Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificates
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Abstract
We study local Nyman–Beurling projection gain in the whole-line Poisson norm. Mixed-order Bernoulli covariances retain all complex phases and diagonalize in divisor coordinates, with a positive cotangent form as their height-scale limit. Building on classical Bernoulli covariance and Farey large-sieve estimates, we control the varying primitive amplitudes and the exact original weight. The resulting complete-tail comparison is uniform for support cutoffs N=o(T) at every fixed pole-free wall gap K∼γ NT, γ>1/(2π). Combining this comparison with a compensated mean gives upper and lower bounds for the actual optimized double-null gain through a corrected finite model, together with a uniform additive error. On every fixed finite geometric module, optimization over arbitrary degrees and coefficients yields gain below each positive threshold on a positive-lower-density set of heights. An evaluation defect quantifies the escape required of successful trial families. Exact evaluation duals, sharp fixed-trial tail capacity and robust constrained certificates complement these results. Full-head examples and Hardy-space continuation turn successful finite certificates into zero-free disks and certified height intervals. The finite model keeps the shared arithmetic head, both coefficient constraints and the complete mean and norm errors in a single coefficient space.
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Nyman–Beurling, Dirichlet polynomials, Bernoulli polynomials, mixed covariance, cotangent sums, gcd matrices, Poisson norms, Möbius inversion
- Manuscript date
- 2026-10-27
- Reader reuse
- No additional permission granted
Scope
Complete complex-q mixed-order arithmetic receivers; pole-free uniform N=o(T) cotangent/Farey return; two-sided and additive corrected optimized-gain comparison; degree-uniform fixed-geometric-module obstruction with positive-lower-density actual-height failure and paid evaluation escape; retained complete finite native applications.
Disclosures
Author-posted preprint. OpenAI ChatGPT and Codex assisted mathematical exploration, derivation, literature comparison, manuscript preparation and internal proof and code checks. The author is responsible for all content and owns Luminary Oddities and the Research Archive. The author declares no other competing interests. This research received no external funding.
Reproduction
The 35-page manuscript and 37-member package provide complete editable sources, eleven verification programs and numerical certificate data. Normal and optimized executions agree in their mathematical outputs. Python 3.12.14 and python-flint 0.9.0 were tested. The manuscript supplies the analytic complete-tail and all-parameter proofs.