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

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.

Physical Transformer

Digital AI systems spanning large language models, vision models, and generative architectures that operate primarily in symbolic, linguistic, or pixel domains. They have achieved striking progress, but almost all of this progress lives in virtual spaces. These systems transform embeddings and tokens, yet do not themselves touch the world and rarely admit a physical interpretation. In this work we propose a physical transformer that couples modern transformer style computation with geometric representation and physical dynamics. At the micro level, attention heads, and feed-forward blocks are modeled as interacting spins governed by effective Hamiltonians plus non-Hamiltonian bath terms. At the meso level, their aggregated state evolves on a learned Neural Differential Manifold (NDM) under Hamiltonian flows and Hamilton, Jacobi, Bellman (HJB) optimal control, discretized by symplectic layers that approximately preserve geometric and energetic invariants. At the macro level, the model maintains a generative semantic workspace and a two-dimensional information-phase portrait that tracks uncertainty and information gain over a reasoning trajectory. Within this hierarchy, reasoning tasks are formulated as controlled information flows on the manifold, with solutions corresponding to low cost trajectories that satisfy geometric, energetic, and workspace-consistency constraints. On simple toy problems involving numerical integration and dynamical systems, the physical transformer outperforms naive baselines in stability and long-horizon accuracy, highlighting the benefits of respecting underlying geometric and Hamiltonian structure. More broadly, the framework suggests a path toward physical AI that unify digital reasoning with physically grounded manifolds, opening a route to more interpretable and potentially unified models of reasoning, control, and interaction with the real world.