Heuristic lower bounds on real Grothendieck constants of finite and infinite order

Abstract

We propose candidate lower bounds on the real Grothendieck constants $K_{\mathrm{G}}(d)$ and their infinite-dimensional limit $K_{\mathrm{G}}$ 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 $K_{\mathrm{G}}\ge 9\pi /16\simeq 1.767146$, improving the rigorous lower bound $6\pi /11\simeq 1.713596$. 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 $K_{\mathrm{G}}(d)$ 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.

arXiv comment21 pages, 2 figures, 3 tables