Sparse-Burst Compression and Mesoscopic Screening in the Shifted Laguerre Converse

Abstract

Let $f$ be a real entire function whose zeros lie in a fixed horizontal strip, and set$$f_\mu(x) = f(x + i\mu) + f(x - i\mu).$$The shifted Laguerre converse asks whether strict positivity of the first Laguerre expression for every nonzero imaginary shift forces $f$ to belong to the Laguerre–Pólya class. We study the sparse-burst screening geometry that remains after earlier density, boundary, and real-axis screening reductions.For the negative real-axis field of a nonreal zero, we introduce canonical $q$-debt cores and an exact layer-cake representation, and use them to prove a descendant-or-annulus relay and a core-compression trichotomy. A quantitative height decomposition shows that a high-population sparse burst either contains a large comparable-height cell or sends at least half of its population to the mean-spacing near-axis scale. We also prove that a locally uniform strict-shifted-Laguerre blow-up cannot converge to a finite-nonreal Laguerre–Pólya-star profile outside the Laguerre–Pólya class.In a direction of finite zero density, the positive screening tail of a zero $w = \alpha + i\beta$ outside horizontal distance $C\beta\sqrt{\alpha}$ can be made arbitrarily small relative to $\beta^{-2}$. Consequently, a separated comparable-height family forces linearly many distinct screening generators in mesoscopic neighborhoods. These results eliminate arbitrary far screening and finite-rank microscopic termination within the stated setting, while cross-family local recycling and extreme real overscreening remain outside the scope of the present paper. No complete proof of the Csordas–Escassut converse is claimed.