Finite-Chain CKN-Bad Scale Counting for Navier-Stokes: Standard PDE Closure and Canonical Detector Realization

Abstract

This paper proves a finite-chain counting theorem for Caffarelli--Kohn--Nirenberg bad scales of suitable weak solutions to the three-dimensional incompressible Navier--Stokes equations. The main standard-PDE result bounds the weighted size of a finite set of CKN-bad scales by nonnegative channel costs consisting of vertical one-component concentration, annular leakage, pressure-tail terms, and pressure--flux--energy residuals. The proved closing mechanism is qualitative one-component compactness under a full local critical bound: small vertical component forces CKN smallness at a smaller radius. The paper then gives a canonical detector realization of the same finite-window counting philosophy. The original abstract detector is not identified with standard PDE channels; instead, a new amended canonical detector is defined using energy, flux, pressure-tail, retained low-pressure-mode, and finite-dimensional residual coordinates. For this amended detector, we prove upper realization, lower audit, CKN extraction, and finite-chain bad-scale counting.