Algebraic Structure of Three-Flavor Neutrino Oscillations in Constant-Density Matter: Cayley--Hamilton Evolution, DMP Resummation, and Closed-Form Uncertainty Propagation

For three-flavor neutrino oscillations in constant-density matter, the Cayley–Hamilton theorem forces the evolution operator into a quadratic polynomial in $\widehat{H}$, with coefficients determined by the three real eigenvalues through a Vandermonde system we write out explicitly. The eigenvalues follow from Cardano’s trigonometric formula, recovering the Zaglauer–Schwarzer expressions. The Denton–Minakata–Parke (DMP) approximation achieves fractional accuracy better than ${10}^{-4}$ because its $1$–$3$ rotation is a resummation: it removes the near-degeneracy that makes the naive expansion diverge at $\widehat{A}\to 1$, replacing the unbounded $(1-\widehat{A}{)}^{-1}$ with an effective parameter ${\epsilon }_0\lesssim 0.015$ bounded uniformly in energy. A density-matrix treatment with a Lindblad term handles open-system decoherence and wave-packet effects in the same language; matter-dressed coherence lengths satisfy $L\mathrm{/}{L}_{\mathrm{c}\mathrm{o}\mathrm{h}}^{ij}\sim {10}^{-3}$–${10}^{-2}$ for terrestrial baselines. The CP asymmetry ${\mathcal{A}}_{\mathrm{C}\mathrm{P}}({\nu }_{\mu }\to {\nu }_e)$ is split into genuine and matter-induced fake contributions. Closed-form Jacobians in the NuFIT~6.0 parameter basis feed Monte Carlo and linearized uncertainty-propagation schemes, the latter validated against a Feldman–Cousins profile-likelihood mapping near physical boundaries. The Denton–Parke NuFast-LBL algorithm [Phys.\ Rev.\ D {\bf 110}, 073005 (2024)] remains the tool of choice for production fits; the analytic expressions here supply what iterative solvers cannot – parameter continuity, transparent limits, and Jacobians in closed form.