A Conditional Structural Closure Framework Under Admissible Low-Mach Realizability for Persistent Concentration Exclusion in Three-Dimensional Incompressible Flow

A persistent concentration of scale-critical quantities is widely regarded as a necessary precursor to finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equations. This paper develops a conditional structural closure framework for excluding a dynamically sustained persistent concentration within an admissible low-Mach realizable class. A localized amplitude-modulation decomposition of High–High transfer activity is introduced together with a localized Littlewood–Paley bridge linking transfer-amplitude concentration to mechanical concentration. A concentration-representation principle and a low-Mach inheritance mechanism then show that a persistent mechanical concentration necessarily generates a nontrivial thermo-acoustic trace. A central result is a localized structural necessity principle establishing that a persistent concentration necessarily generates a localized transfer-amplitude concentration. Combining the localized Littlewood–Paley bridge, the concentration-representation mechanism, the low-Mach inheritance principle, and a thermo-acoustic ε-regularity framework, we show that the resulting thermo-acoustic concentration scenario is excluded within the admissible low-Mach realizable class. Consequently, a dynamically sustained persistent concentration cannot occur in this class. The result is a conditional structural closure theorem: it excludes a persistent concentration only within the admissible low-Mach realizable class and does not constitute an unconditional regularity theorem for arbitrary solutions of the three-dimensional incompressible Navier–Stokes equations. The sole non-universal assumption is admissible low-Mach realizability, which is shown to follow whenever an admissible entropy-coercive, weakly compressible approximation exists. The principal remaining open problem is the universality of admissible low-Mach realizability.