Paper 152I: A Local Coherence Criterion for Ordinary Admissibility Failure and Pinching Payment in High-Vorticity Navier–Stokes Flow

Paper 152I studies a narrow local step in the Navier-Stokes high-vorticity route program. Earlier papers separated positive high-vorticity stretching into admissible, coherent-channel, intermittent-time, moving-support, scale-evasion, residual, and ordinary non-admissible routes. Paper 152I focuses only on the ordinary non-admissible route. Its purpose is to define a non-circular local admissibility gate and to explain what must happen when that gate fails.

The central idea is that ordinary gate failure should become visible as a local defect in the vorticity direction field. If the vorticity direction remains locally coherent while positive stretching persists, then the behavior should be counted as admissible or assigned to a coherent imported route. If the behavior is truly ordinary, non-admissible, and not explained by the imported routes, then failure of admissibility should produce a measurable local direction-coherence defect.

The paper defines this defect by comparing the local vorticity direction to its ordinary local average on a ball or smooth localization window. Once a lower bound on this defect is available, a local Poincare inequality converts the defect into directional-gradient cost. This gives the local bridge: ordinary gate failure implies visible direction incoherence, and visible direction incoherence implies payment by the directional pinching term.

The paper does not prove global Navier-Stokes regularity. It also does not prove final coefficient absorption, close the reduced residual route, or solve the full route-assignment problem. Its contribution is more limited: it states the local gate-failure visibility target clearly, separates definitions from hypotheses, disciplines the lower-order class, and explains why the Poincare step is only the local analytic consequence once visible incoherence has been established. The remaining hard task is to prove that ordinary gate failure really does force visible direction incoherence or else must be assigned to an imported or residual route.