Localized Spectral Nucleus and Orthogonal Fourier-Chain Competition in a Restricted H¹ Perturbation-Growth Problem for the Incompressible Navier-Stokes Equations

We study a restricted three-dimensional H¹ perturbation-growth problem obtained by linearizing the incompressible Navier-Stokes equations about a periodic Taylor-Green-type base flow. The analysis is performed after Leray projection and H¹ conjugation in a divergence-free Fourier basis. The four active base-flow Fourier branches separate into two symmetry-related pairs. One pair preserves kx-ky and the other preserves kx+ky. Introducing reduced coordinates r=(kx+ky)/4 and s=(kx-ky)/4 converts the fixed-kz problem, in the parity sector containing the dominant state, into a nearest-neighbour 2x2 block operator on a Z² lattice. Numerical spectral calculations show a dominant state at kz=1 with growth-rate candidate gamma*=2.17413843730594. The value is unchanged to displayed precision from reduced-lattice half-width M=12 through M=32. The corresponding eigenstate is strongly localized: 99.7268% of its norm lies within lattice radius R<=1, 99.9844% within R<=2, and 99.9998865% within R<=4. Removing the low-mode core gives an exterior spectral edge near 1.94807, leaving a gap of about 0.22607. Embedding the M=32 eigenvector into M=33 produces a residual of 2.49e-14. These observations support a mechanism in which two non-aligned Fourier-chain families feed a localized low-wave-number spectral nucleus rather than simultaneously realizing their separate worst-case growth directions. The result is strong numerical evidence for an isolated spectral state in this restricted problem, but an infinite-dimensional theorem still requires an analytic exterior bound and validated finite-core arithmetic.Keywords: Navier-Stokes equations; Taylor-Green vortex; H¹ stability; Fourier spectral geometry; localized eigenstate; perturbation growth The Python code for verification and the request text for this study are provided in the appendix.