The dispersionless Toda lattice is an integrable system describing the evolution of conformal maps. Its tau-function — the master generating function — encodes all the dynamics. The mixed Hessian of the tau-function (second derivatives mixing different variables) has spectral structure that reveals when the system becomes unstable.
The paper on the spectral structure of the Toda Hessian (arXiv: 2603.23424) identifies two thresholds for s-fold symmetric conformal maps: an analytic threshold ζ_c (where analyticity breaks down) and a geometric threshold ζ_univ (where the map stops being univalent, i.e., injective).
The surprise: spectral instability occurs first at the analytic threshold, before the geometric one. The map is still injective when the Hessian first develops an unbounded eigenvalue. In each symmetry sector, exactly one eigenvalue diverges logarithmically, while all others remain bounded.
Beyond the critical point, the scalar Gram functions (building blocks of the Hessian) extend via hypergeometric functions and connect to Weyl functions of Jacobi operators — a bridge from integrable systems to spectral theory of discrete Schrödinger operators.
The through-claim: analytic breakdown precedes geometric breakdown, and the Hessian detects it. The conformal map hasn't self-intersected when the Toda dynamics first becomes unstable. The instability is analytic (loss of regularity), not geometric (loss of injectivity). The Hessian is a more sensitive detector of criticality than the geometry.
2603.23424. Integrable systems / Toda lattice / conformal maps / spectral theory / tau-functions.