Weighted polynomial localization and theta correction inertia
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Abstract
Let F(z)=ξ((1+iz)/2)/4, G=F'^2-FF'', and let D_0=E_0-Q_0 be the original endpoint correction, where E_0(P)=(4π)^(-1)∫ |(FP)'|^2 and Q_0(P)=(8π)^(-1)∫ GP^2. For every fixed 0<θ<1 and A>0, every sufficiently large degree N, and each polynomial parity, we construct a real subspace with a rational basis, dimension at least c_(A,θ)(log N)^A(log log N)^2, and Q_0≥(2-θ)E_0 on its entire span. The proof combines classical unconditional short-interval zero moments with bounded-population microscopic seed selection, a two-scale root-product budget, seed-averaged logarithmic root energy and higher fixed-order weighted approximation in a derivative and absolute-potential norm. All local complex zeros, multiplicities, exterior roots and mixed polynomial Gram entries are retained. Companion spaces have 0<Q_0/E_0≤θ at the same dimension scale, yielding the corresponding necessary repair-rank bounds for scalar-centered fixed strict contractions. A finite Gaussian zero-ideal approximation theorem and an exact generic current certificate delineate the scope of the method. The constructed correction-negative spaces have positive original Q_0.
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Spectral Theory
- Keywords
- Riemann theta form, correction inertia, weighted polynomial approximation, logarithmic root energy, rank obstruction
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Rational polynomial spaces in each parity with correction-negative and original-square-positive full spans of dimension at least a fixed log-degree power times squared log-log degree, with corresponding scoped repair-rank consequences.
Disclosures
AI assistants were used in mathematical development, checking, source comparison, code and manuscript preparation. This preprint does not claim external human refereeing or formal proof-assistant verification.
Reproduction
25-page manuscript and 15 source-package members. Use build.py and Tectonic to compile main.tex and its included sections. Standard-library support_verify_gaussian.py checks the finite Gaussian controls. Install support_requirements.txt for support_verify_current.py and support_verify_current_independent.py, which independently check the generic interval-current certificate.