A conditional bound $\mathrm{\Lambda }\le 0.13963$ for the de Bruijn--Newman constant

Assuming that $\zeta (s)$ has no nontrivial zero with $\mathrm{R}\mathrm{e}s>7\mathrm{/}8$, as asserted in a recent unrefereed preprint of OpenAI, we obtain $\mathrm{\Lambda }\le 13963\mathrm{/}100000=0.13963$ for the de Bruijn-Newman constant. A published Polymath boundary computation supplies a finite-region strip of half-width $1\mathrm{/}5$. We derive an approximation on $3\le \mathrm{I}\mathrm{m}z\le 6$ from its general Riemann-Siegel coefficient bounds. At height $61\mathrm{/}20$, a fixed cutoff permits two differentiations, and comparison with $\zeta$ bounds both the zero-density kernel and its vertical derivative. A new conjugate-pair inequality and a twenty-four-step strip comparison give the stated bound. The finite scalar and polynomial certificates are included in the paper; earlier comparisons giving $0.17973$, $0.17172$, and $0.16408$ are retained to show the source of the gain. The argument uses neither Gordon’s analytic inputs nor Gomila’s new computations. The seven-eighths hypothesis remains an assumption, and the published numerical boundary input is retained.