Deterministic Trust Regions for Finite-Window Black-Hole Spectroscopy in GW250114

We study finite-window black-hole spectroscopy in the loud-event regime and ask when a multimode ringdown fit supports a stable common-remnant Kerr interpretation. Starting from whitened, tapered detector-frame data, we prove a deterministic frequency-extraction theorem for a projected sampled Prony--matrix-pencil pipeline with explicit statistical, algorithmic, omitted-tail, and mismatch terms. We then construct a local inverse atlas for the Kerr $(\ell,m,n)=(2,2,0)$ map on an event-local detector-frame remnant box for GW250114 and propagate the resulting primary uncertainty into $(2,2,1)$ and $(4,4,0)$ consistency tests. These ingredients yield a detector-frame trust criterion for individual windows. We calibrate mismatch and colored-noise radii on a GW250114-like synthetic waveform bank built from public surrogate, CCE, and numerical-relativity information, and we apply the resulting bounds to the public H1/L1 strain and public parameter-estimation products for GW250114. The accepted windows form an intermediate post-peak band: earlier windows remain sensitive to start-time drift and structured nuisance fits, whereas later windows become variance dominated. Within that band, the recovered remnant remains consistent with the public inspiral--merger--ringdown estimates and supports a common-remnant Kerr interpretation that survives the full preprocessing and robustness checks. For loud events, the relevant question is therefore which finite detector-frame windows sustain spectroscopy, not whether some multimode fit can be made in isolation.