Weighted Hilbert inequality: Schur tests with power weights cannot work

For |sum_{r!=s} x_r conj(x_s)/(l_r - l_s)| <= K sum |x_r|^2/delta_r (Montgomery-Vaughan, delta_r = distance from l_r to its nearest neighbour), the eigenvector identity still works with weights. It reduces mu^2 to the norm of a nonnegative kernel with off-diagonal entries ~ (delta_s delta_t)^{1/2}(delta_s+delta_t)/(l_s-l_t)^2. Any Schur test with weights delta^alpha, even a two-weight version applied to each half of the kernel, fails. alpha < 1/2 diverges on a dense far cluster (n points with spacing eps at distance D). alpha = 1/2 diverges like the number of points on the geometric configuration l_j = 2^j. The kernel is bounded on both examples, so the obstruction is in the technique. A proof needs weights adapted to local density, or a different argument; do not spend time tuning exponents.