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Energy-Helicity Compensated Spectral Depletion for the Three-Dimensional Navier-Stokes Equations with Exact Cone-Mellin Identities

Year: 2026
Type: preprint
Abstract: We introduce an energy-helicity compensated spectral defect for the three-dimensional incompressible Navier-Stokes equations, K_EH(u) = inf_{a,b in R} ||(Lambda - aI - b curl)u||_{H^{1/2}}, which projects the critical derivative away from the gradients of the two quadratic Euler invariants. Using the energy and helicity cancellations, we prove the nonlinear depletion estimate | | <= C K_EH(u) ||u||_{H^1}^2. Consequently, finiteness of the time integral of K_EH(u(t))^2 is a scale-critical continuation criterion for maximal strong solutions. We compute the optimal projection coefficients and obtain exact Gram-matrix, spin-resolved variance, and positive three-frequency determinant representations. The defect allows different spectral centers for the two helicity sectors and is no larger than the corresponding one-parameter radial spread. We also derive an exact cone-Mellin superposition theorem for Gaussian heat-scale stresses. Mellin integration of the viscous-cone transport laws recovers, with explicit constants, the homogeneous H^sigma balance for 0 < sigma < 1. At sigma = 1/2, the signed Gaussian flux reproduces the nonlinear H^{1/2} production, showing that the cone and Mellin descriptions are exact resolutions of the same balance rather than independent coercive estimates. Finally, a scaling argument rules out an instantaneous bound of either the compensated defect or the positive cone-flux density solely by the Leray energy and enstrophy. This manuscript is a preprint and has not undergone peer review. It establishes conditional continuation criteria and structural obstructions; it does not prove global regularity for the three-dimensional Navier-Stokes equations.
DOI: 10.5281/zenodo.21430333
Language: English
Open Access status: green
OpenAlex ID: W7169655288
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