Uniform Resolvent Asymptotics for Growing Theta-Null Source Families

Abstract

Inverse Gram limits for a fixed collection of sources can lose their meaning when the collection grows and its leading matrix becomes ill-conditioned. We prove a relative matrix asymptotic for growing derivative-null source families in a reciprocal-theta realization of the Weil form. With U = log(1/ε), the actual constrained inverse Gram approaches its explicit outer-Hardy/Cauchy leading matrix uniformly when the source count is o(log U); a fixed small multiple of log U is also admitted eventually. The estimates control every complex source combination in the full leading metric. The proof uses a positive inverse-factor kernel with a support gap, logarithmic plateau leverage, exact polar recovery and complete dual-leakage bounds. A separate displaced-wall argument gives the finite bound 2²³ n I whenever U ≥ 64n², and yields an explicit original-metric null-elimination estimate. A same-span perturbation construction shows how individually small source errors can produce an exponentially amplified relative Gram defect while preserving positivity, independence and the polar constraint. Together these results identify both a uniform limiting regime and the source control needed to extend it.