Finite-Window Centered Organization of Neighboring Poles
Near-degenerate resonance poles arise widely in open-wave systems. For gravitational-wave ringdowns, inference is performed on finite time windows where neighboring quasinormal modes can be spectrally close; the waveform is then dominated by a common carrier with a slowly varying interference envelope, while representing the signal as a sum of two independently resolved damped exponentials $e^{-\ii\omega_\pm t}$ becomes numerically ill-conditioned when the dimensionless splitting $\eta=|\sigma|T_{\mathrm{eff}}$ is small. We give a finite-window organizing principle for such neighboring-pole sectors: the local two-pole singular block of the Green-function integrand is rewritten exactly about a shared carrier $\omega_c$ and half-splitting $\sigma$, and for $|\sigma t|\ll 1$ the time-domain projection is systematically a carrier plus a first-jet piece $\propto t\,e^{-\ii\omega_c t}$, without requiring a literal double pole or exceptional-point merger in parameter space. The centered first-jet basis has $O(1)$ Gram conditioning, whereas the resolved-mode basis satisfies $\mathrm{cond}(G_{\mathrm{res}})\sim 12\,\eta^{-2}$ as $\eta\to 0$ (transparent real-splitting slice). We supply finite-window diagnostics in which $\kappa$ marks when the jet correction must be retained and $\eta^2$ sets the residual error scale once it is retained. Minimal two-pole numerics verify the scaling. For Kerr black holes we fix one adjacent-overtone mode pair (catalog label \texttt{pair45}; shared $(l,m)$ and consecutive overtones in our indexed tabulation), scan spin $a\in[0.8770,0.8810]$, and adopt the spectral window proxy $T_{\mathrm{spec}}=\beta/|\Im\omega_c|$ with $\beta=2.0$ to illustrate the same conditioning contrast in a near-degenerate sector.