Phase Propagation Topology and the Navier–Stokes Existence and Smoothness Problem

We introduce a geometric framework — Phase Propagation Topology (PPT) — for organising potential singularities of the three-dimensional incompressible Navier–Stokes equations. The central object is a phase map φ : M → C, where M is a Möbius strip (the singularity domain) and C is a cylinder (spacetime). We prove that the singular set S(φ) forms a well-founded fractal hierarchy under a geodesic metric d = R(φ)·Δθ, that the phase space is Cauchy-complete, and that the velocity field recovered from φ satisfies all three Leray–Hopf conditions away from S(φ). A speed-of-light bound ‖v‖_L∞ ≤ c emerges geometrically at the second singularity level. We further prove that the singular set carries a non-trivial first Stiefel–Whitney class w₁(S(φ)) ≠ 0, making it topologically non-removable yet globally bounded. The framework targets the Clay global regularity formulation: singularities exist, are identifiable, and never cause ‖∇v‖_L² → ∞ in finite time.