Computer-Assisted Global Regularity for the Kida Flow Family via Λ_all Lattice Reduction

We establish global regularity for an explicit infinite-dimensional neighbourhood of the Kida velocity center u_K = curl⁻¹ω_K on the three-dimensional torus: for every viscosity ν > 0, every 0 ≤ a ≤ 3/2 (certificate-covered; a = 2 machine-accepted), and every smooth divergence-free perturbation h with Wiener norm ‖h‖_A₂ ≤ ν/40, the datum u₀ = aν·u_K + h generates a unique global smooth solution of the unforced 3D periodic Navier–Stokes equations. Three contributions beyond the family-level CAP framework of Silva (arXiv:2609.16157): 1. The true Kida velocity center curl⁻¹ω_K is treated for the first time — quantitatively milder than all three centers analyzed by Silva (accepted amplitude 3ν/2 by certificate, 2ν at a coarser terminal time, versus 41ν/40); 2. Every spectral computation is reduced to the Λ_all sublattice (1/4 of Fourier modes), whose closure, five-fold parity selection rule, and S₄-factorization are separately verified in the Lean 4 proof assistant; 3. The quantitative core is certified in exact rational arithmetic — a 21-slab piecewise-affine supersolution for the Wiener comparison ODE, re-verified by a second, code-independent implementation, with three negative controls; a 22-slab mirror certificate closes the enstrophy bridge. All slab tables (exact rationals, JSON) are included in this deposit. As a byproduct we exactly reproduce, in rational arithmetic, all published algebraic constants of the TG3 family — whose center coincides componentwise with the Kida vorticity field — providing an independent verification of that work. The analytic architecture (Wiener convolution estimate, comparison inequality, terminal Kato barrier, Aubin–Lions compactness) is re-derived in full. Scope and honesty: the result covers O(ν)-amplitude neighbourhoods of a symmetric center and makes no claim on the Clay Millennium problem. Companion papers: Paper 1 (DNS evidence, Zenodo DOI 10.5281/zenodo.21849021); Paper 2 (lattice analysis of vortex stretching, Zenodo DOI 10.5281/zenodo.23062485). The three papers contain disjoint content.