Schur Visibility and Anti-Phantom Reduction in One-Component Navier-Stokes Degeneration

Abstract

We study the finite-scale one-component degeneration problem for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under a scale-invariant bound and smallness of the vertical component. Qualitative compactness gives convergence, in the harmonic-pressure quotient, toward the strict two-and-a-half-dimensional boundary, but it does not provide a quantitative rate. This paper proves, in an explicitly abstract trace-obstruction skeleton associated with the old observable closure, that the standard old observable package is insufficient to force a logarithmic or power selected-trace rate. The negative result is an envelope/skeleton theorem, not a Navier--Stokes counterexample. After excluding elementary high-frequency escape by parabolic trace drop and fixed-window analytic obstruction by finite-dimensional Lojasiewicz control, the remaining obstruction is an all-order finite-mode flat branch. We identify the Navier--Stokes-specific mechanism needed to control this branch: strict Schur trace-projectability may fail, but the resulting defect can be visible through the relaxed vertical-pressure channel. In active finite-window models, strict Schur phantoms are relaxed-visible. The final theorem is a conditional dichotomy: either relaxed anti-phantom closure holds and yields conditional logarithmic strict-shadow selection, or there exists an NS-realizable, cleaned, relaxed-invisible, unaligned left-singular cascade.