Growing null subtraction and weighted Weil zero detection

Abstract

Exact zero detectors for the Weil form can have a small negative Rayleigh quotient in a reciprocal-theta norm even after subtraction of known null vectors. We quantify this effect when both the zero multiplicity and the derivative-null prefix grow with height. For the divided-theta detector pair, the optimal negative depth is at most exp(-pi T/4) when (m+N) log T <= c_* T; on the strict subscale its upper logarithmic rate is -pi/2. The proof pays factorial derivative growth, canceled primitive norms, coefficient extraction and the complete two-column Gram matrix. An explicit finite-remainder norm upper gives a complementary sufficient exclusion inequality. We also give a norm-costed map from compact negative Weil spaces into the original constrained realization. Applied to an existing Gaussian cluster construction, it supplies an amplitude-free weighted depth guarantee of rate -1/2 at fixed displacement. The two rates concern different detector families.