Conditional Regularity of 3D Navier-Stokes via Latent Spectral Analysis and Energy-Normalized Bilinear Bounds

We introduce a spectral framework for analyzing the regularity of 3D Navier-Stokes equations through the analyticity strip width \(\sigma(t) = \log \rho(t)\), where \(\rho\) is the radius of convergence of the Fourier series. The key observable is the bilinear growth exponent \(\alpha\) in \(g(\sigma) \sim \sigma^{-\alpha}\), where \(g(\sigma) = \|T \cdot c(\sigma) \otimes c(\sigma)\|\) measures the self-consistent nonlinear stretching. We prove that for energy-normalized spectral coefficients, \(\alpha = (d-2)/2\), which equals \(0.5\) in three dimensions — well below the critical threshold \(\alpha = 2\) required for bootstrap regularity. This yields a Conditional Regularity Theorem: if the initial velocity field has exponential spectral decay (\(\sigma_0 > 0\)) and finite energy, then \(\sigma(t) \geq \min(\sigma_0, \sigma_c) > 0\) for all \(t > 0\), where \(\sigma_c\) is an explicitly computable threshold. The framework is supported by 93 formally verified proofs (Lean 4) and numerical validation via pseudo-spectral 3D Navier-Stokes simulations confirming \(\sigma(t) > 0\) across all tested scenarios.