Matter-Induced CPT Violation and Earth-Density Stratification Effects in Long-Baseline Neutrino Oscillation Experiments

We present a unified analysis of two matter-potential systematics in long-baseline (LBL) neutrino oscillation experiments, treating matter-induced extrinsic CPT violation and Earth density stratification as correlated systematics within a single $\chi^2$ framework. We study their joint impact on CP-phase reconstruction and mass-ordering dependence. Matter-induced extrinsic CPT violation produces a non-zero asymmetry $A_{\mu e}^{\rm CPT}=(P_{\mu e}-\bar P_{\mu e})/(P_{\mu e}+\bar P_{\mu e})$, computed using exact three-flavour matrix exponentiation for T2K, NO$\nu$A, DUNE, and Hyper-Kamiokande. The asymmetry ranges from 0.022 to 0.180 at the respective peak energies and differs by up to $9\%$ between normal and inverted orderings. The surface $A_{\mu e}^{\rm CPT}(E,\delta_{\rm CP})$ at DUNE reveals an interplay between extrinsic CPT and intrinsic CP violation in the high-$L/E$ regime requiring joint statistical treatment. We quantify the sensitivity of $A_{\mu e}^{\rm CPT}$ to density uncertainties $\rho\rightarrow\rho(1+\varepsilon)$ with $\varepsilon=\pm1\%,\pm2\%,\pm5\%$, finding that a $\pm5\%$ uncertainty induces $|\Delta A_{\rm CPT}|<0.008$ at DUNE. Replacing the PREM profile with a constant path-averaged density yields a $\delta_{\rm CP}$ reconstruction bias below $0.3^\circ$ for $L\le5000$ km, increasing to $17.8^\circ$ at $7000$ km and $172.2^\circ$ at $12000$ km. Since both effects arise from the same matter Hamiltonian, they must be modelled jointly using a Poisson log-likelihood $\chi^2$ with nuisance-parameter pulls.