TEBAC Navier--Stokes Program I: Spectral Stokes Foundation and Critical Cascade Ledger

This preprint is the first entry module of a TEBAC-based modular program toward the three-dimensional incompressible Navier--Stokes existence and smoothness problem. The manuscript works on the periodic domain \ (T³\) and develops a rigorous foundational layer for the program: the Leray projection, the divergence-free Hilbert space, the Stokes operator \ (A=- P\), the Galerkin approximation scheme, the classical \ (L²\) -energy identity, the vorticity/vortex-stretching ledger, dyadic spectral packets, triadic Fourier interactions, and the first TEBAC cascade-defect formalism. The main purpose of the paper is not to claim a complete solution of the Navier--Stokes Millennium problem. Instead, it closes the first foundational/export module and isolates the decisive large-data obstruction: the critical high-frequency cascade mechanism. The manuscript proves a small-critical Galerkin closure theorem and records the exact large-data residue appearing in the direct \ (H^1/2\) -critical energy method. In the Stokes scale, the first critical estimate targeted by the program is \₀ ₓ ₓ\|A^1/4uN (t) \|₋ℂ^2+₀T\|A^3/4uN (t) \|₋ℂ^2\, dt C (T, , u₀), \ uniformly in the Galerkin cutoff \ (N\). The present module proves this estimate in the small-critical regime and identifies the remaining arbitrary-data cascade absorption problem as the central target for the later modules NS-II--NS-V. The paper is therefore intended as a theorem-bearing foundational roadmap, with explicit claim-safety statements, non-circularity rules, acceptance tests, and a dependency ledger for the full TEBAC--Navier--Stokes program.