Orbit-Level Transfer Matrix for the 3D Fourier-Galerkin Navier-Stokes System on the Periodic Torus: Explicit Orbit-Triad Incidence Bounds and Deterministic Row-Sum Estimates

Abstract

I study the cubic Fourier-Galerkin truncation of the three-dimensional (3D) incompressible Navier-Stokes equations on the periodic torus after reduction by the full octahedral symmetry group $O_h$. The nonlinear interaction is encoded by a state-dependent orbit-level transfer matrix $M_N(u)$, and the main discrete problem is to estimate orbit-triad incidences in shell slices of translated cubes. Using a face-normalized decomposition, I reduce the local counting problem to the classical two-squares representation function and obtain an incidence bound of order $N^{4+\varepsilon}$ by the shell-counting argument developed in this manuscript. I also derive the exact orbit-level enstrophy identity, the algebraic decomposition $M_N(u)=A_N(u)+V_N(u)$, and deterministic Sobolev row-sum bounds for the raw matrix $M_N(u)$ in the stated range of exponents. These results give an orbit-level description of nonlinear transfer in the truncated system.