The hump between two close zeros: an exact small-gap law for random unitary matrices and a tilt ladder for 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), with U uniform and XN−2 the value of |ΛN−2(1)| under the |ΛN−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, XN−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, as a finite-height correction would. 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.