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. The associated mathematical supplement gives complete height-adapted arithmetic and localization proofs, including whole-line tail payment, an explicit compact-band countermodel, the exterior-weight law and the prefix-Schur signal identity. Its localized criteria retain their own stated norms.
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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 each fixed packet size and compact complex real-part and height-offset range inside the critical strip, the exactly gauged arithmetic step kernel has an explicit mixed-derivative limit, an effective collision-uniform relative Gram comparison and an all-height regularized native zero-jet certificate. The exact target phases and multiplicity limits are retained. The associated mathematical supplement develops complete arithmetic and localization bounds in their stated whole-line norms.
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Reproduction
The complete publication package contains the 14-page paper, its 7-page mathematical supplement, editable TeX sources, exact rational trial inputs, six portable verification programs, citation metadata and SHA256SUMS. Install the pinned requirements, run python verify_all.py and python -O verify_all.py, and build both PDFs with python build.py --tectonic /path/to/tectonic. The interval certificates use python-flint 0.9.0; symbolic floor controls use SymPy 1.14.0.