The best periodic box retains 29.3% of ideal signal-to-noise for 14 TOI-4504 c times with fixed cycles, equal eight-hour transits, and white noise

We test the loss of expected signal-to-noise when a periodic box template is applied to 14 published transit centers of TOI-4504 c. Cycle labels stay fixed. The model has equal eight-hour box transits, equal depths, a known baseline, and equal white noise in complete independent event records. Period, phase, and template width are free. We prove that a global maximum occurs in a finite set of box contact intersections. We check that set with exact integer arithmetic. The best periodic template retains 0.2931265613 of the expected signal-to-noise of a filter that knows all centers. Its width is 29.78912 hours and it overlaps eight of the 14 events. A fixed eight-hour template retains 0.2494194805. Allowing independent center shifts within three times each printed timing error plus rounding raises the best ratio to 0.2957859789. Changing the assumed transit duration from four to twelve hours gives ratios from 0.222270 to 0.340435. A separate seven-event test for planet d gives 0.797821. Source checks, exact witnesses, independent numerical checks, code, provenance, figures, and reproduction instructions are included. This is a controlled timing benchmark. It does not measure detection efficiency in TESS data or test a new planet.

TOI-4504 · transit timing variations · periodic transit search · box template · exact optimization · signal-to-noise · reproducible benchmark