AI System Verifies Partial Progress on Riemann Hypothesis

This research explores the capability of a verifiable AI-assisted reasoning system, named Verifiable Growing Physical Transformer with Recursive Self-Improvement (VGPT-RSI), to make reliable and formally checked partial progress on complex mathematical problems, specifically those related to the Riemann Hypothesis (RH). The system aims to explicitly identify the unresolved mathematical challenges rather than claiming a full proof. The VGPT-RSI system was applied to two RH-adjacent certification tasks. First, it successfully constructed and formally verified a finite RH-boundary certificate for an inequality over a specific region. This involved converting a numerical boundary curve into a certificate-backed lower curve, which was then audited using advanced interval arithmetic and formally checked in proof assistants like Rocq/CoqInterval. Second, the system initiated a formal Lagarias-route certificate, which is based on a criterion stating that the Riemann Hypothesis is equivalent to a global inequality. The VGPT-RSI formalized a finite quantity of this criterion and produced a Coq-checked finite certificate. Crucially, the system precisely pinpoints the remaining mathematical bottlenecks, such as formalizing the Lagarias equivalence globally and proving tail theorems beyond finite cutoffs, demonstrating its ability to contribute certified progress while clearly delineating the limits of its current capabilities.