A Zero-to-Zero Runner for the Riemann Zeta Function: the Truncated Explicit Formula Tested on Odlyzko's Tables to the nd Zero

A runner is described that steps from one non-trivial zero of the Riemann zeta function to the next. From the zero just found it steps forward one Riemann–von Mangoldt mean gap and solves the counting equation , where is the explicit formula for truncated at , either with a sharp cut-off (the speed road) or with a cosine taper (the temperature road). On 1,331,550 zeros near the runner lands a mean 0.0445 of a gap from each true zero, against 0.3282 for the density law alone, and places 99.558% within 0.1 in height. On Odlyzko's published blocks of 9,996 zeros near the th, st and nd zero it places 98.19%, 98.95% and 99.17% within 0.1 with the raw truncated sum, and 99.52%, 97.33% and 97.36% with the height-dependent correction. The lead over the density law narrows at the greatest heights, where the fixed cut-off falls behind the scale. Read together, the two roads place 96.5% of zeros within 0.01 in height, against 74.5% for a scrambled control. Further controls are random-matrix surrogates and the Davenport–Heilbronn function, which has zeros off its critical line. The work is set out as a proof structure of twelve steps, each labelled standard, measured, picture or open, and one step is named as open: showing that no zero lies off the 0.5 line. This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.