Geometric Regularity Proof | Bakry-Émery Curvature

GEOMETRIC REGULARITYPROOF

Bakry-Émery curvature of the eddy viscosity manifold

Step 1 — Given (Eq 10 closure)

Step 2 — Log-viscosity

Step 3 — Hessian of ψ

Both eigenvalues negative → ψ strictly concave

Step 4 — Bakry-Émery Ricci

Positive definite! (negative of negative = positive)

Step 5 — Spectral gap (κ > 0)

Step 6 — Wasserstein-LSV transfer

Ric_BE ≥ κ ⟺ κ-displacement convexity of H along W₂ geodesics (Villani 2009) ⟹ weighted Poincaré via HWI (Otto-Villani 2000)

Step 7 — Enstrophy inequality (Z² supercritical)

CONCLUSION

Global regularity by Sobolev bootstrap

ν_t MANIFOLD

Height = smoothness s

Radius = ν_t scale

Angle θ = vorticity phase (spin)

Blue = negative K (spectral gap)

Red = positive K

Geodesics diverge at the pinch

Gaussian Curvature K

K≪00K≫0

● Gaussian K(s) ●

● Spectral gap κ(s,|v|)

● dZ/dt = CZ^1.5 − γ'sZ²

|v|:3.0 s_min:0.05 C/γ':1.0