A computer-assisted upper bound of 1.7813 for the real Grothendieck constant
Abstract
We give a computer-assisted proof that the universal real Grothendieck constant satisfies KGℝ≤ 1.7813. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the scalar coefficient head through degree $301$ and bounding the entire remaining tail using a weighted Gaussian trace estimate. The finite integral is bounded on a complete interval partition; its spatial exterior is controlled analytically. Exact parameters and source code that regenerate all accepted numerical inputs accompany the paper. The bound improves both the explicit bound $1.7818666069360661$ in the recent literature and the subsequently reported, system-tested value $1.7813319810625639$.
Related
- A computer-assisted lower bound for Landau's constant
- An Improved Lower Bound for the Complex Grothendieck Constant
- Certified Collision Exclusion and an Unconditional Upper Bound Λ<0.156821 for the de Bruijn--Newman Constant
- A Half-Constant Upper Bound for Chowla's Cosine Problem
- An exact computer-assisted lower bound for Fisher-information convexity along the two-dimensional heat flow
BibTeX
@misc{peng2026computerassisted,
title = {{A computer-assisted upper bound of 1.7813 for the real Grothendieck constant}},
author = {Bo Peng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20074},
eprint = {2609.20074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20074},
}
RIS
TY - UNPB
TI - A computer-assisted upper bound of 1.7813 for the real Grothendieck constant
AU - Bo Peng
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20074
UR - https://arxiv.org/abs/2609.20074
AB - We give a computer-assisted proof that the universal real Grothendieck constant satisfies K_G^ℝ≤ 1.7813. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the scalar coefficient head through degree $301$ and bounding the entire remaining tail using a weighted Gaussian trace estimate. The finite integral is bounded on a complete interval partition; its spatial exterior is controlled analytically. Exact parameters and source code that regenerate all accepted numerical inputs accompany the paper. The bound improves both the explicit bound $1.7818666069360661$ in the recent literature and the subsequently reported, system-tested value $1.7813319810625639$.
ER -