The smallest eigenvalue of windowed Weil forms decays like e^{-4πμ} under RH and a local pair hypothesis

Let λ_min(λ) be the smallest eigenvalue of Weil's quadratic form on test functions supported in the multiplicative window [λ⁻¹, λ], and put μ = λ². In previous papers we proved unconditionally that λ_min ≤ 1.24×10²³ μ⁵ (log μ)⁵ e^{−4πμ} for μ ≥ 50, and, under the Riemann Hypothesis (RH) and a hypothesis (LP) on close pairs of zeros, that −log λ_min ≤ Cμ for large μ. Here we show that the exponent 4π is sharp under RH and Hypothesis (LP′): lim_{μ→∞} (−log λ_min(√μ))/μ = 4π. (LP′) requires that, for every interval J and every small h > 0, the number of pairs of zeros in J at distance less than h be at most a constant times h|J|L_J² + L_J², where L_J is the logarithm of the maximal height in J; this is (LP) without the term |J|, hence stronger. Under RH and (LP) alone we obtain limsup ≤ 4π e^{2πC_P}, with C_P the constant of (LP). The proof bounds λ_min from below by a sampling constant of the zeros, adds finitely many virtual sampling points where the zeros are too sparse, and removes them with entire functions (multipliers) whose zeros follow the optimal density of an extremal problem for logarithmic potentials. We show that its value is exactly 2π: the extremal measure is explicit through the Joukowski map, and the matching lower bound comes from the harmonic measure of the plane slit along (−∞, −2] ∪ [2, ∞). Under these hypotheses, the result proves the exponent 4π in Connes's numerical observation that λ_min decays like the prolate quantity 1 − χ₂, and shows that Zhu's conjecture −ln λ* ∼ 2π²μ, for a closely related quantity λ*, would fail. Notes: conditional on RH and (LP′) (Theorem 2 on RH and (LP)); not formalized; not reviewed by independent human experts. Written with extensive use of generative AI tools under the author's direction; see the section "Use of generative AI" in the paper.