The Spectral Window as Dissipation Operator in a Dyadic Navier–Stokes Model: What a Physical Ultraviolet Cutoff Does and Does Not Say about the Navier–Stokes Regularity Problem (SQG Navier–Stokes Note, v1)

This note inserts the window operator of Spectral Quantum Gravity (SQG), H_ω = −κ∂²_ω + c₀/ω² with Dirichlet conditions on [ω_min, ω_max], as the linear dissipative part of an energy-conserving dyadic shell model of Navier–Stokes type. Two structural facts are established. First, by the Weyl asymptotics of the window, the eigenvalues at dyadic indices n_j = 2^j satisfy λ₂ₙ/λ_n → 4, the shell ratio of the Laplacian on dyadic Fourier shells, so the window reproduces Navier–Stokes dissipation on shells without adjustment. Second, for every finite number J of shells, i.e. for any physical ultraviolet cutoff such as the Planck lattice of the SQG measure construction, the model has a unique global smooth solution for all non-negative initial data, but the bounds on higher norms grow with J.In a scan registered before it was run (five nonlinearity exponents, three data sizes, two viscosities, cutoffs up to J = 13, exact Bessel eigenvalues; 150 runs in 30 parameter pairs), the maximal enstrophy converges in J in every pair and no blow-up signature appears.The note states explicitly why none of this bears on the Clay Millennium problem: the problem concerns the continuum equations without cutoff, a cutoff makes regularity trivial rather than proving it, dyadic models are not the Navier–Stokes equations, and Tao's averaged-equation blow-up shows that energy identity and generic harmonic-analysis bounds cannot suffice. In fluid dynamics "SQG" denotes the surface quasi-geostrophic equation; in this note SQG always means Spectral Quantum Gravity.All claims are graded [proved], [computed], [imported], [argued], or [open]. The code archive contains the pre-registration, the scan script and the full result log. Derivations, verification code and drafting were carried out in collaboration with Claude (Anthropic).