Detectability and Systematic Bias from First-Order Phase-Transition Dephasing in Kerr EMRIs

We study gravitational-wave dephasing induced by an effective first-order phase transition in a Kerr extreme mass-ratio inspiral (EMRI). The transition is modeled phenomenologically as a finite-width restructuring of the dissipative flux sector, and its observational consequences are quantified with standard LISA matched-filter diagnostics. For a representative system with $M=2\times10^{5}M_\odot$, $\mu=1.4M_\odot$, and $\hat a=0.90$, we obtain $\rho_{\rm B}=5.064$, $\rho_{\rm T}=4.073$, $\rho_{\rm R}=1.051$, and a mismatch $\mathcal M=2.986\times10^{-3}$ after maximization over extrinsic time and phase shifts. Although the normalized mismatch remains small, the accumulated phase difference grows to $\Delta\Phi_{22}^{\rm SF}\sim 5\times10^{3}\,\mathrm{rad}$, indicating that a narrow transition window can generate a large coherent deformation of the inspiral clock while leaving the waveform globally close to the baseline branch in detector-weighted norm. The resulting signal therefore lies in a bias-sensitive regime, characterized by small mismatch, order-unity residual norm, and large cumulative dephasing. Our results suggest that the dominant consequence of the transition sector is not loss of detectability, but loss of faithfulness for precision inference. This motivates future LISA EMRI waveform models that incorporate parameterized transition sectors directly into the waveform manifold.