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
Explicit Fourier-band observations control the logarithmic reference and its square root on the entire kernel of the specified minimal weighted Weil operator with subpolynomial inverse-tolerance rank and all-Schatten reference-output maps. Finite deflation, complete residual-Gram receiving bounds and the stated weighted lower and multiplicity conclusions follow.
Disclosures
AI systems assisted with mathematical exploration, literature comparison, proof checking and manuscript preparation. These checks are not external human refereeing or formal verification of this manuscript. The author takes responsibility for its statements, proofs, citations and limitations.
Reproduction
The 15-page manuscript and nine TeX sources are supplied with build.py, citation metadata, README.md, MANIFEST.sha256 and three standard-library controls in this 17-member package. README.md gives Tectonic/Python commands. verify_lower.py checks rational lower-bound constants; verify_gram.py checks finite signed-Gram/residual identities; verify_compression.py checks representative parameter arithmetic without constructing the enormous observation matrix. The universal operator arguments are analytic proofs.