Critical-tail localization and growing theta correction inertia
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Abstract
For the Riemann theta transform F(z)=ξ((1+iz)/2)/4, let E(P)=(4π)^(-1)∫|(FP)'|^2, Q(P)=(8π)^(-1)∫(F'^2-FF'')P^2 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 Cexp(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
Critical-tail joint-norm polynomial localization producing positive-original-form channels of both correction signs with square-root-logarithmic dimension growth in each parity.
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. Internal AI review is not external peer review.
Reproduction
14-page manuscript and 13 package members. Run python build.py --tectonic tectonic, or compile main.tex with its five included TeX components. Run python verify_weighted.py and python -O verify_weighted.py using the Python 3 standard library; each reports 1169 exact checks of the MP Gram, derivative, weighted-norm, pole-bin and convolution identities.