TEBAC Navier--Stokes Program IIIa: Terminal Axial-Envelope Coercivity and Arithmetic Packing Closure

This preprint is a terminal companion module to the TEBAC Navier--Stokes Program III. It addresses the final axial-envelope obstruction isolated in the NS-III reduction module and develops an arithmetic-packing route toward terminal coercivity on the periodic domain \(\mathbb T^3\). The manuscript continues the spectral-cascade and critical-resonance framework of NS-I, NS-II, and NS-III. Its central purpose is to close the terminal packing/coercivity input associated with the axial-envelope defect \(\mathfrak B_Q^{\mathrm{env}}\). The approach reformulates the remaining high-frequency obstruction as an arithmetic-geometric packing problem on Stokes spheres in the integer lattice. The paper develops the following terminal mechanism: \[\begin{aligned}&\text{terminal axial-envelope defect}\longrightarrow\text{active Stokes caps}\longrightarrow\text{weighted additive-energy depletion}\\&\longrightarrow\text{weighted BSF core extraction}\longrightarrow\text{spherical curvature pruning}\\&\longrightarrow\text{rank descent}\longrightarrow\text{Beltrami-rigid cancellation or dissipative leakage}.\end{aligned}\] The exact constant-envelope Beltrami branch is shown to be harmless because its leading nonlinear interaction is annihilated by the Leray projection. The nonconstant-envelope branch is controlled through leakage weights, additive-energy decomposition, weighted BSF transfer, spherical curvature pruning, slab localization, and a final arithmetic-packing audit. The final result is a terminal companion closure statement: NS-IIIa supplies the arithmetic-packing coercivity mechanism needed to upgrade the NS-III terminal axial-envelope obstruction into a controlled input for the critical-resonance absorption theorem. This manuscript does not claim a standalone proof of the Navier--Stokes Millennium problem. It should be read as a terminal companion module whose role is to provide the final arithmetic-packing input required by the NS-III reduction module.