A Scale-Critical Dual-Barrier Framework for Spectral Runaway in the Three-Dimensional Navier–Stokes Equations

We study high-wavenumber spectral runaway in the three-dimensional incompressible Navier–Stokes equations by separating two mechanisms: propagation of nonlinear feed across scales and growth of a scale-critical high-frequency tail. For cutoff R we define a dyadic feed ratio γ_R=F(2R)/F(R) and a critical-tail ratio Ψ_R=I_R/D_R, where I_R is positive nonlinear injection into the |k|≥R part of the homogeneous H1/2 energy and D_R is viscous dissipation. A half-scale triadic restriction shows that an output |k|≥R with k=p+q requires |p|≥R/2 or |q|≥R/2. Same-output projected triad contributions lie in k⊥⊂C³ and therefore admit a real Gram representation of rank at most four. Across 36 finite Fourier–Galerkin configurations, γ<1 and Ψ<1 held simultaneously over the tested trajectories. An exact amplitude-scaling argument, however, gives Ψ_R(Au)=AΨ_R(u), ruling out a universal state-independent Ψ<1 bound for arbitrary smooth data. Nested tails likewise imply F(2R)≤F(R), but not a universal strict contraction. The contribution is therefore a Navier–Stokes-centered conditional spectral-runaway framework, a low-dimensional same-output cancellation geometry, and an explicit obstruction that any unconditional large-data theory must overcome.