Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators
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Abstract
We obtain a joint finite-rank approximation law for the complete Birman–Schwinger family of a constrained theta-weighted Weil operator. At each fixed absolute error tolerance, the required rank is at most C exp(C√log(1/ε)) as the positive resolvent parameter ε tends to zero. The construction pays the complete paired arithmetic tail, high-frequency block and mixed localization errors. It gives a corresponding bound on the entire negative spectral projection and implies that the negative eigenvalue magnitudes decay as C exp[−c log²(n+1)]. Thus the full negative part belongs to every Schatten class. We also give effective simultaneous prime-phase returns and prove signed high-frequency saturation uniformly in a complementary growing-cutoff regime, retaining the exact polar constraints and the inverse-theta packet norm. The joint rank and saturation regimes have different parameter scales.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Analysis and Spectral Theory
- Keywords
- Weil operator, negative spectrum, Schatten classes, joint cutoffs, finite-rank approximation, theta kernel, prime-phase recurrence
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
The results use the specified two-row constrained minimal-form realization H=A+B. The joint approximation-rank law is at fixed absolute tolerance. Signed saturation is uniform under a prescribed envelope log M=o(log log log L), a different parameter regime.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.
Reproduction
The complete publication package includes the searchable manuscript PDF, self-contained LaTeX source, citation data, reader instructions, SHA-256 manifest and a standard-library Python verifier for exact finite scalar controls. The operator and asymptotic theorems are proved analytically in the manuscript.