The Enforceability of Distinction: Admissible Possibility Spaces, Continuation Algebra, and the Sep/IJC Representation Theorem

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,