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 (U^{1/3}, X_{N−2}/4), with U uniform and X_{N−2} the value of |Λ_{N−2}(1)| under the |Λ_{N−2}(1)|⁴-tilted Haar measure; the mean number of such gaps is N²(N²−1)ε³/(72π) to leading order, and all moments of order 2k>−5 and all log-moments converge. Summed over all gaps, E Σ_n M_n² ~ ((e²−5)/2) N², by an identity that reduces it to a joint moment evaluated by Winn. By the product formula of Bourgade, Hughes, Nikeghbali and Yor, X_{N−2} is a product of independent factors, so moments and log-cumulants are explicit: in unfolded units the log-hump has mean → 2 log π + 2γ − 10/3, variance ½ log N + O(1) and a negative third cumulant (→ −0.1271), and is asymptotically normal. For the Riemann zeros we cast the analogue as a tilt ladder (conditioning on b = 0, 1, 2 zeros) and report pre-registered tests at L = log(t/2π) ≈ 9–22: the arithmetic shift of the log-variance is close to the Keating–Snaith value −0.088 on every rung; the shift of the third cumulant falls from +0.28 to +0.15 and +0.086 at b = 2 (predicted blind within 1.6σ) and flattens with height. On the lowest rung the finite-height values are derived: from the ratios conjecture we obtain the second and third cumulants of log|ζ| at finite height, including a three-point average of ζ′/ζ whose identity term reproduces the constant of Fazzari and Gerspach; without free parameters they match the data at all seven heights (χ² = 6.4 and 2.5 for 7 points). Through the density of zeros the same averages give the variance of log|ζ′(ρ)| on the second rung (χ² = 10.9 for 7 points, against 1780 for the a_k model). In a further test, single zeros follow Landau's formula for the phases of the prime waves to four decimals, and the midpoints of close pairs lock to these phases 3.87 ± 0.02 times as strongly, close to the factor 4 of a local-GUE heuristic.