Primitive Axiomatics of Finite Obstruction Calculus

Finite evaluation systems — distributed logs, constraint solvers, sensor networks, proof traces, and multi-agent pipelines — often produce local comparisons that cannot be globally reconciled. This paper gives a finite, replayable axiomatics for diagnosing such failures.

The framework begins with a declared finite carrier and typed local differences. A chosen 2-truncated cochain structure separates an observed inconsistency into what can be removed by admissible local reference change and what survives as quotient residual. The resulting obstruction signature (Φ₁, Γ₂, R_cl) distinguishes three cases:

1. Removable gauge artifacts — inconsistencies absorbed by local reference changes.

2. Closed obstructions — irreducible residual classes with no active closure failure.

3. Closure failures — non-closed residuals carrying active d₁-closure charge.

Here Φ₁ measures irreducible quotient mass after repair, Γ₂ measures d₁-closure charge, and R_cl = Γ₂ / Φ₁ records closure charge per unit residual when Φ₁ > 0.

The central separation is algebraic versus observational. The residual class is determined by the declared carrier, repair map, and closure map. Its measured magnitude is not universal: it is fixed by the declared observer gauge, aggregation law, and compatible transformation class. Thus the paper gives conditions under which additive L1 diagnostics are valid, while explicitly refusing any universal L1 assumption.

Limits and scope. This foundation is finite, degree-1, and separated. It does not derive quantum mechanics, the Born rule, physical gauge groups, observer-independent authority, higher cohomology, or a universal norm. Vector-valued and quantum observer geometries are developed only in companion papers.