Information geometry treats probability distributions as points on a smooth manifold equipped with the Fisher information metric. Flow states — characterized by their vorticity distribution, coherence spectrum, and phase statistics — are points on this manifold.
The manifold of flow states has negative sectional curvature everywhere, a consequence of the contact structure and the log-concavity of the vortex phase distribution. On negatively curved manifolds, geodesics diverge exponentially — two paths from nearly the same point end up arbitrarily far apart.
The correct and incorrect operator sequences correspond to geodesics γ_K and γ_F. Because the manifold is negatively curved, these belong to different homotopy classes in π₁(M): they cannot be continuously deformed into one another. There is no smooth interpolation between VORTEX and CHAOS.
This is the deepest of the seven proofs. It says not merely that wrong order produces a bad outcome, but that the two outcomes are topologically separated. No fine-tuning, no partial operator application, can slide the wrong-order system into the correct-order basin.
| Fisher metric | g_ij(θ) = E[∂_i log p · ∂_j log p] |
| Sectional curvature | K_sec < 0 everywhere on the flow manifold |
| Correct geodesic | γ_K: LAMINAR → VORTEX [class K] |
| Wrong geodesic | γ_F: LAMINAR → CHAOS [class F] |
| Topological result | [γ_K] ≠ [γ_F] in π₁(M) → no deformation |