The Hausdorff Dimension of the Intense-Vorticity Set in Three-Dimensional Navier–Stokes Turbulence: A Universality Class Identification

We close two gaps identified in a companion work [2] in the proposed identification of the intense-vorticity set of the three-dimensional incompressible Navier–Stokes equations with the percolation backbone at the infrared fixed point η∗ = 2.52299 of the threedimensional percolation universality class.Gap 1 (functional-analytic identification). We show that three constraints imposed by the Navier–Stokes equations in R3 — three-dimensional embedding, local conservation of vorticity flux, and local dissipation — uniquely select the three-dimensional percolation universality class as the infrared fixed point of the vorticity renormalization-group flow. The identification is conditional on a uniqueness lemma whose proof strategy — via the operator product expansion of the vorticity field at the percolation threshold — we formulate precisely as a target for subsequent work.Gap 2 (coarse-graining scale to physical time). We construct an explicit map between the RG coarse-graining scale ℓ and physical time t via the Kolmogorov eddy-turnover time, yielding an unconditional falsifiable prediction: with an intermittency correction shifting the finite-Reynolds-number exponent toward −0.951 at current direct-numerical-simulation (DNS) resolution (Reλ ∼ 103). This prediction is testable against existing DNS data.Combined with the Gruj´ı´c sparseness program [3, 4], the identification implies global regularity of suitable weak solutions at any finite kinematic viscosity ν > 0, conditional on the uniqueness lemma and the analytical verification that η∗ = 2.52299 closes the harmonic-measure scaling gap.