Initial growth of a vorticity-pair moment with decreasing enstrophy and finite-time bounds

We study an amplitude-weighted moment of the cross product of the vorticity at two points for the unforced three-dimensional Navier–Stokes equations in the whole space, and ask whether it can increase while the enstrophy decreases. We construct an explicit smooth, compactly supported, divergence-free initial velocity, obtained by mollifying a columnar profile with azimuthal and meridional parts, whose solution has both properties on one nonempty initial interval: the pair moment increases at a positive rate given by a certified margin, and the enstrophy decreases. The proof combines a norm-based growth criterion, which pays the viscous contribution and the vanishing radial fibres and passes from an initial margin to an interval by lower semicontinuity, with finite inequalities for the fixed datum verified by exact rational arithmetic and outward interval inclusions. The length of the interval is not quantified. Separately, classical energy estimates give an explicit horizon of strong existence and a bound for the full pair moment on it, uniformly over the admissible mollifiers, and the static estimates give an explicit lower bound for the initial moment. A comparison with a corrected heat reference bounds the moment, including the self-interaction of the error and the whole radial tail; sharper localized budgets are recorded only as conditional statements. No conclusion about global regularity is drawn. This deposit brings together the English manuscript, its Portuguese counterpart, mathematical supplements in both languages, compilable sources and reproducible computational certificates. Project connection: this work was developed within the research programme “A Geometric–Multiscale–Temporal Conjecture of Depletion through Localized Enstrophy for the 3D Navier–Stokes Equations”, represented by https://doi.org/10.5281/zenodo.23073201. It is a separate contribution and is not a new version of that research note. The results here do not establish the project conjecture or global regularity. Deposit status: files are restricted, with no automatic embargo expiry. This record preserves the reconciled scientific version dated 1 November 2026 and its supporting materials. This is a preprint; it has not been accepted for journal publication.