Source-Resolved Resolvent Growth under Reciprocal-Theta Confinement
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Abstract
We prove two-sided bounds for the complete inverse-energy Gram of prescribed theta-derivative sources in the original polar-constrained archimedean realization of the Weil form. For every finite source size d + 1, diagonal normalization by explicit powers of U = log(1/ε) gives positive degree-dependent lower and upper bounds whenever U ≥ 64(d + 2)². For each fixed family the normalized Gram is bounded above and below by constants, and source i has exact logarithmic exponent 4i + 13/2. The upper estimate uses exact half-line polar cancellation and a uniformly priced Fourier primitive, retaining the complete exterior polar tail. The lower estimate combines an exact polynomial coefficient map, all-atom theta bounds and a common-weight shell covariance with exact polar shorting. A two-metric residual theorem receives non-square-integrable divided sources, and explicit finite receiving bounds accompany the analytic estimates.
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Available editions
- Edition 1 · 2026-10-27 · public
Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- resolvent, theta functions, spectral theory, constrained quadratic forms
- Manuscript date
- 2026-10-27
- Reader reuse
- No additional permission granted
Scope
Complete constrained source-Gram bounds for all d ≥ 0 and U ≥ 64(d + 2)², constant normalized fixed-family bounds and exact logarithmic exponents, with two-metric residual and finite polar receiving estimates.
Disclosures
AI assistance was used in mathematical development, manuscript preparation and verification-code review. The author is responsible for the statements and presentation.
Reproduction
The publication package includes editable LaTeX, a portable build script and Python finite-control code with python-flint 0.9.0. Directed 512-bit scalar enclosures and exact rational joint/shorted moment checks accompany the analytic proofs. See README.md and REPRODUCING.md.
References and related resources
- Quantitative Null-Space Compression for a Weighted Weil Operator, edition 2
- Observation norms under double-exponential confinement, edition 2
- Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators, edition 3
- Suzuki, Spectral convergence associated with Weil's quadratic form
- NIST DLMF, Recurrence Relations and Derivatives
- Johansson, Arb interval arithmetic