Collision-uniform kernel asymptotics for Báez–Duarte zero packets
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Abstract
We determine the high-height kernel of the harmonic-step space underlying the Báez–Duarte approximation problem. For packets with unequal real parts in a fixed compact rectangle inside the critical strip, an exact arithmetic gauge gives an explicit limiting Mellin kernel. Convergence holds with every fixed number of mixed derivatives. Complex Newton–Hermite divided differences then give a relative Gram comparison uniform through arbitrary collisions, without a minimum-separation condition. A literal finite-height error and an explicit separation-free Gram floor make the comparison effective. A regularized variational certificate also applies at every real height T≥4, without a small-error onset or an independence assumption on the native functionals. The resulting native zero-jet projection bound retains the exact unequal powers and oscillatory phases of the target. We derive a two-point interference formula and a multiple-zero lower bound with a logarithmic multiplicity factor. The proof works directly with the arithmetic step kernel and pays both ends of its infinite series.
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- Collision_uniform_arithmetic_packets.pdf140.1 KiB · pdf
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Báez–Duarte approximation, Mellin transform, reproducing kernels, divided differences, multiple zeros
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
For fixed-size zero packets in a compact rectangle inside the critical strip, the exact harmonic-step kernel has collision-uniform relative Newton–Hermite Gram asymptotics and explicit finite-height bounds. The regularized native variational bound holds for all real T≥4; the supplement proves separate whole-line and localized norm interfaces.
Disclosures
AI-assisted mathematical exploration, manuscript preparation and computational checks.
Reproduction
The 18-member package contains the 14-page main manuscript, its 7-page mathematical supplement, main.tex, supplement.tex, build.py, README.md, citation and checksum metadata, requirements.txt, rational Poisson inputs, verification.json and seven Python entry points. README.md gives Tectonic build instructions and verify_all.py replay commands with Python 3.12, SymPy 1.14.0 and python-flint 0.9.0. Six mathematical controls cover exact packet algebra, error constants, regularized variational bounds, Gram floors, complete Poisson tails and localization; verify_all.py runs them together. Infinite-series, collision-uniform and density results have analytic proofs. The supplied Poisson example lies within established verified-height coverage.