Mechanizing the Identity-Persistence Forcing Architecture: From Declared Regimes to Independently Reconstructible Verdict Theorems

Abstract

Mechanizing the Identity-Persistence Forcing Architecture develops a formal theory of when a theorem corpus is independently reconstructible from declared proof routes, verified imports, correspondence obligations, and a trusted formal base. The paper separates proof-architecture auditing from proof reconstruction: an acyclic dependency graph can certify route structure without establishing the correctness of the mathematical derivations carried by those routes.

The central object is a base-relative reconstructibility closure. For a finite acyclic registered proof architecture with verified base \(B\), each theorem is associated with alternative conjunctive proof routes. A monotone closure operator determines the least fixed point \(S_B^\star\) of claims reconstructible from \(B\). The theory is then refined into a symbolic provenance calculus. Each unresolved obligation is represented as a variable, conjunctive dependencies compose multiplicatively, and alternative routes compose additively. Boolean evaluation of the resulting provenance expression \(P_t\) exactly characterizes whether a target theorem \(t\) becomes reconstructible after a declared set of obligations is resolved.

The provenance object induces two dual certificate families. Minimal completion sets are the inclusion-minimal unresolved obligation sets whose resolution closes some support route; minimal blocking cuts are the minimal transversals that intersect every completion set. Under the finite acyclic hypotheses, blocker duality recovers each family from the other. This distinguishes two questions that must not be conflated: what is sufficient to complete a theorem, and what set of unresolved obligations is sufficient to keep every route blocked.

Source-to-formal correspondence is treated by the same factorization principle used elsewhere in the program for bounded verification. A correspondence property is certifiable from a declared evidence interface exactly when its truth factors through that interface, equivalently when the interface kernel is contained in the correspondence kernel. The result replaces any claim of a permanent human semantic boundary with an exact regime-relative condition: verified enrichment of the interface may enlarge the reconstructible corpus, while no downstream verifier can recover distinctions absent from its declared evidence.

The paper also develops a trust-base split between logical and operational trust and a computational complexity profile for reconstruction tasks. Fixed-set replay and closure are linear in the route DAG. Enumeration of completion and blocker families may have exponential output and belongs to hypergraph-transversal territory. Minimum Verification Cost—finding a least-weight set of unresolved obligations sufficient to reconstruct a target—is NP-hard in the unrestricted finite provenance class by reduction from weighted hitting set, already at a two-level AND/OR architecture. A read-once subclass provides a syntactically checkable sufficient condition for exact linear min-plus evaluation.

The resulting theory yields a precise status object for mechanized reconstruction: a theorem’s provenance, reconstructibility under a declared base, its minimal completion certificates, its blocking certificates, and the declared cost of closing it. The paper proves the theory of this reconstruction boundary; the full mechanized instance for the Identity-Persistence Program remains a separate implementation obligation until the registered proof artifact is built.