Navier--Stokes Regularity via Critical Norm Tracking

We study 3D incompressible Navier--Stokes flow on a periodic box and design a diagnostic suite that links classical Prodi--Serrin mixed norms, weak-$L^p$ (Lorentz) proxies, scaling-invariant weighted criteria, and sparseness-based geometry. The goal is to evaluate how these regularity indicators behave in direct numerical simulation (DNS) across Reynolds-number sweeps and to quantify numerical sensitivities due to de-aliasing. The study matters for both mathematical regularity theory and practical turbulence modeling because it formalizes computable indicators that can be tracked in simulations and compared to validated datasets. We provide a formal problem setting, an algorithmic pipeline, and a hypothesis-driven evaluation protocol, with explicit acceptance criteria and uncertainty procedures. Empirical results are not yet available because the required scientific Python stack and external dataset access are currently unavailable in the execution environment; we therefore report the full experimental design and the evidence plan that will be used once runs are enabled.