The Certificate Decomposition Theorem and Primal-Dual Verification Algorithms

This paper establishes the algorithmic and computational foundation of the L1 coboundary framework for AI verification. Building on prior work, it proves the Certificate Decomposition Theorem, which guarantees that the total structural defect of an AI system output on a domain graph decomposes uniquely into irreducible and repairable components. Given a section over a graph-structured domain, the total defect I(s) admits an exact decomposition: I(s) = Phi + R(s) where Phi is the irreducible cohomological defect and R is the repairable component. The irreducible defect is formulated as an L1 quotient norm and computed via linear programming. The paper introduces a primal coordinate descent algorithm with an implicit dual interpretation, combining: Linear programming (L1 Hodge decomposition) for computing the irreducible defect Iterative median-based updates (Autopoietic Gauge Iterator) for repairing the section A convergent optimization process with monotone descent guarantees The algorithm provably converges to the minimal achievable defect Phi, providing a constructive realization of the decomposition theorem. Complexity analysis shows that the full verification pipeline runs in polynomial time, with near-linear scaling in practical graph settings. Additional contributions include: Formal proofs of existence, uniqueness (for scalar components), and stability of the decomposition A distributed and asynchronous variant of the coordinate descent algorithm Case studies demonstrating behavior on acyclic, cyclic, and mixed graph structures A graph signal processing interpretation of L1 projection as a robust alternative to L2 methods This work provides the computational backbone for certificate-based AI verification, enabling exact decomposition, efficient optimization, and scalable implementation across graph-structured domains.