Equivalence classes of finite-time transitions in optimal control and non-equilibrium relaxation

What the math gives to ML

The paper provides a constructive classification of finite-horizon stochastic optimal-control problems by the sign of the determinant of a Hamiltonian matrix, yielding parabolic, hyperbolic, and elliptic control regimes. Its transferable asset is the prediction that finite training horizons can cross a computable conjugate-point or bifurcation time at which the optimal trajectory changes qualitatively, rather than merely degrading smoothly. A practical neural-network adaptation is a mode-wise Hamiltonian optimizer that estimates local curvature and damping, computes the corresponding finite-horizon Riccati flow, and changes the horizon or learning-rate policy before the predicted singularity. The mechanism is falsifiable through a sharp divergence or mode-switch boundary in the optimizer gain as the horizon crosses the calculated critical time.