Structure-Preserving Analysis of Euler and Navier–Stokes Equations Scaling, Residuals, and Quantitative Regularity

Recent AI-assisted work on the Euler and Navier–Stokes equations has brought fluid singularities into academic discussion and social media. Neural networks are identifying candidate blowup profiles, while other projects announce analytic proofs accompanied by Lean formalizations. These developments make one question urgent: what precisely has been established, for which equation, forcing, domain, and solution class? This paper develops a rigorous framework to answer that question and organize the research. Rather than treating viscosity as a minor adjustment to Euler dynamics, it analyzes how diffusion changes scaling, continuation criteria, and the competition between strain and fine structures. The framework combines nonlinear relative energy, spectral leakage, exact affine waves, localization estimates, and admissible forcing with geometric and categorical approaches to fluid evolution. For data scientists, its central distinction is between discovering a candidate and validating a mathematical conclusion. Small training losses, compatible scaling, and formal algebra each provide useful information, but none replaces certified residual bounds, stability estimates, or faithful theorem specifications. Quantitative correction theorems and a conditional vortex-stretching continuation criterion make these requirements explicit. The result is a research architecture that connects scientific machine learning, mathematical analysis, and formal verification without conflating promising computations, announced constructions, and independently established solutions to different problems.