Proof VI of VII

Numerical / Constructive

Grid simulation confirmation of P_ON
Grid simulations confirm P_ON > 0 under correct operator sequencing, and P_ON → 0 under reversed sequencing. The constructive proof closes the loop between algebra and physics.

The algebraic and geometric proofs (I–V, VII) establish the existence of the stable attractor and the conditions for reaching it. Proof VI closes the constructive loop: explicit finite-difference grid simulations confirm that predicted behavior actually appears in numerical computation.

Simulations are run on a discretized dm³ contact manifold with the full operator sequence applied explicitly at each timestep. Under correct order, the vortex coherence probability P_ON converges to approximately 0.94 within 5 turnover times and remains stable across all tested storm scenarios.

Under reversed order, P_ON decays to zero within 3 turnover times regardless of inflow strength. The system settles into turbulent non-rotating flow — no energy harvesting, no flood peak attenuation.

Sensitivity analysis across 10-year, 50-year, and 100-year design storm inflows shows P_ON under correct order is robust: the vortex forms faster under higher inflow and steady-state coherence does not degrade. The system performs better at scale.

Key relations
P_ONP_ON = lim_{t→∞} P(vortex coherent at time t)
Correct orderP_ON(K before F) → ~0.94 within 5 T*
Wrong orderP_ON(F before K) → 0 within 3 T*
Grid resolutionΔr=0.01, Δθ=π/180, Δt=0.001 T*
Storm scenariosP_ON stable: 10-yr, 50-yr, 100-yr inflows
t / T* P_ON 0 1 10-yr / 50-yr / 100-yr storm band (K→F robust across all) K→F ≈0.94 F→K →0 5T*
Engineering consequence
HVEH modules designed for 10-year storm flows will outperform their rated capacity during 50-year and 100-year events. This inverts the typical infrastructure vulnerability curve — the system gets more stable as the event gets more severe, up to the basin overflow limit.