Collective Contraction in the de Bruijn--Newman Heat Flow: Off-Zero Logarithmic Derivatives and Certified Barrier Refinements
Abstract
We study the de Bruijn--Newman family Hτ(z), with heat-deformation parameter τ and H0(z)=1/8,ξ(1/2+iz/2). The moving zeros considered here are therefore zeros of Hτ, not directly of ζ(s). Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero z=x+iy of maximal imaginary height, [ \frac{d}{dτ}y^2\le -2-4y^2\mathcal G_τ(z), ] where 𝒢τ≥0 is a projected interaction sum. A second exact inequality bounds 𝒢τ below by an off-zero logarithmic derivative −Im(Hτ'/Hτ)(x+iη), providing a direct interface with the effective Polymath approximation Hτ=BτFτ. At the frontier X=6000000185827, τ0=129/800, and y02=87677/2500000, a directed Cauchy certificate gives 𝒢τ>3/2 for τ0≤τ≤0.178, x≥ X, and |y|≤ y0. Combining this contraction with certified upper and lower zero-free barriers, a two-envelope argument reduces the maximal nonreal height to $0.08$ by time 0.1747532546428610755…; the classical de Bruijn contraction then completes the landing. Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ Λ\le0.1779532546428610755\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-x certificates are independent contributions.
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BibTeX
@misc{planat2026collective,
title = {{Collective Contraction in the de Bruijn--Newman Heat Flow: Off-Zero Logarithmic Derivatives and Certified Barrier Refinements}},
author = {Michel Planat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37164},
eprint = {2609.37164},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37164},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Collective Contraction in the de Bruijn--Newman Heat Flow: Off-Zero Logarithmic Derivatives and Certified Barrier Refinements
AU - Michel Planat
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.37164
UR - https://arxiv.org/abs/2609.37164
AB - We study the de Bruijn--Newman family H_τ(z), with heat-deformation parameter τ and H_0(z)=1/8,ξ(1/2+iz/2). The moving zeros considered here are therefore zeros of H_τ, not directly of ζ(s). Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero z=x+iy of maximal imaginary height, [ \frac{d}{dτ}y^2\le -2-4y^2\mathcal G_τ(z), ] where 𝒢_τ≥0 is a projected interaction sum. A second exact inequality bounds 𝒢_τ below by an off-zero logarithmic derivative −Im(H_τ'/H_τ)(x+iη), providing a direct interface with the effective Polymath approximation H_τ=B_τF_τ. At the frontier X=6000000185827, τ_0=129/800, and y_0^2=87677/2500000, a directed Cauchy certificate gives 𝒢_τ>3/2 for τ_0≤τ≤0.178, x≥ X, and |y|≤ y_0. Combining this contraction with certified upper and lower zero-free barriers, a two-envelope argument reduces the maximal nonreal height to $0.08$ by time 0.1747532546428610755…; the classical de Bruijn contraction then completes the landing. Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ Λ\le0.1779532546428610755\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-x certificates are independent contributions.
ER -