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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 e^(−πT/4) when (m+N) log T ≤ c_* T; on the strict subscale its upper logarithmic rate is −π/2. The proof bounds 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.