τ-Navier–Stokes Regularity
τ-Navier–Stokes regularity (`IV.R161`) is the τ-categorical statement that the τ-NS flow on τ-admissible fluid data preserves regularity globally in time: the defect-transport operator on the T² fiber of the holomorphic state space is compact, so no finite-time singularity can develop. The classical Navier–Stokes equations are the chart-shadow projection (`V.P43`); the Clay Millennium NS regularity problem reduces to this categorical fact.
τ-Definition
τ-Navier–Stokes regularity (`IV.R161`) is the τ-categorical statement that the τ-NS flow on τ-admissible fluid data preserves regularity globally in time: the defect-transport operator on the T² fiber of the holomorphic state space is compact, so no finite-time singularity can develop. The classical Navier–Stokes equations are the chart-shadow projection (`V.P43`); the Clay Millennium NS regularity problem reduces to this categorical fact.
Categorical invariant. On τ-admissible fluid data, the defect-transport operator on the T² fiber is compact; ‖u_n‖_{H^s} stays bounded for all refinement n and all t > 0.
Primary registry anchor:
IV.R161
τ-Derivation Chain
-
I.K0— Universe Postulate -
IV.D223— Navier–Stokes regime — τ-admissible fluid data class -
IV.D232— τ-Navier–Stokes flow on the T² fiber -
IV.R161— Navier–Stokes regularity — compactness of T² fiber prevents singularity formation -
V.T71— Macro τ-NS regularity — chart-shadow lift to the macroscopic Navier–Stokes equations
Lean modules referenced:
TauLib.BookIV.ManyBody.DefectFunctionalExt2
SI Translation
Calibration anchor: PG-P01-neutron
Calibration chain:
- T² fiber compactness on the τ-holomorphic state space
- viscosity ν read off from the B/D-sector cascade
- SI bridge via m_n anchor for density and length scales
Manuscript reference: manuscript-sources/book-04/part06/ch52-defect-functional.tex