Explain the anomalous post-Newtonian resonance-duration spike

Identify the precise cause of the anomalous spike in the PN5-computed duration of the prograde $(4,0,-2)$ resonance at spin $a=0.9$ near eccentricity $e\approx0.35$.

Background

The paper compares resonance durations computed with the exact Kerr radiation-reaction module and the fifth-order post-Newtonian PN5 module in FEW. For the prograde (4,0,−2)(4,0,-2) resonance at high spin, PN5 produces an anomalous spike near e≈0.35e\approx0.35, which the authors attribute tentatively to a breakdown of the PN5 approximation near the separatrix but do not explain precisely.

References

The one striking exception is the $(4,0,-2)$ resonance at $a=0.9$ for prograde orbits, where the PN5 duration spikes near $e \approx 0.35$; we verified that the other $n=4$ resonances show a similar peak. We could not pin down its precise cause, but attribute it to the breakdown of the PN5 approximation at this combination of high spin and proximity to the separatrix.

— Not All Resonances Are Created Equal: Prioritizing Tidal Resonances in EMRIs  (2609.03732 - Bonga et al., 3 Sep 2026) in Section 3, Resonance durations