Certified Collision Exclusion and an Unconditional Upper Bound Λ<0.156821 for the de Bruijn--Newman Constant
Abstract
The de Bruijn--Newman constant Λ is the transition time beyond which all zeros of the heat-deformed Riemann ξ-function are real. We develop a computer-assisted collision-exclusion method and obtain the bound Λ<156821/106=0.156821, in the normalisation of Polymath. The proof combines the published finite-height verification and the starting inequality Λ≤1/5 with a compact-interior continuation theorem, five stable prime-polynomial Gram sources, certified moving-index recovery, a balanced real-jet transfer and a uniform Euler-product barrier at infinity. The resulting closed time-band exclusion, together with uniform remote reality, forces strictness at the proposed endpoint. The finite certificate uses 664,505 structural Fourier coefficients, 44,182 closed torus boxes, 184 separator moments and 17,385 separator rectangles. We isolate a Cauchy--Schwarz obstruction intrinsic to the four-prime source family, explaining the narrow feasibility margin of the certified endpoint. We state the proof-to-certificate interface and distinguish the present unconditional logical claim from other recent, as yet unrefereed, computational bounds.
Related
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- A conditional bound Λ≤0.13963 for the de Bruijn--Newman constant
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BibTeX
@misc{planat2026certified,
title = {{Certified Collision Exclusion and an Unconditional Upper Bound $Λ<0.156821$ for the de Bruijn--Newman Constant}},
author = {Michel Planat and Dale James Raymond},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06245},
eprint = {2611.06245},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06245},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Certified Collision Exclusion and an Unconditional Upper Bound Λ<0.156821 for the de Bruijn--Newman Constant
AU - Michel Planat
AU - Dale James Raymond
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06245
UR - https://arxiv.org/abs/2611.06245
AB - The de Bruijn--Newman constant Λ is the transition time beyond which all zeros of the heat-deformed Riemann ξ-function are real. We develop a computer-assisted collision-exclusion method and obtain the bound Λ<156821/10^6=0.156821, in the normalisation of Polymath. The proof combines the published finite-height verification and the starting inequality Λ≤1/5 with a compact-interior continuation theorem, five stable prime-polynomial Gram sources, certified moving-index recovery, a balanced real-jet transfer and a uniform Euler-product barrier at infinity. The resulting closed time-band exclusion, together with uniform remote reality, forces strictness at the proposed endpoint. The finite certificate uses 664,505 structural Fourier coefficients, 44,182 closed torus boxes, 184 separator moments and 17,385 separator rectangles. We isolate a Cauchy--Schwarz obstruction intrinsic to the four-prime source family, explaining the narrow feasibility margin of the certified endpoint. We state the proof-to-certificate interface and distinguish the present unconditional logical claim from other recent, as yet unrefereed, computational bounds.
ER -