Conditional Continuation and Small-Data Regularity for 3D Navier–Stokes on 𝕋³, with the Frohmanian Symplectic Tether as a Metric (Non-Poisson) Correction

Review draft. This paper is conditional and gives no proof of the Clay Millennium problem; large-data global regularity for 3D Navier–Stokes remains open. Theorems A (Beale–Kato–Majda / Kozono–Taniuchi continuation: T* = ∞ if ∫‖ω‖B0∞,∞ dt < ∞), B (Kato local existence, Tloc ≥ 1/(Cs‖u0‖Hs), s > 5/2) and C (Fujita–Kato small data, ‖u0‖Ḣ1/2 ≤ ε0ν, mean-zero) are classical results, cited from Beale–Kato–Majda (1984), Kozono–Taniuchi (2000), Kato, and Fujita–Kato (1964); they are not claimed as new or as consequences of the tether. The Frohmanian Symplectic Tether term B(F,G) is symmetric, so it is a metric (metriplectic-type) correction, not a Poisson correction; no Jacobi identity is claimed. In the velocity-slot form (Πu applied to ∇×δF/δω), B(F,H) = 0 because ∇×δH/δω = u. The cubic inequality M′ ≤ CM² − κ″M³ remains an open hypothesis.