有限阶与无限阶实Grothendieck常数的启发式下界
Heuristic lower bounds on real Grothendieck constants of finite and infinite order
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中文总结 AI 辅助
本文通过旋转不变三次核构造实Grothendieck常数的启发式下界，提出半球策略全局最优猜想，改进下界至9π/16，并给出有限维度4至24的数值支持及维度见证。
中文摘要 AI 辅助
我们提出使用旋转不变三次核来构造实Grothendieck常数 $K_{\mathrm G}(d)$ 及其无限维极限 $K_{\mathrm G}$ 的候选下界。对于连续构造，我们确定了一个显式阈值，超过该阈值时半球二元策略在三次边界变形下不稳定。在数值证据的支持下，我们猜想在该阈值处半球策略是全局最优的。这一猜想将意味着 $K_{\mathrm G}\geq9\pi/16\simeq1.767146$，改进了严格下界 $6\pi/11\simeq1.713596$。我们还从维度至多24的单位向量构造了有限对称系数矩阵。大规模see-saw优化支持连续预测，并在所研究的维度4至24中建议了改进的 $K_{\mathrm G}(d)$ 下界，但前提是所猜想的二元最优是精确的。选定的矩阵在解释为二分关联贝尔函数时，还产生了有限设置的候选维度见证。如果相应的二元上界成立，这些见证将证明每一方的局部希尔伯特空间维度至少为5。
英文摘要
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 $K_{\mathrm G}\geq9π/16\simeq1.767146$, improving the rigorous lower bound $6π/11\simeq1.713596$. 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.
发表机构
- HUN-REN Institute for Nuclear Research（匈牙利科学院核物理研究所）
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