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Abstract

We classify inverse energies of fixed frequency sources in a reciprocal-theta realization of the Weil form and compare two positive receivers on identical constrained spaces. Regular profiles have an exact archimedean inverse-Gram limit given by their full Fourier-dual matrix, unchanged by any fixed finite collection of known derivative-null and polar rows. A nonzero frequency-zero trace produces exactly one divergent direction, of order log log(1/ε) on the original polar space. For the complete positive part of every fixed arithmetic cutoff, every direction of an independent negative-band source family instead outgrows every fixed power of log log(1/κ), even after those known rows are removed. We give a complete eight-feature application at cutoff 13. For comparison with derivative sources, we identify their fixed-family normalized leading matrix by an exact outer-Hardy and Cauchy formula and sharpen the admitted growing-family range to an explicit onset asymptotic to 4n log n. The proofs preserve complete theta tails, mixed covariances and the original minimal form domains.