Weighted polynomial localization and theta correction inertia

Overview

Abstract

Let F(z)=xi((1+iz)/2)/4, G=F′²-FF″, and let D₀=E₀-Q₀ be the original endpoint correction, where E₀(P)=(4π)⁻¹∫|(FP)′|² and Q₀(P)=(8π)⁻¹∫GP². 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 Q₀≥(2-θ)E₀ 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<Q₀/E₀≤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 Q₀; the all-order theta sign problem remains open.

Keywords Riemann theta form · correction inertia · weighted polynomial approximation · logarithmic root energy · rank obstruction