This is a pre-registration of the analysis protocol for searching arithmetic frequencies (related to ln p and ln pq) in the long-range deviation of the spectral rigidity Δ₃ (L) of Riemann zeta zeros from GUE universality at large heights T. The protocol fixes in advance: - Data provenance requirements and verification procedures (including independent T-range validation via the Riemann–von Mangoldt formula) - Analysis windows (W1, W2, W3) with pre-specified T-ranges- Range of L (2 to 40) - Statistical decision criteria (BIC thresholds, block bootstrap with N=2000, Holm–Bonferroni correction) - Pre-specified sensitivity analyses- Rules for reporting any deviations from the protocol **Research question: ** Does the spectral rigidity statistic Δ₃ (L) of Riemann zeta zeros at large heights T exhibit a periodic component compatible with the logarithms of primes or semiprimes, beyond what is explained by GUE universality? **Theoretical motivation: ** According to Berry–Keating and Bogomolny–Keating, the amplitude of arithmetic corrections to Δ₃ (L) is suppressed and becomes potentially detectable only at sufficiently large T. All analysis decisions were made prior to examining new data at T > 10⁶. Results will be reported regardless of outcome (including the case of no detection). This document is published with a DOI before any analysis on new data is performed. **Related resources: **- Project repository: https: //github. com/serhii-kanivets/riemannᵦetaDelta3 (to be updated with results) - Verification that Δ₃ (L) implementation passes Poisson (Δ₃ (L) ≈ L/15) and GOE/GUE stability tests is documented in the project.
Serhii Kanivets (Sun,) studied this question.