Frequency-Source Resolvent Limits and Positive-Receiver Instability
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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.
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Available editions
- Edition 1 · 2026-10-27 · public
Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Weil form, resolvent, theta function, Hardy space, Cauchy matrix, constrained inverse, frequency source
- Manuscript date
- 2026-10-27
- Reader reuse
- No additional permission granted
Scope
Exact fixed-frequency inverse-Gram limits after fixed known rows; rank-one frequency-zero singularity; all-direction complete positive-receiver divergence on the same source; fixed-family derivative leading matrix with exact outer-Hardy/Cauchy formula, inverse, determinant and exponential conditioning bounds; certified eight-column limit matrices and fixed-row Blaschke prices.
Disclosures
AI assistants supported derivation, literature comparison, manuscript preparation and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 33-page paper is accompanied by complete editable LaTeX sources, five portable programs and exact mathematical data in a 31-member publication package. The programs check the derivative estimates, endpoint and admission controls, eight-feature approximation, directed outer-factor integrals, all entries of the eight-by-eight limit matrix and its inverse, and fixed-row identities. Normal and optimized outputs agree. The manuscript supplies the complete analytic proofs and all-frequency tail bounds.