The Navier–Stokes Millennium Problem in Geometry-Attached Time

This work reformulates the Navier–Stokes Millennium Problem using geometry-attached time, making explicit an implicit assumption of the classical formulation: that time remains a globally valid evolution parameter whenever standard energy bounds hold. By conditioning time on admissible geometry, the paper shows that the traditional binary framing—global smoothness versus finite-time blow-up—is incomplete. Finite-time breakdown admits a third, mathematically distinct regime (admissibility collapse), in which energy may remain bounded while the geometry required for time-parametrized evolution degenerates. This reframing clarifies the logical structure of the Millennium Problem without claiming a solution.