A second harmonic at a fixed beat period reduces forecast errors for 19 TOI-2525 transits in 2023–2025

We compare timing curves for the two known TOI-2525 planets. We use 32 historical times and an existing verified SCI dataset of 19 later times. Weighted fits use only the historical data from 2018–2022. The beat period is fixed at 427.8211 days from published orbital periods. With all historical times, two harmonics give smaller later errors than one harmonic. On 13 later b times, the unweighted root mean square error falls from 31.85 to 18.61 minutes. On six later c times, it falls from 12.39 to 6.33 minutes. The later times cover 2023–2025. A single-wave period search from 250 to 650 days uses early data only. Its later errors are 61.46 minutes for b and 15.74 minutes for c. Alternative later flux fits retain the two-harmonic gain. However, removing one historical time reverses the comparison in one of 21 b fits and one of 11 c fits. Two harmonics also increase error for the three 2023 c events. A third harmonic gives further descriptive improvement. These results apply to the complete early sample and accepted later events. The source search for events uses a two-planet prediction. The later data were visible during study design. This is an exploratory benchmark, not a blind forecast. It supplies no new transit measurements or physical orbit solution. The package includes source hashes, code, event forecasts, sensitivity results, figures, and offline repeat instructions.

TOI-2525 · TIC 149601126 · transit timing variations · harmonics · transit prediction · TESS · reproducible benchmark