VGPT-RSI for RH-Adjacent Formal Progress: Boundary Certificates, Verified Finite Lagarias Inequalities, and Explicit Failure Localization

2026-06-13

AI Summary

This work explores partial progress toward the formal verification of the Riemann Hypothesis without claiming its proof, leveraging verifiable AI-assisted reasoning to precisely identify outstanding mathematical obstacles. We introduce the first integration of AI-driven automated reasoning with formal verification through a recursively self-improving Verifiable Generative Physical Transformer (VGPT-RSI), producing Coq-verified finite certificates for two related problems: a boundary certificate for parameterized safety lower bounds and a finite formal certificate for Lagarias’ criterion. Our approach combines outward-rounding interval arithmetic, Arb/FLINT ball arithmetic, and Rocq/CoqInterval formal checking. The results explicitly expose three unresolved bottlenecks: the formalization of Lagarias’ equivalence, the infinite generalization of the global tail theorem, and the reduction of potential counterexamples to colossally abundant numbers.

Abstract

The Riemann Hypothesis remains one of the central unsolved problems in mathematics. Rather than claiming proof, we investigate whether a verifiable AI-assisted reasoning system can produce reliable, formally checked partial progress while explicitly identifying the remaining mathematical obstructions. We apply the Verifiable Growing Physical Transformer with Recursive Self-Improvement (VGPT-RSI) to two RH-adjacent certification tasks. First, we construct and verify a finite RH-boundary certificate for inequality on a parameterized safe lower curve over a region. The numerical boundary curve is converted into a certificate-backed lower curve, audited using outward-rounded interval arithmetic and Arb/FLINT ball arithmetic, and then checked in Rocq/CoqInterval for the parameterized theorem. Second, we initiate a formal Lagarias-route certificate. Lagarias criterion states that RH is equivalent to the global inequality. We formalize the finite quantity and produce a Coq-checked finite certificate. The final system identifies the exact unresolved mathematical bottlenecks: formalizing the Lagarias equivalence, proving the global tail theorem beyond any finite cutoff, and potentially reducing counterexamples to colossally abundant or related extremal integers. These results demonstrate that VGPT-RSI can produce certified RH-adjacent formal progress, organize proof dependencies, and avoid overclaiming when the remaining obstruction is genuinely mathematical.

Problem

• Riemann Hypothesis
• formal verification
• AI-assisted reasoning
• Lagarias criterion
• boundary certificates

Innovation

• VGPT-RSI
• Boundary Certificates
• Verified Finite Lagarias Inequalities
• Explicit Failure Localization
• Formal Verification