Collision-uniform kernel asymptotics for Báez–Duarte zero packets
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Abstract
We prove collision-uniform high-height asymptotics for logarithmically growing zero packets in the harmonic-step space underlying Báez–Duarte approximation. An exact arithmetic gauge converges to an explicit complex Mellin kernel, with literal mixed-derivative errors at every finite order. A positive Hardy subkernel and an explicit rational Newton–Hermite interpolant give a separation-free Gram floor whose price is exponential in packet cardinality. Combining these bounds yields relative Gram comparison uniformly for arbitrary collisions and for cardinality at most a sufficiently small explicit multiple of the logarithm of height. When the real-part rectangle approaches both open walls of the strip at an inverse logarithmic rate, packets of size proportional to the logarithm of height divided by its iterated logarithm remain admissible, with an explicit restriction on the proportionality constant. At admitted zeta-zero packets, exact native zero-jet projection retains the target's unequal powers and oscillatory phases. We also give a literal onset, a vanishing sublogarithmic packet budget, all-height regularized certificates, and fixed-multiplicity consequences. Complete arithmetic tail estimates and supporting norm comparisons are included; the argument uses the original harmonic-step kernel throughout.
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- Collision_uniform_arithmetic_packets.pdf169.0 KiB · pdf
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7b28a20cb26aea9ccdfbc80ace3c49abf0f1a0dd291a2bd0673c03f68af7a7ff - Collision_uniform_arithmetic_packets_publication.zip314.3 KiB · package
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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
- none
Scope
The exact harmonic-step kernel admits collision-uniform relative Newton–Hermite Gram comparison for packet cardinality up to an explicit sufficiently small multiple of log T. A separation-free exponential Gram lower bound and literal complete arithmetic error also permit a paid simultaneous approach to both open real-part walls. Native zero-jet projection, all-height regularization and a sublogarithmic certificate bound use the same kernel and exact target.
Disclosures
Author-posted preprint. OpenAI ChatGPT and Codex assisted mathematical derivation, proof development and checking, literature comparison, verification-code development and writing. Scott Schubr is the sole author and is responsible for the content. The author owns Luminary Oddities and the Luminary Research Archive and declares no other competing interests. This research received no external funding.
Reproduction
The source package contains the complete main manuscript and mathematical supplement, editable sources, seven mathematical verification programs, rational input data, reader/build instructions and a complete member manifest. The controls cover exact Newton–Hermite algebra, complete error budgets, Gram floors, regularized variational identities, directed interval Poisson tails and localization. The logarithmic packet control checks exact rational-complex interpolation and Hardy norms, including repeated and mixed nodes. Universal analytic assertions are proved in the manuscripts. Python 3.12.14, SymPy 1.14.0 and python-flint 0.9.0 were tested.