where the queried interface fits inside one shared classical record. The IJC branch is the precise failure of every faithful commuting continuation completion. IJC is not a synonym for “quantum.” It is an interface-relative algebraic verdict on whether a single Boolean-record completion exists for the tested word behavior. Quantum interfaces are an empirical instance of the IJC branch — but the IJC classification is independent of any specific physical realization, and there is no postulated identification of IJC with quantum mechanics anywhere in the paper. Surrounding results Refinement stability. Sep is monotone under coarsening of 𝒯; IJC can be revealed by refinement but cannot be erased by it. Branch verdicts are stated relative to both 𝒯 and the chosen operational resolution. Directed finite-test limits. A coherent projective family of finite-stage defenders is required for a global Sep limit; Sep at every finite stage alone is not sufficient. Compactness and bounded-atom hypotheses recover a projective-limit defender. ε-faithful defenders and finite-margin robustness. An approximate-defender theory with explicit operational-norm tolerances; the IJC margin m is preserved under finite-table perturbations of size below m/2. Continuation-word quotient algebra. Typed partial-product structure, incompatible-concatenation rule, contextual congruence, and the two-sided ideal property proved before the quotient is invoked. No-added-resource criterion. Operational sufficient tests and an explicit checklist precede the cost-monotonicity theorem; passive post-processing is the canonical witness. Worked models. A fully worked Sep defender (two classical bits); a Sep with positive interface cost (shared bus); a sequential IJC example (the explicit two-query table); a coupled-but-classical example (shared classical bath). A numerical ledger/floor case study and an explicit κΓ-calibration sensitivity table accompany the foundational sections. Process-category axiom block. A finite APF process-category axiom block; the word algebra realised as a typed path algebra equivalent to a free word algebra modulo contextual-equivalence relations; the defender restated as a partial Boolean-record functor. Trigger-condition discipline Every load-bearing structural theorem is paired with the explicit conditions under which it holds, named at point of use: Cost monotonicity — no-added-resource coarse-graining; Ledger additivity — factorized support; Independent counting — ledger-independent resolution; Robust-floor extension beyond explicitly finite protocols — compact robust-interface conditions R1–R4 (continuity, lower-semicontinuity, bounded atoms, projective coherence); ε-faithful defenders — explicit operational-norm tolerances and finite-margin half-perturbation regime. Each theorem-local condition is stated alongside an explicit failure-mode description and conservative-counting examples (shared coarse record, independent records, coupled discriminator). Reviewers see what the conclusion depends on without searching the paper. Bridges to standard frameworks Section 14 supplies a full finite-static bridge from APF to the published landscape of foundations programmes: a generalized-probabilistic-theory translation (effect algebras, ordered ∗-algebras, accessible GPT fragments, cone equivalence) and finite-static equivalence propositions for Spekkens-style noncontextuality, Fine-style joint distributions, joint-measurability and commuting-dilation criteria, sheaf-theoretic contextuality, sequential instruments and order algebras, the resource-theoretic hierarchy, epBA minimal Boolean embeddings, and the NPA hierarchy with finite-level commuting certificates. In the static finite case: Sep ⇔ commuting dilation ⇔ noncontextual model ⇔ faithful Fine joint distribution ⇔ global Bochner section. The Sep/IJC verdict is a single algebraic invariant that all of these standard frameworks witness from different sides. The novelty boundary is explicit: this paper’s contribution is the four-fold combination — continuation identity + positive cost + ledger bookkeeping + word quotienting — not the standard equivalence between finite Boolean models and commutative algebras. What this paper does not claim This paper does not derive Hilbert space, complex amplitudes, the Born rule, tensor products, density matrices, or the Standard Model. Promoting the commutative record representation to a complex-Hilbert-space construction requires record-positivity and composite-tomography assumptions that are isolated and named in §15. Those assumptions are deferred to a companion paper on quantum structure. This paper does not assert that quantum systems are the only IJC instance, or that classical systems are the only Sep instance. The classification is interface-relative. This paper does not commit to a substance-level reading of admissibility. The Admissible Possibility Space is set-theoretic; the framework’s commitment is structural, not metaphysical. Code and reproducibility The companion repository contains the v8.8 manuscript, the bundled apf/ codebase (every theorem traces to a named check function, including the Phase 21 inseparable-IJC bridge check check_T_inseparable_IJC), an interactive D3.js dependency DAG for the full Admissibility Physics Framework, and a one-click Colab walkthrough notebook. The repository and Colab notebook are being refreshed alongside this v8.8 deposit and will carry the updated v8.8 vocabulary plus the Phase 22 codebase rev (Sep/IJC representation theorem and continuation-word algebra checks) as those land. The links below are stable across the refresh; their content updates in place. GitHub repository — full codebase, manuscript .tex/.pdf, AI-onboarding bundle, audit-native context pack, theorem-by-theorem traceability. One-click Colab walkthrough — reproduces every Paper 1 result from first principles, no setup required, runs end-to-end in your browser. Interactive dependency DAG — explore the bank-registered theorem graph; click any node to surface its check function and dependency chain. Why this paper matters for the series Paper 1 is the bedrock of the Admissibility Physics Framework. v8.8 fixes that bedrock as a finite-interface algebraic theorem with an explicit primitive boundary — positive enforcement cost — and explicit theorem-local triggers, in language an external reviewer can audit without prior APF exposure. The Sep/IJC dichotomy, stated entirely within finite linear algebra over ℝ, is the regime classifier that downstream APF papers (Paper 2 on Standard Model gauge structure, Paper 5 on quantum reconstruction, Paper 6 on dynamics and gravity, Paper 8 on the Admissibility-Capacity Ledger) inherit and refine. The companion quantum-structure paper promotes the commutative record representation to a complex-Hilbert-space construction by adding record-positivity and composite-tomography hypotheses. The argument-first companion volume to this paper — the Paper 1 main body, currently v4.8 at 10.5281/zenodo.18604678 — is being brought up to v4.9 alignment with the v8.8 spec on a parallel track. About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, building physics from finite enforcement of distinction taken seriously as a constitutive commitment. Paper 1 (this paper) is the formal bedrock; downstream papers extend the chain through the Standard Model gauge group, fermion content, quantum formalism, Loren
A finite joint-query interface admits a single shared classical record if and only if its tested continuation-word algebra embeds faithfully into a commutative algebra of records — if and only if it admits a finite Boolean-record defender — if and only if a finite system of unit, product, recovery, positivity, and state–effect equations is solvable. The failure of every commuting completion is the precise definition of irreducible joint constraint. This is the Sep/IJC Representation Theorem, and it is the centerpiece of The Enforceability of Distinction. What this paper is A standalone formal-foundation paper for finite-interface physics. Self-contained: no prior exposure to the Admissibility Physics Framework is required. The mathematical core is a single biconditional theorem about finite joint-query interfaces, stated and proved in §11 using only finite linear-algebraic content over ℝ. The paper is built on a deliberately small primitive spine of three commitments: Physical identity is finite admissible continuation identity. A physical state is what it can still do. A physical distinction is a finite enforceable separator of continuation profiles. Distinctions are operational, not abstract. Physical distinctions carry positive enforcement cost. The positive cost floor is the only foundational scalar input. Everything else — the physical quotient, descent of distinctions and filters, the enforcement ledger, convex mixtures from finite classical control, and the continuation-word quotient algebra — is a forced construction. The finite-interface ingredients (finite query family Q, finite tested word set 𝒯, operational resolution, positive resolved floor) are not postulates: they are consequences of a finite act of interrogation plus the positive-cost spine. The Sep/IJC Representation Theorem Let (Γ, Q, 𝒯) be a finite joint-query interface with finite query family Q, finite tested word set 𝒯, finite operational resolution, and positive resolved floor. Let 𝒜Q,𝒯 be the tested continuation-word quotient algebra. Then the following are equivalent: (Γ, Q, 𝒯) admits a commuting-extension defender; 𝒜Q,𝒯 embeds faithfully into ℝΛ as a commutative record representation, for some finite labelling Λ; The associated finite Boolean-feasibility system over unit, product, recovery, positivity, and state–effect equations is solvable. The Sep branch is exactly the case where the queried interface fits inside one shared classical record. The IJC branch is the precise failure of every faithful commuting continuation completion. IJC is not a synonym for “quantum.” It is an interface-relative algebraic verdict on whether a single Boolean-record completion exists for the tested word behavior. Quantum interfaces are an empirical instance of the IJC branch — but the IJC classification is independent of any specific physical realization, and there is no postulated identification of IJC with quantum mechanics anywhere in the paper. Surrounding results Refinement stability. Sep is monotone under coarsening of 𝒯; IJC can be revealed by refinement but cannot be erased by it. Branch verdicts are stated relative to both 𝒯 and the chosen operational resolution. Directed finite-test limits. A coherent projective family of finite-stage defenders is required for a global Sep limit; Sep at every finite stage alone is not sufficient. Compactness and bounded-atom hypotheses recover a projective-limit defender. ε-faithful defenders and finite-margin robustness. An approximate-defender theory with explicit operational-norm tolerances; the IJC margin m is preserved under finite-table perturbations of size below m/2. Continuation-word quotient algebra. Typed partial-product structure, incompatible-concatenation rule, contextual congruence, and the two-sided ideal property proved before the quotient is invoked. No-added-resource criterion. Operational sufficient tests and an explicit checklist precede the cost-monotonicity theorem; passive post-processing is the canonical witness. Worked models. A fully worked Sep defender (two classical bits); a Sep with positive interface cost (shared bus); a sequential IJC example (the explicit two-query table); a coupled-but-classical example (shared classical bath). A numerical ledger/floor case study and an explicit κΓ-calibration sensitivity table accompany the foundational sections. Process-category axiom block. A finite APF process-category axiom block; the word algebra realised as a typed path algebra equivalent to a free word algebra modulo contextual-equivalence relations; the defender restated as a partial Boolean-record functor. Trigger-condition discipline Every load-bearing structural theorem is paired with the explicit conditions under which it holds, named at point of use: Cost monotonicity — no-added-resource coarse-graining; Ledger additivity — factorized support; Independent counting — ledger-independent resolution; Robust-floor extension beyond explicitly finite protocols — compact robust-interface conditions R1–R4 (continuity, lower-semicontinuity, bounded atoms, projective coherence); ε-faithful defenders — explicit operational-norm tolerances and finite-margin half-perturbation regime. Each theorem-local condition is stated alongside an explicit failure-mode description and conservative-counting examples (shared coarse record, independent records, coupled discriminator). Reviewers see what the conclusion depends on without searching the paper. Bridges to standard frameworks Section 14 supplies a full finite-static bridge from APF to the published landscape of foundations programmes: a generalized-probabilistic-theory translation (effect algebras, ordered ∗-algebras, accessible GPT fragments, cone equivalence) and finite-static equivalence propositions for Spekkens-style noncontextuality, Fine-style joint distributions, joint-measurability and commuting-dilation criteria, sheaf-theoretic contextuality, sequential instruments and order algebras, the resource-theoretic hierarchy, epBA minimal Boolean embeddings, and the NPA hierarchy with finite-level commuting certificates. In the static finite case: Sep ⇔ commuting dilation ⇔ noncontextual model ⇔ faithful Fine joint distribution ⇔ global Bochner section. The Sep/IJC verdict is a single algebraic invariant that all of these standard frameworks witness from different sides. The novelty boundary is explicit: this paper’s contribution is the four-fold combination — continuation identity + positive cost + ledger bookkeeping + word quotienting — not the standard equivalence between finite Boolean models and commutative algebras. What this paper does not claim This paper does not derive Hilbert space, complex amplitudes, the Born rule, tensor products, density matrices, or the Standard Model. Promoting the commutative record representation to a complex-Hilbert-space construction requires record-positivity and composite-tomography assumptions that are isolated and named in §15. Those assumptions are deferred to a companion paper on quantum structure. This paper does not assert that quantum systems are the only IJC instance, or that classical systems are the only Sep instance. The classification is interface-relative. This paper does not commit to a substance-level reading of admissibility. The Admissible Possibility Space is set-theoretic; the framework’s commitment is structural, not metaphysical. Code and reproducibility The companion repository contains the v8.8 manuscript, the bundled apf/ codebase (every theorem traces to a named check function, including the Phase 21 inseparable-IJC bridge check check_T_inseparable_IJC), an interactive D3.js dependency DAG for the full Admissibility Physics Framework, and a one-click Colab walkthrough notebook. The repository and Colab notebook are being refreshed alongside this v8.8 deposit and will carry the updated v8.8 vocabulary plus the Phase 22 codebase rev (Sep/IJC representation theorem and continuation-word algebra checks) as those land. The links below are stable across the refresh; their content updates in place. GitHub repository — full codebase, manuscript .tex/.pdf, AI-onboarding bundle, audit-native context pack, theorem-by-theorem traceability. One-click Colab walkthrough — reproduces every Paper 1 result from first principles, no setup required, runs end-to-end in your browser. Interactive dependency DAG — explore the bank-registered theorem graph; click any node to surface its check function and dependency chain. Why this paper matters for the series Paper 1 is the bedrock of the Admissibility Physics Framework. v8.8 fixes that bedrock as a finite-interface algebraic theorem with an explicit primitive boundary — positive enforcement cost — and explicit theorem-local triggers, in language an external reviewer can audit without prior APF exposure. The Sep/IJC dichotomy, stated entirely within finite linear algebra over ℝ, is the regime classifier that downstream APF papers (Paper 2 on Standard Model gauge structure, Paper 5 on quantum reconstruction, Paper 6 on dynamics and gravity, Paper 8 on the Admissibility-Capacity Ledger) inherit and refine. The companion quantum-structure paper promotes the commutative record representation to a complex-Hilbert-space construction by adding record-positivity and composite-tomography hypotheses. The argument-first companion volume to this paper — the Paper 1 main body, currently v4.8 at 10.5281/zenodo.18604678 — is being brought up to v4.9 alignment with the v8.8 spec on a parallel track. About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, building physics from finite enforcement of distinction taken seriously as a constitutive commitment. Paper 1 (this paper) is the formal bedrock; downstream papers extend the chain through the Standard Model gauge group, fermion content, quantum formalism, Loren
A finite joint-query interface admits a single shared classical record if and only if its tested continuation-word algebra embeds faithfully into a commutative algebra of records — if and only if it admits a finite Boolean-record defender — if and only if a finite system of unit, product, recovery, positivity, and state–effect equations is solvable. The failure of every commuting completion is the precise definition of irreducible joint constraint. This is the Sep/IJC Representation Theorem, and it is the centerpiece of The Enforceability of Distinction. What this paper is A standalone formal-foundation paper for finite-interface physics. Self-contained: no prior exposure to the Admissibility Physics Framework is required. The mathematical core is a single biconditional theorem about finite joint-query interfaces, stated and proved in §11 using only finite linear-algebraic content over ℝ. The paper is built on a deliberately small primitive spine of three commitments: Physical identity is finite admissible continuation identity. A physical state is what it can still do. A physical distinction is a finite enforceable separator of continuation profiles. Distinctions are operational, not abstract. Physical distinctions carry positive enforcement cost. The positive cost floor is the only foundational scalar input. Everything else — the physical quotient, descent of distinctions and filters, the enforcement ledger, convex mixtures from finite classical control, and the continuation-word quotient algebra — is a forced construction. The finite-interface ingredients (finite query family Q, finite tested word set 𝒯, operational resolution, positive resolved floor) are not postulates: they are consequences of a finite act of interrogation plus the positive-cost spine. The Sep/IJC Representation Theorem Let (Γ, Q, 𝒯) be a finite joint-query interface with finite query family Q, finite tested word set 𝒯, finite operational resolution, and positive resolved floor. Let 𝒜Q,𝒯 be the tested continuation-word quotient algebra. Then the following are equivalent: (Γ, Q, 𝒯) admits a commuting-extension defender; 𝒜Q,𝒯 embeds faithfully into ℝΛ as a commutative record representation, for some finite labelling Λ; The associated finite Boolean-feasibility system over unit, product, recovery, positivity, and state–effect equations is solvable. The Sep branch is exactly the case where the queried interface fits inside one shared classical record. The IJC branch is the precise failure of every faithful commuting continuation completion. IJC is not a synonym for “quantum.” It is an interface-relative algebraic verdict on whether a single Boolean-record completion exists for the tested word behavior. Quantum interfaces are an empirical instance of the IJC branch — but the IJC classification is independent of any specific physical realization, and there is no postulated identification of IJC with quantum mechanics anywhere in the paper. Surrounding results Refinement stability. Sep is monotone under coarsening of 𝒯; IJC can be revealed by refinement but cannot be erased by it. Branch verdicts are stated relative to both 𝒯 and the chosen operational resolution. Directed finite-test limits. A coherent projective family of finite-stage defenders is required for a global Sep limit; Sep at every finite stage alone is not sufficient. Compactness and bounded-atom hypotheses recover a projective-limit defender. ε-faithful defenders and finite-margin robustness. An approximate-defender theory with explicit operational-norm tolerances; the IJC margin m is preserved under finite-table perturbations of size below m/2. Continuation-word quotient algebra. Typed partial-product structure, incompatible-concatenation rule, contextual congruence, and the two-sided ideal property proved before the quotient is invoked. No-added-resource criterion. Operational sufficient tests and an explicit checklist precede the cost-monotonicity theorem; passive post-processing is the canonical witness. Worked models. A fully worked Sep defender (two classical bits); a Sep with positive interface cost (shared bus); a sequential IJC example (the explicit two-query table); a coupled-but-classical example (shared classical bath). Process-category axiom block. A finite APF process-category axiom block; the word algebra realised as a typed path algebra equivalent to a free word algebra modulo contextual-equivalence relations; the defender restated as a partial Boolean-record functor. Trigger-condition discipline Every load-bearing structural theorem is paired with the explicit conditions under which it holds, named at point of use: Cost monotonicity — no-added-resource coarse-graining; Ledger additivity — factorized support; Independent counting — ledger-independent resolution; Robust-floor extension beyond explicitly finite protocols — compact robust-interface conditions R1–R4; ε-faithful defenders — explicit operational-norm tolerances and finite-margin half-perturbation regime. Bridges to standard frameworks Section 14 supplies a full finite-static bridge from APF to the published landscape of foundations programmes: a generalized-probabilistic-theory translation (effect algebras, ordered ∗-algebras, accessible GPT fragments, cone equivalence) and finite-static equivalence propositions for Spekkens-style noncontextuality, Fine-style joint distributions, joint-measurability and commuting-dilation criteria, sheaf-theoretic contextuality, sequential instruments and order algebras, the resource-theoretic hierarchy, epBA minimal Boolean embeddings, and the NPA hierarchy with finite-level commuting certificates. In the static finite case: Sep ⇔ commuting dilation ⇔ noncontextual model ⇔ faithful Fine joint distribution ⇔ global Bochner section. The Sep/IJC verdict is a single algebraic invariant that all of these standard frameworks witness from different sides. The novelty boundary is explicit: this paper’s contribution is the four-fold combination — continuation identity + positive cost + ledger bookkeeping + word quotienting — not the standard equivalence between finite Boolean models and commutative algebras. What this paper does not claim This paper does not derive Hilbert space, complex amplitudes, the Born rule, tensor products, density matrices, or the Standard Model. Promoting the commutative record representation to a complex-Hilbert-space construction requires record-positivity and composite-tomography assumptions that are isolated and named in §15. Those assumptions are deferred to a companion paper on quantum structure. This paper does not assert that quantum systems are the only IJC instance, or that classical systems are the only Sep instance. The classification is interface-relative. This paper does not commit to a substance-level reading of admissibility. The Admissible Possibility Space is set-theoretic; the framework’s commitment is structural, not metaphysical. Code and reproducibility The companion repository contains the v8.8 manuscript, the bundled apf/ codebase, an interactive D3.js dependency DAG, and a one-click Colab walkthrough notebook. The repository and Colab notebook are being refreshed alongside this v8.8 deposit and will carry the updated v8.8 vocabulary plus the Phase 22 codebase rev as those land. The links below are stable across the refresh; their content updates in place. GitHub repository — full codebase, manuscript .tex/.pdf, AI-onboarding bundle, audit-native context pack. One-click Colab walkthrough — reproduces every Paper 1 result from first principles, no setup required. Interactive dependency DAG — explore the bank-registered theorem graph in your browser. Why this paper matters for the series Paper 1 is the bedrock of the Admissibility Physics Framework. v8.8 fixes that bedrock as a finite-interface algebraic theorem with an explicit primitive boundary — positive enforcement cost — and explicit theorem-local triggers, in language an external reviewer can audit without prior APF exposure. The Sep/IJC dichotomy, stated entirely within finite linear algebra over ℝ, is the regime classifier that downstream APF papers (Paper 2 on Standard Model gauge structure, Paper 5 on quantum reconstruction, Paper 6 on dynamics and gravity, Paper 8 on the Admissibility-Capacity Ledger) inherit and refine. The companion quantum-structure paper promotes the commutative record representation to a complex-Hilbert-space construction. The argument-first companion volume to this paper — the Paper 1 main body, currently v4.8 at 10.5281/zenodo.18604678 — is being brought up to v4.9 alignment with the v8.8 spec on a parallel track. About the APF series. The Admissibility Physics Framework is a ten-paper derivation chain plus core infrastructure, building physics from finite enforcement of distinction taken seriously as a constitutive commitment. Paper 1 (this paper) is the formal bedrock; downstream papers extend the chain through the Standard Model gauge group, fermion content, quantum formalism, Lorentzian spacetime, Einstein field equations, the cosmological constant, and the minimum quantum of action. Each paper has a companion GitHub repository with the vendored apf/ codebase, a one-click Colab walkthrough, and an interactive D3.js dependency DAG. Engine — Admissibility Physics Unified Theorem Bank & Verification Engine — 10.5281/zenodo.18604548 · GitHub Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — 10.5281/zenodo.18605692 · GitHub Paper 1 (main body) — The Enforceability of Distinction — 10.5281/zenodo.18604678 · GitHub Paper 1 (this deposit) — The Enforceability of Distinction: Admissible Possibility Spaces, Continuation Algebra, and the Sep/IJC Representation Theorem — standalone formal-foundation paper Paper 2 — Finite Admissibility and the Failure of Global Description — 10.5281/zenodo.18604839 · GitHub Paper 3 — Entropy, Time,
Duplicate record. The canonical deposit for this paper is APF Paper 1 — The Enforceability of Distinction (Technical Supplement) (DOI 10.5281/zenodo.19714957). This record is retained because published Zenodo DOIs cannot be removed; please cite the canonical record. Title: Formal Supplement: End-to-End Derivation from Finite Enforcement Capacity to Quantum Structure Authors: Brooke, E. S. Description: Self-contained formal supplement to "The Enforceability of Distinction: Quantum Structure from Finite Enforcement Capacity" (Paper 1 of the Admissibility Physics Framework series). Derives finite-dimensional quantum mechanical structure — Hilbert space, C*-algebra, Born rule, Tsirelson bound, no-broadcasting — from a single resource-theoretic axiom (A1: finite enforcement capacity), one regularity condition (MD: uniform per-test cost floor), one richness premise (BW: budget-window triple), and constitutive admissibility semantics (SP, K3, FD1–FD6). The derivation proceeds in three tiers. The core chain (A1 + MD + BW + constitutive semantics) produces: a finite-dimensional real vector space with sector decomposition, a sector-orthogonal bilinear form forced by disjoint-support additivity (K3), self-adjoint enforcement projections, superadditive joint enforcement costs, a noncommutative real *-algebra, Wedderburn–Artin block decomposition, a GNS Hilbert-space representation, and a faithful positive state constructed from enforcement data. Field selection excludes real nonclassical blocks via state-separation completeness (T2c.1, single-interface, proved) and quaternionic blocks via composition closure (T2c.2, multi-interface, conditional). Standard quantum corollaries — Born rule (via Busch's generalized Gleason theorem), Tsirelson bound (via the Cirel'son identity), monogamy, and no-broadcasting — follow within the derived algebraic skeleton. Key features: (1) explicit variational framework for joint enforcement, including admissible defender class, projection reduction lemma, Grassmannian existence proof, and a theorem that all optimal joint defenders have nontrivial pool support when the superadditivity surplus is positive; (2) formal equivalence between nontrivial pool-supported joint threats, superadditive enforcement cost, and algebraic noncommutativity; (3) a fully explicit 11-step qubit model computing every quantity from enforcement costs through Born probabilities; (4) named support-recruitment monotonicity principle (SRM) separating structural from physical selection of orthogonal projections; (5) explicit computable stability thresholds for Wedderburn block structure and field type under approximate disjoint-support additivity; (6) front-matter tables classifying every result as unconditional, conditional, or dependent on external theorems. 71 pages. Three parts: Part I (Core Derivation, §1–7), Part II (Worked Examples and Detailed Verifications, §8–12), Part III (Discussion, Comparisons, and Scope, §13–14). No companion paper result is assumed. Compiled LaTeX source and PDF included. Keywords: quantum reconstruction, finite-dimensional quantum mechanics, C*-algebra, Hilbert space, Born rule, Tsirelson bound, no-broadcasting theorem, enforcement capacity, admissibility physics, Wedderburn–Artin theorem, GNS construction, noncommutativity, field selection, resource-theoretic foundations Related identifiers: Supplement to: Brooke, E. S., "The Enforceability of Distinction: Quantum Structure from Finite Enforcement Capacity" (submitted to Foundations of Physics) Related to: Brooke, E. S., "The Structure of Admissible Physics" (Paper 2), Zenodo DOI 10.5281/zenodo.18867119 About the series. Admissibility Physics derives the Standard Model gauge group, fermion content, quantum formalism, Lorentzian spacetime, the Einstein field equations, the cosmological constant, and the neutrino, strong-CP, and dark sectors from a single assumption — that enforcement is finite at every causally connected region — with zero free parameters. The core derivation runs Papers 0–11; the extension papers carry the individual sectors. Each paper's main text and Technical Supplement is deposited separately, and every quantitative claim traces to a named, executable check in the public theorem bank (over 3,800 bank-registered theorems across roughly 450 modules, 48 quantitative predictions). Website: admissibilityphysics.com. Engine and theorem bank: https://github.com/Ethan-Brooke/APF-Engine. Full series and latest deposits: Zenodo community admissibility_physics. The Engine — Admissibility Physics: An Executable Constraint Framework and Constants Map — DOI 10.5281/zenodo.18529115. Paper 0 — What Physics Permits: A Constraint-First Framework for Physics — DOI 10.5281/zenodo.18439523 Paper 1 — The Enforceability of Distinction — DOI 10.5281/zenodo.18439200 Paper 2 — Finite Admissibility and the Failure of Global Description — DOI 10.5281/zenodo.18439274 Paper 3 — Entropy, Time, and Accumulated Cost — DOI 10.5281/zenodo.18439363 Paper 4 — Admissibility Constraints and Structural Saturation — DOI 10.5281/zenodo.18439397 Paper 5 — Quantum Structure from Finite Enforceability — DOI 10.5281/zenodo.18439433 Paper 6 — Dynamics and Geometry as Optimal Admissible Reallocation — DOI 10.5281/zenodo.18439445 Paper 7 — A Minimal Quantum of Action from Finite Admissibility — DOI 10.5281/zenodo.18439513 Paper 8 — The Admissibility-Capacity Ledger — DOI 10.5281/zenodo.19721384 Paper 9 — The Geometric Substrate as Cost Structure of Comparison Continuations — DOI 10.5281/zenodo.20041675 Paper 10 — The Calculus of Finite Continuability — DOI 10.5281/zenodo.20041680 Paper 11 — Forced Universality from Capacity-Bounded Admissibility — DOI 10.5281/zenodo.20684198 Paper 13 — The Minimal Admissibility Core — DOI 10.5281/zenodo.18361446 Paper 16 — Markov Breakdown and the Hard Problems — DOI 10.5281/zenodo.20684207 Paper 18 — The Electroweak Sector as a Capacity Equilibrium — DOI 10.5281/zenodo.20684209 Paper 20 — The Enforcement Crystal — DOI 10.5281/zenodo.18531732 Paper 24 — The Recruitment-Radius Extension — Foundations — DOI 10.5281/zenodo.20684211 Paper 28 — Absolute Mass Scales from Electroweak Capacity Saturation — DOI 10.5281/zenodo.20684215 Paper 29 — Plaquette Representation Dominance and Confinement — DOI 10.5281/zenodo.20684218 Paper 30 — A Tube Mechanism for the Lattice Mass Gap — DOI 10.5281/zenodo.20684220 Paper 31 — Osterwalder–Schrader Structure of Lattice Yang–Mills — DOI 10.5281/zenodo.20684222 Paper 33 — Trace-to-Scheme Export Architecture — DOI 10.5281/zenodo.20684224 Paper 35 — The Dark Sector as a Two-Role Capacity Decomposition — DOI 10.5281/zenodo.20684228 Paper 40 — The Thermodynamics of Finite Distinction (Between Symmetry and the Void) — DOI 10.5281/zenodo.20684235 Paper 41 — The Horizon as a Continuation Ledger — DOI 10.5281/zenodo.20684241 Paper 42 — The Weak Mixing Angle Is Not Free — DOI 10.5281/zenodo.20684245 Paper 43 — The Neutrino Sector — DOI 10.5281/zenodo.21194191 Paper 44 — What a Finite Theory Owes the Continuum — DOI 10.5281/zenodo.21195874 Paper 45 — What the Vacuum Can Keep — DOI 10.5281/zenodo.21194195 Paper 46 — The Strong CP Angle Is Not Free — DOI 10.5281/zenodo.21194197 Paper 47 — The Confinement Antecedent Is Not Forced — DOI 10.5281/zenodo.21194199 Technical Supplements cross-link to their main paper via isSupplementTo and to the engine via isDocumentedBy. Concept DOIs above resolve to the latest version. 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
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
A finite interface either admits a single classical assignment consistent across all the records it holds at once — a global Boolean section — or it does not. This paper develops the gauge-sector consequences of that dividing line inside a capacity-denominated cost calculus, and it is built to be verified from this document and one public repository alone. We prove exact, finite theorems: a characterization theorem forcing cost = count from three axioms with no continuity hypothesis; an additivity theorem and a non-monotonicity no-go for the superadditive surplus Δ; the classification of sharp gauge-invariant colour records at every N (the unique meson record is the entangled singlet, the unique N-fundamental record is the entangled baryon, with an abelian control showing both are non-abelian features); and a contextuality census of gauge-invariant interfaces, including an exact Kochen–Specker colouring obstruction in the physical SU(3) octet channel and an exact 1/6 robustness floor for the verdict. The centerpiece is a fully worked model: SU(2) Hamiltonian lattice gauge theory at strong coupling, static sources joined by flux strings, where Δ is computed on a precisely defined cut and set beside the standard diagnostic. The verdict is structured — co-movement on the N-ality partition, under-determination beyond sign within a family, divergence on split flux, and a checked witness that the capacity side never recomputes the Casimir — and the structure is the result. One section, clearly bounded, reads these theorems onto confinement; it declares its imports and conditions its conclusions on a named saturation premise. No energy gap is derived, no line-operator algebra is constructed, and no thermal claim is made; the paper says exactly where those boundaries sit and why. Every theorem carries a named machine-verified certificate in exact arithmetic. The Technical Supplement (deposited alongside and linked from this record) is the proof-dense companion and the canonical formal home of the Δ calculus for the series. 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 repository (github.com/Ethan-Brooke) 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
A single axiom — finite enforcement capacity — is shown to force the structural skeleton of quantum mechanics. The enforcement cost of simultaneously maintaining distinctions within any causally connected region is bounded: this constraint alone, applied through standard mathematical results (GNS construction, Gleason's theorem, Wedderburn–Artin, Frobenius, Skolem–Noether), yields Hilbert space, the Born rule, completely positive trace-preserving dynamics, tensor products, gauge symmetry, and von Neumann entropy as committed capacity. No free parameters are introduced and no physics is imported beyond the axiom. The bridge theorem (T1) derives order-dependent enforcement — the operational content of noncommutativity — using only finite sets and exact rational arithmetic on an abstract state set, without assuming Hilbert-space structure. Frobenius' theorem then forces the complex numbers as the unique ground field compatible with non-commutative, trace-preserving capacity accounting. Twenty-three theorems are verified by an executable codebase with zero external dependencies. Supplementary materials: Machine-verifiable codebase: https://github.com/Ethan-Brooke/APF-Paper-1-The-Enforceability-of-Distinction Interactive derivation DAG: https://ethan-brooke.github.io/APF-Paper-1-The-Enforceability-of-Distinction/ Reviewer walkthrough (Google Colab, zero install): https://colab.research.google.com/github/Ethan-Brooke/APF-Paper-1-The-Enforceability-of-Distinction/blob/main/APF_Reviewer_Walkthrough.ipynb Paper 1 of 7 in the Admissibility Physics Framework (APF).
A Kernel-Role Determinacy Theorem for Record Fixation, Fixed-Domain Quotients, and Boundary-Trace Normal Form This paper specializes the AASC kernel to the construction and use of quantum measurement records. Its central question is not initially how outcomes are produced, whether a wavefunction collapses, or which interpretation of quantum theory is correct. It asks what must be fixed for a measurement record to function as the same determinate physical and evidential object across retention, prospective use, deliberation, comparison, and report. For any non-degenerate determinate physical object, state, carrier, transformation, record, comparison, or endpoint, the paper takes the governing kernel to be: NDObj(x) ⇒ K(x) = (Adm, Stand, Ref, Irr). Admissibility, standing, reference, and irreversibility are therefore treated as necessary role conditions of determinate record use, not as an optional interpretive overlay added to an otherwise complete physical description. Measurement-record determinacy The manuscript constructs a fixed-domain measurement-record target in which a record must preserve: reference to the same construction token; standing-bearing evidential eligibility; admissible reuse; original act-time qualification; lawful redescription; retained and prospective record use; deliberative and comparative use; reportability and downstream inheritance. The resulting assignment space is quotiented by equivalence of standing-bearing measurement-record content. A rival quotient claiming exactly the same content collapses to the canonical measurement-record quotient. Representational differences that preserve the complete role content are treated as skin; differences affecting record identity, evidential ordering, admissibility, continuation, or downstream use remain physically relevant. Physically non-selective quantum regimes The paper then studies a physically non-selective quantum regime containing unitary branching, entanglement, decoherence, branch-local records, and stable local reports. The central result is conditional and target-specific: The bare unitary-entangling-decohering package can support multiple stable branch-local records without supplying a unique standing evidential record for the fixed construction-relative target. This is not a claim that branch-local records are unreal, unstable, or physically useless. It is a distinction between: physical occurrence of branch-local record structures; local branch-relative reportability; evidential role fixation for a record reused as the same standing object across the measurement practice. The paper shows that non-selective structures may agree on branch weights, density content, decoherence, correlations, instrument sums, channel closure, effect weights, projector orthogonality, and repeatability without selecting one fine record as the standing evidential record. Fixed-domain quotient and fixation commutation The theorem ladder establishes: fixed-domain assignment-space forcing; exact quotient construction for standing-bearing record content; same-scope equivalence collapse; fixation commutation under admissible transformation; inheritance of the original qualification requirement; persistence-bearing reuse; asymmetry exhaustion; anti-weakening; selector exhaustion. If evidential ordering varies across admissible assignments while the target requires one fixed-domain ordering, the ordering is not defined on that scope merely because numerical values or branch-local reports happen to coincide. The paper therefore treats measurement-record fixation as prior to the theory-level evidential reuse that depends on it. Boundary-trace normal form After internal and provenance/indexical routes have been exhausted, the residual same-domain route is a singleton stable boundary trace. Such a trace is not collapse dynamics, hidden-variable selection, or a new physical mechanism. It is an admissibility-preserving role anchor that can fix a record-bearing assignment only when its channel, incidence, transport law, original authority, and singleton-positive condition are independently warranted. The result distinguishes: sound partial selection from complete role fixation; unique trace value from unique original occupant; class-valued record fixation from raw-token fixation; later discovery of an original qualification from retroactive repair; physical record occurrence from standing evidential identity. Finite quantum specialization The paper occupies the abstract structure with an explicit finite two-branch quantum measurement domain containing: two outcomes; two fine records and one merged record; denominator-two branch weights; a balanced integer-amplitude state; density-table realization; selective and non-selective instrument maps; environment-record maps; projective branch maps; complementary effects; zero-product projectors; repeatability operations. On the declared balanced witness, amplitude weights, density weights, branch-map traces, effect weights, pointer records, instrument records, environment records, and channel records agree on the same branch architecture. Selective surfaces preserve the fine-record distinction. Non-selective surfaces agree only on the merged record. The fine records therefore remain separated, while the declared non-selective surfaces supply no singleton fine-record selector. The finite selector-boundary theorem eliminates the disjunction of candidate hidden selectors across the state, instrument, environment, channel, effect, and repeatability surfaces. The resulting diagnosis is precise: the missing structure is admissible fine-record role fixation, not further branch production, weighting, decoherence, stabilization, or repeatability. Scope and non-claims The manuscript establishes a finite, conditional, manuscript-facing theorem. It does not claim: to derive Hilbert-space quantum mechanics; to derive general POVM theory; to derive full projection-valued-measure theory; to reconstruct the full operator algebra; to provide a general Kraus representation theorem; to reconstruct complete positivity; to derive the continuum Born rule; to derive universal physical decoherence theory; to introduce a new collapse dynamics; to prove that no physical collapse process occurs in nature; to prove physical branch selection; to establish interpretive truth; to classify every possible physical measurement mechanism. The absence of a hidden selector on the declared finite non-selective surfaces is therefore not a theorem that no physical collapse or selection process can occur elsewhere. It is the finite theorem that those declared surfaces do not themselves supply the required fine-record selector. Relation to quantum interpretations The result does not deny the practical success of quantum theory, decoherence, branch-relative reporting, instrumental updates, or decision-theoretic methods. A practice aimed only at prediction, branch-local reporting, or instrumental updating may be a different target from the construction-relative record use studied here. If a theory claims to underwrite retained, prospective, deliberative, and evidential reuse of the same record-bearing construction token, it must preserve the relevant kernel roles and admit a standing-bearing fixation regime. Bare Everettian dynamics, without additional dynamics, ontological privileging, or admissible restriction to one constructional class, supplies no fine-record selector on the declared target. A strategy that preserves non-selective physics while restricting admissible record-fixation assignments belongs to a restriction-to-one-constructional-class route. Instrumentalist and QBist strategies instead reduce or redirect the target explanatory burden. Role in the AASC programme This paper provides the record-fixation layer of the AASC Quantum Determinacy Pair. Its companion Bell-scope work addresses the compatibility and no-signaling side of quantum witness determinacy. Together, the pair treats quantum records and witnesses as kernel-governed standing structures rather than as automatically determinate physical occurrences. The paper’s principal contribution is to identify the first failure point in a physically non-selective measurement regime: The initial problem is not necessarily the production of branch-local outcomes. It is the admissible fixation of the record role through which outcomes are retained, reported, remembered, compared, and used as evidence. The public Lean 4 formalization is synchronized with the finite measurement-boundary release v0.9.1 and provides the finite artifact for the declared effect, projector, compatibility, closed-stack, boundary-certificate, and audit layers. Keywords AASC; measurement problem; quantum measurement; measurement-record fixation; physically non-selective regimes; Everettian non-selection; record-role determinacy; fixed-domain quotient; standing equivalence; boundary-trace fixation; selector exhaustion; decoherence; branch-local records; admissibility and irreversibility; Lean 4 formalization