Weighted polynomial localization and theta correction inertia
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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.
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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
Original endpoint y=0 only. For every fixed A>0 and theta in (0,1), every sufficiently large degree and each parity; constants and onset depend on these fixed parameters. Both channel families have strictly positive original Q0.
Disclosures
AI assistants were used in mathematical development, checking, source comparison, code and manuscript preparation.
Reproduction
The 25-page manuscript and source package include complete analytic proofs, a second first-order weighted approximation certificate, a finite Gaussian ideal model, an exact rational degree-80 generic current witness and two independent ball-arithmetic implementations. Python 3.11+; tested python-flint 0.9.0. Build instructions and complete SHA-256 member manifest are included.