CHIC: Caley-Hamilton, Invariants and Constants for Neutrino Oscillation Probabilities and Gradients

We use the Caley-Hamilton theorem to derive analytical solutions for the three-flavor neutrino propagation amplitude in a constant-density medium and their derivatives with respect to the mixing parameters. This approach avoids the diagonalization of the Hamiltonian and exploits precomputed matrix invariants to separate the dependence of oscillation probabilities on neutrino energy and propagation baseline. The results are implemented in the CHIC software, which provides simple, fast and efficient computation of oscillation probabilities and their derivatives. Finally, we demonstrate the value of probability gradients for neutrino data analyses and introduce a complementary visualization, the oscillograds, to probe underlying features of neutrino mixing.