The Enforceability of Distinction: Finite Capacity, the Quantum Skeleton, and the Correlation Space

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 of in the Admissibility Physics Framework (APF).