The rigidity of Weil's quadratic form on short support: an equilibrium law and the simplicity of the zeros of ζ

For X > 1 let μ_X be the smallest eigenvalue of Weil’s explicit-formula quadratic form restricted to even test functions supported in [−½ ln X, ½ ln X]. We determine its exponential rate. Assuming the Riemann Hypothesis, μ_X ≤ exp(−4πX + O(log² X)) (Equilibrium bound). The constant 4π is the value of an explicit equilibrium problem: the test function follows the zeros up to height 4πX and, above it, vanishes on a sweeping (balayage) measure, whose potential equals −2πX on the whole free region (Sweeping identity). Conversely, a lower bound ln μ_X ≥ −4πX − o(X), even only on average, forces N(T) − N_d(T) = o(T), where N_d counts distinct ordinates (Repetition theorem). This is more than the pair correlation conjecture gives in its usual form. Under RH and regularity hypotheses on the zeros, the matching lower bound holds (Dual lower bound), so the rigidity law ln μ_X ∼ −4πX is equivalent to the simplicity of almost all zeros (Rigidity–simplicity equivalence). The proof uses exact Lagrange interpolation in finite-dimensional layers and a window-cancellation estimate for logarithmic potentials. Numerically, the rate is 5.453–5.479 decades per unit of X, against 4π/ln 10 = 5.4575. We also state a conjectural deficit law for clusters of repeated zeros, supported by computations. None of the results proves the Riemann Hypothesis.