The centered cross-product expectation is more than zero at all 532 ds000030 starts in a 16-trial model with independent responses of probability 0.5.

In a 16-trial staircase model with independent Bernoulli responses of probability 0.5, exact rational calculations found a positive expected centered delay–response cross-product at all 532 starting delays observed in OpenNeuro ds000030 (range 93.75–187.5 ms). The known-probability score had expectation zero. Two sets of 10,000 seeded replays reproduced the selected publication’s numerical results, and the six observed-field slopes differed by less than 1e-12 per 100 ms. These calculations show that adaptive updates and sample centering can produce a positive descriptive slope under the stated synthetic model. They do not establish a biological delay effect: the model omits reaction times, learning, serial behavior changes, training, and delivered cue times, and no stopping-time estimate or biological hypothesis test was calculated.

adaptive staircase · centered cross-product · stop-signal task · synthetic model · exact expectation · Monte Carlo · response probability · OpenNeuro ds000030