The crossover of parametrized Schwarzschild quasinormal-mode overtones: a one-parameter local connection problem, tested numerically

Miyachi, Namba, Omiya and Oshita (arXiv:2512.18631) showed with exact WKB that parametrized corrections δV = f Σj αj r−j to the Regge–Wheeler potential change the high-overtone asymptotics of Schwarzschild quasinormal modes. For j = 4, Re ωN grows as N1/5 once α42|ω| ≫ 1, and the Schwarzschild value ln3/4π is recovered only in the opposite limit. Their published Leaver spectra for α4 = −1 also show two sharp dips of Re ω towards zero, near N ≈ 357 and N ≈ 1.02×104, which the N1/5 law does not describe. We give a leading-order description of the whole crossover. Near r = 0 and at large |ω|, the problem reduces to a local equation with the single parameter κp = αp ω(p−3)/2, and a Motl–Neitzke monodromy argument gives e4πω = C(κ), where C is a connection coefficient computed numerically. C(0) = −3 recovers Schwarzschild. For α4 < 0, |C| dips repeatedly and the N1/5 law is only an envelope. For α4 = −1, |C| < 1 for N ∈ (347.3, 370.7); the full spectrum has no genuine complex root for 357 < n < 375. |C| nearly touches 1 again at N ≈ 10647.4; the full spectrum has a matching near-touch at n ≈ 10182, 4.4% earlier. These fixed-α comparisons are exploratory. We tested the quantization condition against Leaver spectra following an internal written protocol whose timestamps are self-recorded. On a grid of 162 resolved modes up to N = 3136, residuals at fixed y = α4(N+½)1/2 fall as N−1/2 for both signs of α4, but at the largest N they are still 7–28% for α4 < 0, so the protocol's 1% criterion fails and that hypothesis is not confirmed (a post-hoc extrapolation brings them to within 1.6% of zero). For α4 > 0, tracked modes reach the imaginary axis at a point that extrapolates to y = 3.79–3.83, depending on a fit window chosen after review, against the predicted 3.7936. For j = 5 the registered exponent test is met (e = 0.321 ± 0.090 against 1/3), with an uncertainty three times its tolerance. Our original fixed-α roots all coincide with the published ones; near the dips our root search also finds genuine roots the published data omit. Note on versions. Version 2 supersedes version 1, which stays available. Version 1 did not state that Miyachi et al. had already published both α4 = −1 dips, and presented the second as an out-of-sample prediction; version 2 credits the dips and the N* ∝ α4−2 scaling to them and credits axis crossings at a crossover to Daghigh et al. Validation 3 is now done against their public data (Zenodo record 20020134), and fixed-α series are relabelled by overtone counting. Both dips are re-analysed (the second dip of C had been sampled too coarsely). Protocol outcomes are stated by the rules as written (H1 for α4 > 0 not confirmed; H2 not confirmed; H3 confirmed by its rule; precision doubling partial), and "pre-registered" is replaced by "internal protocol". Full list in the paper's Note on versions. Files: paper (PDF) and source bundle (code, data, internal protocol and amendment log, README and build script). Computations and drafting were assisted by an AI system (Claude).