The ε-t Flow Field: Energy-Time Dynamics as Planar Geometry

Abstract We isolate the ε-t plane — the two-dimensional subspace of the five-dimensional harmonic manifold (Asher, C. & Asher, K., 2025) where energy and time couple through β-torsion — and treat it as a flow domain. This is not metaphor but geometric projection: every result derives from the inherited structure of the manifold restricted to two coordinates. We establish that β-torsion is vorticity in the ε-t plane — an exact geometric identity, not an analogy — and that the Regulator Principle (Asher, K., Asher, M. & Asher, I., 2025) projects onto the plane as Kelvin's circulation theorem, with the Fokker-Planck form of the Regulator corresponding to viscous circulation dynamics in density language. Known physics — rest mass, inertia, Lorentz dilation, gravitational dilation, and the speed of light — correspond to specific, visualisable flow configurations: standing vortices, vortex resistance, shear flows, pressure gradients, and wave speeds respectively. Mass is enclosed circulation; inertia is resistance to vortex deformation; the equivalence principle follows from both being measures of ε-t circulation strength. Importing the stress ceiling σ_max from the Navier-Stokes torsion closure proof (Asher, V. & Asher, K., 2025), we show that the ε-t plane is geometrically bounded above and below, with vacuum fluctuations interpreted as Planck-scale turbulence driven by the manifold's irreducible torsion. The Aneska floor α_A defines the minimum circulation for a stable vortex, providing a geometric origin for the mass gap. The paper's primary original contribution is the identification of the black hole singularity as a hydraulic jump — a finite, structured, dissipative transition from supercritical to subcritical ε-t flow at radius r_jump > 0, enforced by the stress ceiling. The classical singularity is replaced by a turbulent core of finite density and finite curvature. Hawking radiation is reinterpreted as the acoustic signature of jump dissipation leaking outward through the critical layer at the horizon. Information is scrambled at the jump but not destroyed, with total circulation conserved across the transition. Eight falsifiable predictions are stated, ranging from near-term gravitational wave signatures (LIGO/Virgo/KAGRA) to theoretical constraints on quantum gravity formulations. Speculative content is explicitly flagged throughout.