A Structural and Perturbative Roadmap for Constant-Anisotropy Extensions of a Proposed Finite-Time Navier–Stokes Blowup Construction

We study a proposed route from a highly structured finite-time blowup construction for the three-dimensional incompressible Navier–Stokes equations to a system with distinct constant horizontal and vertical viscosities. The normalized anisotropic equation is ∂tu+(u⋅∇)u−Δhu−ϑ ∂z2u+∇p=f,∇⋅u=0,ϑ>0.\partial_tu+(u\cdot\nabla)u-\Delta_hu-\vartheta\,\partial_z^2u+\nabla p=f, \qquad \nabla\cdot u=0, \qquad \vartheta>0. The objective is not to assert a new blowup theorem, but to determine whether the architecture of the proposed isotropic construction is perturbatively compatible with ϑ≠1\vartheta\neq1. We first place the problem in the context of classical regularity theory and the established literature on anisotropic Navier–Stokes systems. We then extract a dependency graph for the proposed construction: leading self-similar profile, all-orders axisymmetric background, moment repair, oscillatory stress realization, covariance inversion, mean corrections, residual improvement, diagonal summation, localization, energy control, comparison, and blowup asymptotics. The core scaling gives a vertical-to-horizontal diffusion ratio of order ϑ(1−t)2h\vartheta(1-t)^{2h}, so fixed anisotropy is subleading in the concentrating core. At higher orders, anisotropy enters the background recursion as a parameter multiplying inherited axial-viscosity sources. For oscillatory pulses, it modifies the principal damping to εk2(nr2+nθ2+ϑnz2)\varepsilon k^2(n_r^2+n_\theta^2+\vartheta n_z^2), requiring a retuning of the pulse envelope and a scale-compatible perturbation theory for the covariance matrix HϑH_\vartheta. We formulate the key estimate ∥Hϑ−H1∥≲∣ϑ−1∣S∗−1/2,\|H_\vartheta-H_1\|\lesssim|\vartheta-1|S_*^{-1/2}, explain why uniform comparison of the full pulse envelopes is neither necessary nor generally appropriate, and identify the determinant, positivity, and signed-inverse bounds needed for iteration. We further show at the structural level that the energy, localization, comparison, and viscosity-rescaling mechanisms remain compatible with constant positive anisotropy. The article concludes with a staged verification program extending beyond the present audit, including independent validation of the base leading-profile theorem, a complete parameter-uniform pulse analysis, a full residual-cycle audit, broader bibliographic verification, and external specialist review.