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Abstract

We characterize the reach of complete finite lower certificates for the original minimal weighted Weil operator. The infimum of their certified negative allowances equals the operator's actual negative spectral depth, and a computable constrained core supplies rigorous two-sided approximations to its lower edge at every prescribed positive accuracy. The proof combines logarithmic form-core density with compact domination of a monotone frequency limit. Exact derivative-null vectors allow finite deflation without changing the original form or discarding mixed terms. An executed certificate with eight such vectors proves I+H≥0.999999978600004918685 I, with a retained positive potential on the null complement. All arithmetic, frequency, spatial, quadrature and finite-matrix errors are included; directed ball arithmetic verifies both signed matrices of order 4085. We also prove that, at fixed arithmetic cutoff and positive multiplication reference, boundedly many constraints cannot maintain the required cap as the regularization decreases, even with an arbitrary growing frequency band. The quantitative obstruction forces growing constraint dimension and identifies the continuation mechanism that the adaptive framework must control.