Beyond logarithmic powers: critical-tail localization of theta correction inertia
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Abstract
For the Riemann theta transform F(z)=xi((1+iz)/2)/4, let E(P)=(4π)⁻¹∫|(FP)′|², Q(P)=(8π)⁻¹∫(F′²−FF″)P² and D=E−Q. We obtain ordinary-polynomial subspaces, in either parity, with opposite signs of D and strictly positive original Q on their entire nonzero spans. Their dimensions are at least C exp(c√(log N))√(log N) for every sufficiently large degree N. A critical Meixner–Pollaczek receiver controls the derivative and absolute-potential norm globally. A simultaneous inverse-square window selection and a relative root-product budget improve the cost of localized targets. A finite-order compact-cutoff family with uniform low-order energies and a fully controlled growing differentiation order gives the stated scale. All local complex roots, multiplicities, exterior roots and mixed Gram entries are retained. The same growth is necessary for repair rank in a specified scalar-centered strict-contraction scheme.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions · Complex Analysis · Spectral Theory
- Keywords
- Riemann theta form, correction inertia, critical exponential weight, Meixner-Pollaczek approximation, finite-order cutoffs
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
For every fixed 0<θ<1 and all sufficiently large ordinary degree N, each parity has two rational-basis real polynomial spaces of dimension at least C exp(c√(log N))√(log N), with Q/E≥2−θ and separately 0<Q/E≤θ on their entire nonzero spans. The rank consequence applies only to the specified scalar-centered uniform strict-contraction architecture.
Disclosures
Research preprint. AI tools assisted with mathematical exploration, drafting, symbolic and exact-arithmetic checks, source comparison and internal proof review. Scott Schubr is the author and assumes responsibility for the manuscript.
Reproduction
The supplementary package contains the manuscript, complete editable LaTeX sources, build instructions, citation metadata, member checksums and a standard-library Python verifier. Normal and optimized runs each perform 1169 exact finite checks corroborating the bounded algebra; the general results have analytic proofs.