Three-point correlation of the Riemann zeros at finite height: the ratios conjecture bin by bin on 147 million zeros up to t = 3e10, the amplitude of the deviation from GUE, and smaller sets further up - a reproducible numerical study IV

Fourth study of the series (study I: doi:10.5281/zenodo.23815328; study II: doi:10.5281/zenodo.23928017; study III: doi:10.5281/zenodo.23991165). It measures the three-point correlation R3 of the zeros of the Riemann zeta function at finite height and compares it bin by bin with the formula that Conrey and Snaith (2008) derived from the ratios conjecture, which contains all lower-order terms down to the constant term. Data: 147.2 million zeros computed by D. Platt (LMFDB) in 14 blocks from the first zero to t = 3.06e10, the error of every bin estimated from 40 consecutive pieces of each block, and three sets of 1e4 zeros of A. Odlyzko near t = 2.7e11, 1.4e20 and 1.4e21. (i) The differences that Conrey and Snaith reported for the first 1e5 zeros (standard deviation about 0.03, maximum 0.164) have the size of the counting noise of the triples (0.027 against 0.0265 +- 0.0007 for pure noise); with the error of every bin the formula agrees, chi2 per bin 1.050 against the expected 1.054. (ii) In all 14 blocks the formula agrees bin by bin (chi2 per bin 0.905-1.145, mean 1.038, 784 bins each), while GUE gives 6.4-117, the formula without its arithmetic factors 5.0-90, a model built from pairs alone 2.7-37 and unitary matrices of an effective size 27-446; at t = 3.06e10, R3 in the bin (0.5-0.6)^2 is still about 31 standard errors below GUE and within 1.6 errors of the formula. (iii) Written as one amplitude, R_data - R_GUE = lambda (R_ratios - R_GUE), the deviation from GUE gives lambda3 = 0.993-1.010 in all blocks with errors of 0.2-0.8 %, no trend with height; inside the lowest blocks the amplitude of consecutive pieces falls as the formula averaged over the heights of each piece predicts. (iv) Following S. J. Miller, smaller sets much further up: Odlyzko's three sets give lambda3 = 0.62 +- 0.38 together, consistent with both the formula and GUE; about 1.9e5 zeros near 1e21 would be needed for an error of 0.2. As an addendum to study III, quadratic Dirichlet L-functions in three bands of width 15,000 at |D| = 1e12 (27,355 L-functions computed for this study with PARI/GP, SHA-256 of every block) follow the ratios conjecture (chi2 per bin 0.87-1.21) and lie 15.4 standard errors above the Katz-Sarnak limit USp near the central point (combined density in [0, 1/2]: 0.5040 +- 0.0061 against 0.5054 from the ratios conjecture and 0.4105 for USp). Over eight decades of the conductor the excess over USp falls only as 1/ln(|D|/pi), as the ratios conjecture predicts - a slowness of the kind M. Watkins found for cubic twists; the data alone do not fix the limit, and the ratios conjecture brings the excess below 0.05 only near |D| ~ 3.6e21. The band statistics of consecutive discriminants at 1e12 against control stretches at 1e10 remain undecided under a rule fixed before the last band was computed. All numbers are checked by null models of the pipeline, a sensitivity analysis and an independent recomputation that shares no code with the main one. Agreement is numerical support for a conjecture, not a proof of it; nothing here bears on the Riemann hypothesis. Article in English and Czech (PDF, DOCX, LaTeX, Markdown), figures, data and the scripts that reproduce every number; the zeros are downloaded from the LMFDB and Odlyzko's pages and checked by their md5 sums, and the zeros of the L-functions are regenerated by the generator of study III and checked against the SHA-256 of every block. Text, figures and data CC BY 4.0, code MIT.