A certified unconditional upper bound Λ ≤ 0.1965 for the de Bruijn–Newman constant

Abstract

Abstract

Let Λ denote the de Bruijn–Newman constant. The recorded state of the art is Λ ≤ 0.2: when Platt–Trudgian verified the Riemann hypothesis to the exact height T = 3,000,175,332,800 (≈ 3.0002×10^12), it was noted at once that this improves the Polymath15 bound Λ ≤ 0.22 to Λ ≤ 0.2. A sharper, essentially 0.197-class computation at the same barrier site was publicly reported in a May 2020 comment on T. Tao's blog by Rudolph [Dwars] — uncertified, unrefereed, and never independently verified. This paper supplies the certified, theorem-grade version of that bound and goes beyond it. Subject to the published Polymath15 criterion and the published Platt–Trudgian verification, the attached certificates prove, unconditionally, Λ ≤ 0.1965 = 1965/10000 exactly, a certified improvement below the recorded Λ ≤ 0.2 and below the uncertified 0.1972624050 sighting of 2020, with every numerical ingredient backed by interval-arithmetic certificates (explicit ball enclosures, exact-rational endpoint comparisons, exit-0 standalone verifiers) and every theorem-grade claim verified on two mechanically independent code lines sharing zero code. The route to 0.1965 runs through a certified Euler-3 mollifier bound, certifiably beyond the reach of the Euler-2-class mollifier underlying the 2020 computation; its assembly binds a certified Euler-3 selection chain, gap-free finite sweeps, a box-uniform tail bound, a three-stage normalization-conversion lemma chain, the y-quantifier transfer, and the site bookkeeping lemmas, with an independent second-line sign-off on the assembled statement. The same certificate chain certifies the intermediate record Λ ≤ 0.197 and the 2020 comment's exact row, Λ ≤ 0.197262405. Beyond the bound itself we contribute: certified mollifier results, including the first quantitative saturation map of the finite-Euler-product mollifier family at this site (the family provably runs out of cheap depth); the verification methodology, which addresses the cheap-re-verifiability question raised in the 2020 thread and catches a class of silent rendering defects — three of which we document, with resolutions, in the committed Polymath15 artifacts; and a certified negative/feasibility map for the next rung (where the Euler-2 mollifier provably dies, an exact certified costing of the Λ ≤ 0.19 wall at ≥ 3.4881× the entire Platt–Trudgian computation, a measured GPU pricing of that wall, and a measured kill of the contour-mode verification route). Results currently resting on a single certified verification line are explicitly marked. Contact/updates: Mosaic Intelligence (@111111, https://x.com/111111)