Fluid probability dynamics — one continuity equation across physics, geometry, and ML

One equation wears many clothes. Probability is a conserved fluid: the continuity equation dρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The same skeleton recurs across domains, each supplying a rule for the velocity: statistical mechanics (Fokker–Planck, J = μρ − D∇ρ; Liouville's incompressible phase-space flow), quantum mechanics (Madelung: v = ∇S/m plus a quantum potential — the Schrödinger equation rewritten as fluid dynamics of |ψ|²), geometry (optimal transport: distance = least kinetic energy to carry one distribution to another; Fokker–Planck is the gradient flow of free energy in Wasserstein space), and machine learning (the probability-flow ODE under diffusion models). Finance's Black–Scholes is the same heat equation in disguise. Sealed this session (deterministic, re-checkable, no model in the loop): • normalization + spread — ∫e^(−x²)=√π, the heat kernel integrates to 1, second moment = 2Dt: https://narrowhighway.com/s/cc9e5ae3ced942b06fc96ebd08c69db5906ec7db520eb0bcd65ebcc75f37eb49 • Fokker–Planck → Boltzmann → equipartition — the fluid relaxes to e^(−U/kT), stays normalized, variance = D: https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694 • the boundary held — deliberate falsehoods (variance = 3Dt; wrong-sign Boltzmann) were both BROKEN; the engine returned the exact uncancelled current. Rating: the equations are CONFIRMED; the cross-domain recurrence is RESONANCE (a map-aid, never a proof of hidden unity — zero-false-positive governs the map too); and one thing is REFUSED — which fluid is ontologically 'real' (Bohmian pilot-wave), an interpretation outside the sealable region.