VGPT-RSI: Verified AI certificates for Riemann-adjacent progress

VGPT-RSI produced Coq-checked finite certificates and verified boundary curves in an arXiv submission dated 13 June 2026 by Zhixin Hu, Tao Xu, Xiaodian Sun, Li Jin and Momiao Xiong. The 31-page paper, arXiv:2606.15096, applies the Verifiable Growing Physical Transformer with Recursive Self-Improvement (VGPT-RSI) to two RH-adjacent certification tasks and reports explicit localization of the remaining mathematical obstructions.

What the paper did

The authors applied VGPT-RSI to two certification problems tied to the Riemann Hypothesis. First, they constructed a finite RH-boundary certificate for an inequality on a parameterized safe lower curve over a region. The workflow converted a numerical boundary curve into a certificate-backed lower curve, audited that curve using outward-rounded interval arithmetic and Arb/FLINT ball arithmetic, and then checked the parameterized theorem in Rocq/CoqInterval.

Second, the team initiated a formal Lagarias-route certificate. The paper recalls that "Lagarias criterion states that RH is equivalent to the global inequality." They formalized the finite quantity appearing in that route and produced a Coq-checked finite certificate for the finite portion of the inequality.

Both tasks were presented with formal verification artifacts: numerical curves converted into certificate-backed objects, multiple interval-arithmetic audits, and Coq-checked certificates for finite statements. The submission includes three figures supporting these constructions and proofs.

How VGPT-RSI achieved verification

VGPT-RSI served as an AI-assisted reasoning system that generated and organized the steps needed to turn numerical evidence into formally checked statements. For the boundary-certificate task the pipeline was: produce a numerical boundary curve, convert it into a certificate-backed lower curve, audit arithmetic using outward-rounded interval arithmetic and Arb/FLINT ball arithmetic, and check the parameterized theorem in Rocq/CoqInterval.

For the Lagarias-route task the system formalized the finite quantity in the Lagarias criterion and produced a Coq-checked finite certificate. The paper emphasizes that these outputs are finite, formally checked artifacts rather than a global proof of the Riemann Hypothesis.

The authors stress that VGPT-RSI also tracked dependencies and avoided overclaiming by explicitly naming what remains to be done in purely mathematical terms.

Why it matters

The submission demonstrates a repeatable workflow that transforms numerical artifacts into formal, machine-checked certificates using a chain of interval arithmetic tools and Coq. That matters because it separates what can be verified by computation and formal proof from the genuinely unresolved mathematical steps. Rather than offering an unverified claim about the Riemann Hypothesis, the system produces auditable pieces and an explicit map of remaining bottlenecks.

The paper highlights three concrete unresolved obstacles: formalizing the Lagarias equivalence inside the proof assistant, proving the global tail theorem beyond any finite cutoff, and the potential need to reduce hypothetical counterexamples to colossally abundant or related extremal integers. Naming these bottlenecks makes subsequent work more targeted and measurable.

What to watch

A concrete near-term milestone is a formal proof of the Lagarias equivalence within Coq, which the authors identify as an unresolved item. Another signal will be any extension of the Coq-checked finite certificates to address the global tail theorem beyond finite cutoffs or to link failures to colossally abundant or related extremal integers.

Paper details: arXiv:2606.15096, DOI https://doi.org/10.48550/arXiv.2606.15096, 31 pages, 3 figures, subject cs.AI.