The Navier–Stokes existence and smoothness problem asks whether smooth, globally defined solutions exist for the three-dimensional incompressible Navier–Stokes equations. The obstruction is the finite-time blow-up of vorticity, governed by the Beale–Kato–Majda (BKM) criterion. This paper treats the problem in the vorticity formulation of Pure Temporal Geometry (PTG-ℂ), complementing the primitive-variable treatment of PTG-NS. The velocity field u is identified with the kinematic axis Δ. Vorticity is identified with a signed component of the temporal context axis Φ, not with its magnitude. Viscous diffusion, the dissipative term of the vorticity equation, is identified with the SINK operator Ś; vortex stretching, the non-dissipative, scale-redistributing term, is identified with the PUMP operator P̂. This assignment matches the fixed thermodynamic character SINK and PUMP carry throughout the series and matches the identification already made in PTG-NS. Two separate results are proved. Magnitude boundedness (Theorem 2) shows that Φ cannot exceed the finite JUMP threshold Φcritical while a node remains synchronised, so ‖ω‖∞ does not diverge in finite steps; this is the resolution of the BKM criterion, inherited from the Singularity Prevention Lemma of PTG-NS. Separately, the six-step Chrono-Elastic cycle Ĥ → P̂ → Ś → Ĥ → Ś → P̂ is shown to close exactly (T₆ = T₀) and to be exactly holomorphic over one complete period, with two applications of the Horizon operator Ĥ (complex conjugation) marking two reconnection events per period. Magnitude boundedness and the Horizon crossing are distinct structural results; the first resolves BKM, the second describes the periodic internal structure of the already-bounded turbulent cascade.
Isong Otto Beseka (Fri,) studied this question.