Finite evaluation systems—such as distributed logs, constraint solvers, and multi-agent pipelines—often produce local comparisons that cannot be globally reconciled. This paper states the primitive axiomatics for a finite obstruction calculus, providing a bounded diagnostic framework for evaluating such failures. By modeling local comparisons as an evaluation preorder over a chosen finite simplicial carrier, the framework supports a three-way classification of observed inconsistencies: 1. Removable Gauge Artifacts: Inconsistencies resolvable by local reference changes. 2. Closed Obstructions: Irreducible residual obstructions, measured by the quotient residual mass (Φ₁). 3. Closure Failures: Active non-closed ruptures, measured by the d₁-closure charge (Γ₂). The core diagnostic is the bounded obstruction signature (Φ₁, Γ₂, R₂₋). Crucially, this axiomatics separates the derived algebraic structure from observer-dependent metric choices. It shows in this framework that measured magnitude is selected by the observer's specific scalar gauges, aggregation laws, and transformation monoids, explicitly avoiding universal ℓ¹ assumptions. Limits & Scope: This foundation is strictly finite and separated. It does not support or claim a universal ℓ¹ norm, nor does it make claims regarding quantum mechanics, the Born rule, or physical gauge groups. Vector-valued and quantum branches are relegated to companion papers.
JEREMY H. CARROLL (Fri,) studied this question.