Heuristic lower bounds on real Grothendieck constants of finite and infinite order
Abstract
We propose candidate lower bounds on the real Grothendieck constants KG(d) and their infinite-dimensional limit KG using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply KG≥9π/16≃1.767146, improving the rigorous lower bound 6π/11≃1.713596. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on KG(d) in the studied dimensions from $4$ to $24$, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.
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BibTeX
@misc{bene2026heuristic,
title = {{Heuristic lower bounds on real Grothendieck constants of finite and infinite order}},
author = {Erika Bene and Tamás Vértesi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35399},
eprint = {2609.35399},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35399},
}
RIS
TY - UNPB
TI - Heuristic lower bounds on real Grothendieck constants of finite and infinite order
AU - Erika Bene
AU - Tamás Vértesi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35399
UR - https://arxiv.org/abs/2609.35399
AB - We propose candidate lower bounds on the real Grothendieck constants K_G(d) and their infinite-dimensional limit K_G using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply K_G≥9π/16≃1.767146, improving the rigorous lower bound 6π/11≃1.713596. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on K_G(d) in the studied dimensions from $4$ to $24$, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.
ER -