Quantitative Null-Space Compression for a Weighted Weil Operator
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Abstract
We study a fixed minimal weighted realization H = A + B of the constrained Weil form, with nonnegative logarithmic reference A and the complete bounded paired arithmetic perturbation B. For every 0 < τ ≤ 1, explicit Fourier-band moments cut out a finite-codimensional subspace of the entire operator kernel on which both A and A^(1/2) have norm at most τ. The number n(τ) of observations satisfies log n(τ) = O(√(1 + log(1/τ)) log(3 + log(1/τ))). Consequently AΠ_Y and A^(1/2)Π_Y belong to every Schatten class for every closed Y ⊆ ker H. The proof combines two complete phase-space tails, a finite-rank real-phase Taylor approximation, and the exact null equation. We also give norm estimates for finite null deflation, a full residual-Gram receiving certificate, an explicit weighted bound H ≥ −99I/100, and a multiplicity obstruction to identifying the derivative-null closure with the whole kernel. The observations are explicit; the projection inside the unknown kernel is not computationally constructed.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Functional Analysis
- Keywords
- Weil quadratic form, null space, Schatten classes, finite-rank approximation, resolvent residuals, spectral theory
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
A fixed minimal weighted Weil realization with the complete paired arithmetic perturbation; quantitative entire-kernel observation and reference-output bounds, norm deflation and a conditional signed finite-matrix receiver.
Disclosures
AI systems assisted with mathematical exploration, literature comparison, proof checking and manuscript preparation. The author takes responsibility for the statements, proofs and citations.
Reproduction
The supplement contains all editable LaTeX sections, a portable Tectonic build helper and three standard-library Python controls. Run python build.py with Tectonic, or compile main.tex with pdfLaTeX twice. Run verify_lower.py, verify_gram.py and verify_compression.py normally and with python -O. These check exact finite arithmetic and residual examples and supplement the analytic proofs.