Supervised White-Box Reinforcement Learning for Incompressible Navier--Stokes A--D Discovery: A Supervisor--Discovery--Solver Architecture for Symbolic Normal Forms, Adversarial Falsification, PDE Certification, and Clay-Prize Research

Abstract

**There are now two papers in this zenodo record. One is the Discovery RL NS, the snd is/are the closed form solutions for NS used. This paper records the architecture and research history of a supervised white-box reinforcement-learning discovery program developed for the four alternatives A--D in the official three-dimensional incompressible Navier--Stokes Millennium Problem. The central methodological claim is not that a single neural policy ``solved'' the equations. The system is hierarchical. A \emph{Supervisor/meta-RL layer} manages research state, allocates search effort, freezes or reopens claims, and rewards falsification, compression, source completeness, and theorem-relevant progress. Beneath it, a \emph{white-box discovery layer} proposes explicit mathematical objects---coordinates, invariants, normal forms, similarity ansatzes, tensor decompositions, stochastic representations, spectral recurrences, symbolic residual corrections, and counterexamples---using a typed pseudo-wheel grammar. Beneath discovery, multiple \emph{solver families} execute exact algebra, asymptotics, Fourier/pseudospectral evolution, Fokker--Planck/SDE transforms, geometric/Lagrangian analysis, q-eLM symbolic search, PINN-style residual fitting, special-function matching, and theorem-level verification. Every candidate remains explicit and is promoted through a status ladder rather than being converted directly from numerical success into proof. For the unforced alternatives A and B, the discovery history begins with the migration from compressible density-based Fokker--Planck coordinates to the true incompressible obstruction: trace-free strain, vorticity stretching, nonlocal pressure, deformation geometry, and concentration. White-box search then isolated the repeated-eigenvalue strain edge, a pressure--vorticity--viscous repeated-plane splitter, exact-edge Riccati dynamics, localization constraints, strict non-edge defects, reset dynamics, and finally a stretch-envelope continuation architecture. The current internal research state treats the later A/B result as stabilized and frozen for adversarial testing. The periodic B branch has survived two explicit adversarial rounds without meeting the reopen threshold. At the same time, the source audit identifies a remaining archival packaging gap: several nonclassical sector-closure arguments are not yet reproduced line-by-line in one standalone citable artifact. The manuscript therefore distinguishes \emph{frozen research state} from \emph{source-complete external theorem package}. For the forced alternatives C and D, the white-box program is presented as a post-announcement reconstruction and mechanism-discovery exercise, not as a pre-announcement priority claim for the singular vortex. The released construction is compressed into a physics-first normal-form hierarchy: anisotropic similarity core, protected inner region, annular stress defect, exact heat exterior, finite moment matching, divergence-free covariance channels, residual-order improvement, and shrinking-support summation. A large symbolic tournament and recurrence experiments are interpreted as mechanism probes that help reveal why the constructive proof has its particular architecture; they are not the source of the theorem. The resulting A--D record motivates a general scientific-discovery protocol: propose with flexible machine intelligence, express with white-box mathematics, attack with adversaries, verify independently, audit quantifiers and sources, and promote only at the evidentiary level actually earned.