Numerical Spectral Correspondence Between a Non-Autonomous Quadratic Map and the Riemann Zeros: An Exploratory Study

This study examines a non-autonomous quadratic map driven by a logarithmic cooling schedule (μn∼1/ln2n, a phenomenological ansatz), building on our recent published result that the logistic map’s symbolic dynamics at its band-merging point is isomorphic to the prime sieve. From its trajectories, we construct an empirical, non-normal, dissipative transfer matrix and compares its complex eigenphases to the non-trivial Riemann zeros after calibrating a few free parameters against the same low-order zeros—an in-sample numerical correspondence, not an independent prediction. We quantify this gap directly: fitting on the first M∈{50,70,80} zeros and evaluating on the rest gives a held-out MSE one to two orders of magnitude larger than the training error, with the fitted coupling drifting across M but remaining comparatively stable across ten random seeds at fixed M=70 (CV ≈4.7%). At low order (N≲20), an unselected re-computation shows a qualitative rank correlation (ρ=0.56, p=0.010) with, but no significant joint co-location (p=0.099) of, a residual feature reported in recent ion-trap quantum simulations of the same zeros. At larger N (N≥1000), the model matches a globally rescaled GUE surrogate’s mean counting-function trend better once conjugate eigenphases are restored, though this partly follows from the construction’s own symmetry; a separate, standard unfolded local-statistics test shows the model’s own eigenphase spacings do not match GUE, unlike the true zeros. These are numerical observations on a heuristic model, not a proof or Hilbert–Pólya-type operator construction; a dedicated table tabulates the epistemic status of every main claim.