A Three-Layer Dependency-and-Logic Audit for Mathematical Proofs and AI-Generated Claims: The S/M/P × D/L Framework

We introduce a structured audit framework for mathematical proofs, conjectures, and AI-generated mathematical claims. The framework operates on three layers—Semantic, Mathematical, and Programming—and for each layer examines two complementary dimensions: internal Dependency relations and internal Logic-structure discipline. The resulting S/M/P × D/L matrix provides six elementary audit cells (S*D, S*L, M*D, M*L, P*D, P*L) plus cross-layer mapping checks. We demonstrate the framework on five cases spanning correct proofs (Euclid's infinitude of primes), classical fallacies (the all-triangles-are-isosceles proof, Kempe's four-color proof), a research-level computer-assisted proof (Appel–Haken four-color theorem), and an open diagnostic (the Riemann Hypothesis). We also outline a nearest-proven-theorem map heuristic for open conjectures and situate the framework against the Lean formalization equivalence spectrum.