Resolution of Navier-Stokes Singularity Problem in ROG-H

The Navier–Stokes existence and smoothness problem asks whether the three-dimensional incompressible Navier–Stokes equations admit smooth, globally defined solutions. The obstruction is finite-time blow-up of vorticity, governed by the Beale–Kato–Majda (BKM) criterion. This paper states the problem in the vorticity formulation of Pure Temporal Geometry, in its quaternionic realisation (PTG-ℍ). We map the Navier–Stokes variables to the four quaternionic axes: velocity to the Now (Δ), vorticity to Memory (Φ₁), topological ordering to Sequence (Φ₂), and vortex-stretching potential to Expectation (Φ₃). The carrier of the flow is a transient gaseous bubble: a node that nucleates, grows, reconnects, and dissolves, rather than a permanent fluid parcel. Two results are established and kept separate. Magnitude boundedness (Theorem 2) shows that Φ₁ cannot exceed the finite JUMP threshold Φ_critical while a node remains synchronised, so ‖ω‖∞ does not diverge in finite steps. This is the resolution of the BKM criterion. Separately, we show that the four-dimensional Horizon operator Ĥ_ℍ fails to close the turbulent cascade, because the Heisenberg torsion is a single directed channel rather than a symmetric term. Closure is instead supplied by the gauge-shift mechanism already proved for the general Heisenberg Cycle and Heisenberg Wave: the six-step Heisenberg Bubble Lifecycle closes exactly, and the cascade achieves net gauge restoration over one period.