{"slug": "vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite", "title": "VGPT-RSI for RH-Adjacent Formal Progress: Boundary Certificates, Verified Finite Lagarias Inequalities, and Explicit Failure Localization", "summary": "Researchers applied the Verifiable Growing Physical Transformer with Recursive Self-Improvement (VGPT-RSI) to produce formally verified partial progress on problems adjacent to the Riemann Hypothesis, including a finite boundary certificate and a Lagarias-route certificate checked in Coq. The system explicitly identifies remaining mathematical obstructions, demonstrating AI-assisted formal verification without overclaiming proof of the hypothesis.", "body_md": "arXiv:2606.15096v1 Announce Type: new\nAbstract: 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.", "url": "https://wpnews.pro/news/vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite", "canonical_source": "https://arxiv.org/abs/2606.15096", "published_at": "2026-06-16 04:00:00+00:00", "updated_at": "2026-06-16 04:21:09.057244+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "ai-safety", "large-language-models", "ai-tools"], "entities": ["VGPT-RSI", "Riemann Hypothesis", "Lagarias criterion", "Coq", "Arb", "FLINT", "Rocq/CoqInterval"], "alternates": {"html": "https://wpnews.pro/news/vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite", "markdown": "https://wpnews.pro/news/vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite.md", "text": "https://wpnews.pro/news/vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite.txt", "jsonld": "https://wpnews.pro/news/vgpt-rsi-for-rh-adjacent-formal-progress-boundary-certificates-verified-finite.jsonld"}}