Collective Contraction in the de Bruijn--Newman Heat Flow: Off-Zero Logarithmic Derivatives and Certified Barrier Refinements

Source abstract

We study the de Bruijn--Newman family $H_τ(z)$, with heat-deformation parameter $τ$ and $H_0(z)=\tfrac18,ξ(\tfrac12+\tfrac{iz}{2})$. 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 $\mathcal G_τ\ge0$ is a projected interaction sum. A second exact inequality bounds $\mathcal G_τ$ below by an off-zero logarithmic derivative $-\operatorname{Im}(H_τ'/H_τ)(x+iη)$, 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 $\mathcal G_τ>3/2$ for $τ_0\leτ\le0.178$, $x\ge X$, and $|y|\le y_0$. 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 $0.1747532546428610755\ldots$; 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 Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ Λ\le0.1779532546428610755\ldots .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.