Chapter 33: The Hartogs Flow Operator

We construct the central operator of the Navier–Stokes treatment: the Hartogs flow operator H_flow, which extends τ-admissible initial data into the τ³ bulk via Local Hartogs continuation. The operator is linear, tower-coherent, and sector-preserving. A structural surprise emerges: the split-complex codomain H_τ forces the natural differential operator to be the wave operator, not the Laplacian. This operator polarity swap—from elliptic to hyperbolic—is architecturally forced by the bipolar structure of the lemniscate boundary, and it explains why τ-internal Navier–Stokes has hyperbolic character while the classical formulation is parabolic.