Structural–Temporal Rigidity in Fluid Dynamics A Unified Framework for the Exclusion of Critical Cascades

This project develops a unified analytic framework explaining why certain singularity and blow-up scenarios are structurally forbidden in nonlinear fluid systems. The core idea is that regularity is protected not by isolated inequalities, but by a coupled rigidity between: (i) Structural closure of nonlinear interactions across frequency scales, and (ii) Temporal persistence, which enforces a nonzero time cost for energy transfer between dyadic shells. The framework is developed in five complementary papers: 1. Paper I establishes dyadic structural closure for the 3D incompressible Navier–Stokes equations, ruling out analytic energy leakage. 2. Paper II introduces time–frequency rigidity, excluding critical cascade scenarios through minimal persistence constraints. 3. Paper III abstracts these ideas into the No Fast Cascade Principle, independent of specific PDE structure. 4. Paper IV extends the rigidity framework to systems with asymmetric or fractional dissipation, including MHD and SQG. 5. Paper V synthesizes the results into a conceptual theory of Structural–Temporal Rigidity, reframing global regularity as a geometric and temporal obstruction rather than an inequality optimization problem. Together, these works propose a structural explanation for the robustness of regularity across a wide class of dissipative fluid and plasma models.