How classical equations hide quantum integrability inside their hidden symmetries

Classical integrable equations like sinh-Gordon arise from compatibility conditions on matrix differential problems. The Wronskian coefficients connecting different solutions encode quantum data: they are eigenvalues of transfer matrices and Baxter operators that satisfy Y-systems and thermodynamic Bethe Ansatz equations, all without invoking scattering theory. This framework includes N=4 super Yang-Mills at strong coupling and dual Wilson loops in AdS_3, and directly proves two Zamolodchikov conjectures.