The Navier–Stokes Singularity Problem as a Discrete Temporal-Relational Phase Transition

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 Φ_critical 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.