General Theory of Finite Obstruction Calculus

This preprint presents the General Theory of Finite Obstruction Calculus (GTFOC), the finite local obstruction-verification layer of the Omega Tower of Layered Authority. GTFOC formalizes how a finite system detects difference, identifies admissible repair, passes to quotient residuals, measures closure, and returns typed obstruction terminals without converting local evidence into object-level authority. The core construction is a finite cochain repair complex C⁰ → C¹ → C² with d₁d₀ = 0. Exact repairs form im(d₀), quotient obstruction lives in Q¹ = C¹ / im(d₀), and closed non-exact residuals live in the corresponding H¹ sector. A declared finite gauge induces a weighted ℓ¹ quotient obstruction magnitude Φ, with a primal-dual certificate identity under explicit finite LP-duality hypotheses. Together with the closure diagnostic Γ, this yields the native Exact / Survivor / Open obstruction trichotomy, plus typed refusal and budget terminals when the declared finite regime cannot close the claim. The paper is a unified proof-light publication spine. It consolidates the GTFOC mathematical core, theorem registry, status discipline, bridge quarantine policy, and mechanization boundary. Version 1.3 reconciles prior status drift against the encyclopedia and adds verified Lean4 support for three scoped kernels: the Φ–Γ separation lemma, the decidable native classifier core, and the primitive-difference Local-Green / Global-Red quotient obstruction. These mechanizations support finite classifier and quotient-boundary claims; they do not claim production-scale scientific discovery, physical truth, cognitive authority, or unrestricted target-domain realization. GTFOC’s contribution is an audit discipline for finite claims: when a system says “exact,” “repaired,” “residual,” “open,” “refused,” or “budget unknown,” the calculus asks what finite carrier, repair image, quotient class, closure diagnostic, and replayable certificate support that status. Its central boundary is that obstruction evidence never self-promotes to authority. Authority belongs to the downstream finite descent authority layer, not to the local obstruction verifier itself.