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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.