A Tether That Holds Nothing: Energy-Null Metriplectic Corrections and the Vanishing Enstrophy Contribution on T^3
Year: 2026
Type: preprint
Abstract: We record the Frohmanian geometric tether, a symmetric, negative-semidefinite quadratic correction to the Lie-Poisson bracket on the vorticity coadjoint orbit of ideal incompressible fluid, in both its stream-function (psi) and velocity-slot forms. The velocity-slot kernel annihilates the kinetic energy by construction (condition C2), restoring consistency of the metriplectic sum with the classical Euler equations. That consistency, however, is precisely the obstruction: by symmetry the tether contributes nothing to the enstrophy balance, so no cubic damping of enstrophy appears. We state the open analytic hypothesis FrohmanianTetherRate, distinguish its inequivalent differential forms, and delimit the scope of the associated product theorem: unconditional only for a discontinuous step field, zero data, and constant flow. This is a negative/structural result. Lean 4 status: tetherKernelVel_C2 is kernel-checked conditional on the named hypothesis KineticEnergyCurlIdentification. No claim is made on the Clay Millennium problem; global regularity of 3D Navier-Stokes remains open. Files: paper PDF, LaTeX source, and figure PNGs.
Author: Frohman, Benjamin Stanley
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225325472