The hump between two close zeros: random unitary matrices and the Riemann zeta function

Let M be the maximum of |Λ_N|, the characteristic polynomial of a Haar unitary N×N matrix, on the arc between two consecutive eigenangles at distance s. For fixed N we prove that, conditionally on s<ε, the pair (s/ε, M/s²) converges in law as ε→0 to independent variables (U1/3, XN−2/4), where U is uniform and XN−2 is |ΛN−2(1)| under the |ΛN−2(1)|⁴-tilted Haar measure. Such gaps occur at rate N²(N²−1)ε³/(72π), and by the product formula of Bourgade, Hughes, Nikeghbali and Yor all moments and log-cumulants of the hump are explicit; in unfolded units the log-hump has mean → 2 log π + 2γ − 10/3 and a negative third cumulant → −0.1271. For the Riemann zeros we cast the analogue as a tilt ladder, conditioning on b = 0, 1, 2 zeros, and test it on zeros at heights L = log(t/2π) ≈ 9–22. On the first two rungs, log|ζ(1/2+it)| and log|ζ′(ρ)|, we derive the variance and the third cumulant at finite height from the ratios conjecture, using three- and four-point averages of ζ′/ζ and the density of zeros; the identity term reproduces the constant of Fazzari and Gerspach. Without free parameters the four predictions match the data at all seven heights (χ² between 2.5 and 11.6 for 7 points), while the random-matrix model with the Keating–Snaith arithmetic factor, which is right in the limit, fails at these heights (χ² up to 16 826). Pre-registered tests also show that midpoints of close pairs lock to the phases of the prime waves about four times as strongly as single zeros.