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We develop a proof program that targets a quantitative bridge between scale-invariant local energy control and global critical $L^\infty_t L^3_x$ bounds for the three-dimensional incompressible Navier--Stokes equations. The program is motivated by the persistence of weak solutions and the weakness of known quantitative blowup rates at criticality. Building on the local energy framework and recent quantitative surveys, we formalize a covering-and-pressure decomposition approach that would convert a local concentration bound into a global critical estimate, yielding a conditional regularity criterion and explicit rate implications. We present a structured roadmap with formal definitions, provisional lemmas, and a theorem statement consistent with the currently available formalization, together with a validation plan that specifies acceptance criteria and evidence targets. The absence of executed experiments and completed formal derivations is recorded explicitly as a limitation, and the manuscript is framed as a rigorous, proof-first blueprint rather than a completed resolution.

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.