Growing null subtraction and weighted Weil zero detection
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Abstract
Exact zero detectors for the Weil form can have a small negative Rayleigh quotient in a reciprocal-theta norm even after subtraction of known null vectors. We quantify this effect when both the zero multiplicity and the derivative-null prefix grow with height. For the divided-theta detector pair, the optimal negative depth is at most e^(−πT/4) when (m+N) log T ≤ c_* T; on the strict subscale its upper logarithmic rate is −π/2. The proof bounds factorial derivative growth, canceled primitive norms, coefficient extraction and the complete two-column Gram matrix. An explicit finite-remainder norm upper gives a complementary sufficient exclusion inequality. We also give a norm-costed map from compact negative Weil spaces into the original constrained realization. Applied to an existing Gaussian cluster construction, it supplies an amplitude-free weighted depth guarantee of rate −1/2 at fixed displacement. The two rates concern different detector families.
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Research details
- Subjects
- Mathematics · Number Theory · Functional Analysis · Spectral Theory
- Keywords
- Weil quadratic form, Riemann xi function, weighted zero detection, null-space subtraction, factorial derivative bounds, negative spectral subspaces
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
Uniform bounds for the specified divided-theta pair hold when zero multiplicity and derivative-null prefix grow within (m+N) log T≤c_*T; a complete finite-norm exclusion inequality and norm-costed compact-space transfer give separate detector-family depth estimates.
Disclosures
AI assistants supported derivation, literature comparison, manuscript preparation and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 11-page manuscript is supplied with main.tex, build.py, CITATION.cff, README.md, MANIFEST.sha256 and verify.py in a 7-member package. README.md gives Tectonic build instructions and Python 3.8+ standard-library commands for 7,350 finite algebra controls, including optimized Python. The growing-parameter estimates and compact-space transfer are analytic proofs; the Gaussian cluster application uses its cited external theorem.