A contraction diagnostic on a next-zero iterator recovers the Berry–Keating cut-off from the prime weights alone

A next-zero predictor for the Riemann zeta function may be built by solving the Riemann–von Mangoldt counting equation with the Selberg prime sum truncated at a cut-off . The natural iteration divides the residual by alone, making it a chord method whose fixed-point multiplier at the iterated point is , with the local mean gap. Its root-mean-square proxy is computable from the tone weight table alone, before any zero is evaluated, and is linear in with . Setting therefore returns . This is the typical-case scale, fixed from the weights alone; the uniform pointwise bound is the step above it and stands open. Measured at 200 zeros per height, 30% of sites carry where , and 18% where . Solving numerically for the sharp cut-off at which recovers that scale to within one percent, monotonically improving with height: evaluated at gives 0.9736 at , 0.9891 at and 0.9943 at . The scale is Berry and Keating's resolution threshold; the contribution here is the route, a contraction diagnostic on an iterator reaching the same wall a semiclassical argument gives, with no zeros involved at any stage. We also report, with its full statistics, the result of a criterion we proposed and tested — that a tapered window improves placement while . The data go the other way. A fixed cut-off table shows overlapping confidence intervals and no consistent threshold, and a fixed-height sweep separating from height reverses the ordering outright: the taper wins at low . Two prior errors of our own are corrected: an expression for that was quadratic where is linear, and an identification of the r.m.s. tone energy with , which differ by a window-dependent factor. 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.