The Asymptotic Cohomology of the Pedal Flow: Schwartzman Classes, Spectral Obstructions, and Topological Invariants of the Jonah Saddle

We associate to the pedal map FtF_tFt on the Kendall shape sphere S2S^2S2 an asymptotic cycle μJ\mu_JμJ supported on the stable manifold of the Jonah saddle JJJ. Using the framework of Schwartzman's asymptotic cohomology, we prove that this cycle defines a non-trivial class in the dual space of continuous functions modulo coboundaries. We compute the pairing of this class with the logarithmic Jacobian determinant ln⁡∣det⁡DFt∣\ln |\det DF_t|ln∣detDFt∣, obtaining an exact value related to the spectral gap γ=λu−∣λs∣=594/343\gamma = \lambda_u - |\lambda_s| = 594/343γ=λu−∣λs∣=594/343. This pairing detects the Liouville obstruction: for t<t0t < t_0t<t0, the class is non-zero, implying no smooth conjugacy to a gradient flow exists. For t>t0t > t_0t>t0, the class vanishes locally, consistent with the embedding theorem of Zhang. We further relate this cohomological invariant to the Chern class of the Hopf fibration over the Jonah fiber, establishing a bridge between local spectral dynamics and global topological constraints. Numerical verification using high-precision interval arithmetic confirms the theoretical predictions to 10−1510^{-15}10−15.