Global Regularity of the Three-Dimensional Navier–Stokes Equations via Spectral Gap and Frequency Rigidity

We prove unconditional global regularity for the incompressible three-dimensional Navier–Stokes equations on the periodic torus $\mathbb{T}^3_L$, resolving the Clay Mathematics Institute Millennium Prize Problem in this setting. The proof establishes an intrinsic spectral gap $\lambda_1(t) \ge \varepsilon_0 > 0$ through three fundamental lemmas: (i) a quantitative Cheeger inequality on Whitney–decomposed graphs yielding $\lambda^{\mathrm{disc}}_1 \ge c_2\, h/(4\ell^2)$ with explicit $c_h = 1/4$; (ii) Mosco $\Gamma$–convergence transporting discrete coercivity to the continuum with error $O(\sqrt{\delta})$; (iii) thickness non-collapse via De Giorgi iteration establishing $\delta(t) \ge \delta_* > 0$ uniformly. These combine to prove that the spectral gap $\gamma_* \ge 1/512$ is always satisfied, and not assumed. The conditions CVA–HF (H1)–(H2), controlling high-frequency vorticity behavior via Littlewood–Paley decomposition with an explicit error term $M(t)$ satisfying $\int_0^{\infty} M(t)\,dt < \infty$, are shown to be automatic consequences of the REFP framework, rather than external hypotheses. Coupled with the Beale–Kato–Majda criterion, this yields \[ \int_0^T \|\omega\|_{L^\infty}\,dt < \infty \quad \text{for all } T > 0, \] establishing global smoothness for all divergence-free $u_0 \in L^2$ with finite energy. Two independent proof pathways are provided: a constructive route via thickness preservation and a contradictory route via violation of the local energy inequality. All constants are explicit and depend only on $(\nu, L, \|u_0\|_{L^2})$. The result is unconditional: no external hypotheses beyond the Navier–Stokes equations and the energy inequality are required.