Papers

Predictive claims across several fields are validated by reporting the fraction of held-out points falling within a factor of T of the prediction. That statistic has a null model which is almost never reported: a CONSTANT predictor ignoring the inputs entirely. We give the null in closed form. If log10 of the held-out target has standard deviation s, the constant's absolute log error is half-normal, so its expected pass fraction is p_null(T,s) = 2*Phi(log10(T)/s) - 1. Monte Carlo over 21 (T,s) cells reproduces this to a maximum absolute error of 0.0007 against a 0.0020 tolerance derived from the Monte Carlo standard error rather than chosen. Two usable outputs follow. First, a requirement table: at tolerance factor 2, a constant scores at or above 0.90 unless the held-out target spans more than 0.183 dex, and at or above 0.60 unless it spans more than 0.358 dex. A study whose held-out target is narrower than that cannot distinguish its law from a constant however good the law is, and the pass fraction it reports is uninformative rather than merely weak. Second, a sample-size table: separating a law that is right 95% of the time from its constant baseline requires 191 held-out rows when the target spans 0.20 dex, and 33 when it spans 0.30 dex. Corpora in this area typically hold tens. Applied to a published round that reported 24/25 = 0.960 inside a factor of two against a committed bar of 0.60, the constant scored 21/25 = 0.840 on the same rows, implying a held-out spread of 0.214 dex - 1.67x too narrow for the 0.60 bar to be falsifiable, and 3.9x too few rows to separate the two figures. We also report a methodological incident: the derivation's first validation failed by 25 sigma because of a floating-point defect in a linear congruential generator, and was caught only because the acceptance threshold had been derived from the standard error instead of set to a round number.