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

# VGPT-RSI for RH-Adjacent Formal Progress: Boundary Certificates, Verified Finite Lagarias Inequalities, and Explicit Failure Localization
Source: [https://arxiv.org/abs/2606.15096](https://arxiv.org/abs/2606.15096)
[View PDF](https://arxiv.org/pdf/2606.15096)

> 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\.

## Submission history

From: Momiao Xiong \[[view email](https://arxiv.org/show-email/f778f593/2606.15096)\] **\[v1\]**Sat, 13 Jun 2026 04:12:25 UTC \(1,169 KB\)