The accurate modeling of Earth matter effects is a critical systematic consideration for precision measurements of the $CP$-violating phase ${\delta }_{CP}$ in long-baseline neutrino oscillation experiments. We investigate the extent to which constant and path-averaged density approximations are sufficient for reproducing the ${\nu }_{\mu }\to {\nu }_e$ appearance probabilities predicted by the preliminary reference Earth model (PREM) across baselines ranging from $L=1000\mathrm{km}$ to $L=12000\mathrm{km}$. Using a full three-flavor matrix exponentiation framework with spatially resolved PREM density profiles, we demonstrate that the constant density approximation introduces a negligible bias of less than 0.3° in the reconstructed ${\delta }_{CP}$ for baselines up to $L\approx 5000\mathrm{km}$, but undergoes a sharp increase beyond this threshold, reaching 17.8° at $L=7000\mathrm{km}$ and 172.2° at $L=12000\mathrm{km}$. We show that this bias arises from energy-dependent distortions in the oscillation probability produced by the Earth’s internal density layering, which generate degeneracies between matter-induced and intrinsic $CP$-violating contributions to the appearance channel that cannot be removed by marginalizing over a single effective density. These results demonstrate that the constant density approximation is not a conservative simplification but a source of fundamental systematic error at long baselines, and motivate the incorporation of spatially resolved Earth density treatments, such as PREM, in the analysis frameworks of next-generation long-baseline neutrino oscillation experiments.

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• Neutrino oscillations
• Neutrinos
• CP violation