A Spectral Conditioning Corollary for the Oscillatory Pulse Hierarchy in a Smoothly Forced Navier-Stokes Blow-Up Construction

This independent technical note derives a spectral-conditioning corollary from the estimates of the recently released smoothly forced Navier–Stokes blow-up construction. For admissible pulse labels (\\gamma=(\\ell,a,\\sigma)), the analysis shows that the condition number of the homogeneous first-harmonic pulse propagator satisfies (\\log \\kappa(P_\\gamma)=\\Theta(\\ell^2)), while the leading two-family covariance realization remains uniformly well-conditioned, with (\\kappa(H_{\\ell,\\beta})=O(1)). Their comparison yields a dynamical–covariance conditioning separation: increasingly anisotropic pulse dynamics coexist with a nondegenerate leading covariance geometry. In compact similarity regions where the active dyadic scale is comparable to the remaining physical time (\\tau=1-t), the propagator result admits the bandwise reformulation (\\log \\kappa(P_\\gamma)=\\Theta(\\log^2(1/\\tau))). The underlying Navier–Stokes construction, its pulse equations, and covariance estimates are results of the cited source authors. Exponential-separation theory, Kelvin/shearing-wave mechanisms, and singular-value analysis of propagators are established prior art and are not claimed as new. The contribution of this note is the source-derived condition-number formulation and paired asymptotic comparison for this particular pulse hierarchy. The author is not aware of a prior statement of this paired condition-number asymptotic for the construction after literature searches through September 14, 2026, but no claim of absolute priority is made.