Geometric and Information-Theoretic Reformulations of the Beale–Kato–Majda Criterion for Navier–Stokes

Description This work documents the first systematic case study in which a general-purpose large language model (GPT-5.1) independently generated mathematically coherent conjectures about the Beale–Kato–Majda (BKM) blow-up criterion for the 3D incompressible Navier–Stokes equations. Importantly, the model received no specialized mathematical finetuning, no PDE datasets, and no training related to fluid dynamics.All reasoning was performed inside a consumer-facing UI (UI-GPT), guided only by a structured inference prompt (“Truth-Manifold mode”). Under these constraints, the model produced: A geometric reformulation (“Geo-BKM”) centered on strain eigenvalue growth and its relation to curvature-like quantities in fluid dynamics; An information-theoretic reformulation (“Info-BKM”) based on Dirichlet energy, enstrophy, Fisher-type quantities, and passive-scalar information concentration; A unified action-style perspective linking these two formulations conceptually. These conjectures are not proven, nor are they claimed as new mathematical results.They are presented as evidence that modern LLMs can autonomously reorganize established PDE theory into new, structurally meaningful research programs—suggesting a new paradigm for “AI-assisted theoretical discovery,” where language models act not merely as computational tools but as generative engines for mathematics itself. The paper further demonstrates that the AI-generated conjectures are consistent with classical facts: Global regularity in 2D Navier–Stokes Shock formation in 1D Burgers Modern geometric and dissipative viewpoints This compatibility indicates that LLM-driven theoretical synthesis may become a viable complement to traditional mathematical research, even in areas previously assumed inaccessible to automated reasoning. The corresponding inference prompt—the method by which the LLM produced these structures—is intentionally not included here and will be published separately as part of a follow-up study on “prompt-based mathematical cognition in large language models.”