Lower bound for the complex Grothendieck constant

The paper proves $$K_G^{\mathbb{C}}>1.35584631827168.$$ The previously recorded Davie lower bound is approximately $1.33807$, while Haagerup's upper bound is approximately $1.40491$. Thus the result closes more than one quarter of the remaining lower-to-upper-bound gap, but it does not determine the exact value of the complex Grothendieck constant. The construction uses finitely many additional complex Hermite projections together with a common radial weight to obtain a dimension-independent $L_{\infty}$-to-$L_1$ estimate. The numerical inequalities needed for the final bound are certified by interval arithmetic. The paper also analyzes the limit of its particular weighted criterion. If $\mathcal{K}_*$ denotes the supremum obtainable within that framework, it proves $$1.35584631827168<\mathcal{K}_*<1.35584697425050.$$ The upper endpoint here is a limitation of this specific criterion, not an upper bound on $K_G^{\mathbb{C}}$ itself.

The paper proves $$K_G^{\mathbb{C}}>1.35584631827168.$$ The previously recorded Davie lower bound is approximately $1.33807$, while Haagerup's upper bound is approximately $1.40491$. Thus the result closes more than one quarter of the remaining lower-to-upper-bound gap, but it does not determine the exact value of the complex Grothendieck constant. The construction uses finitely many additional complex Hermite projections together with a common radial weight to obtain a dimension-independent $L_{\infty}$-to-$L_1$ estimate. The numerical inequalities needed for the final bound are certified by interval arithmetic. The paper also analyzes the limit of its particular weighted criterion. If $\mathcal{K}_*$ denotes the supremum obtainable within that framework, it proves $$1.35584631827168<\mathcal{K}_*<1.35584697425050.$$ The upper endpoint here is a limitation of this specific criterion, not an upper bound on $K_G^{\mathbb{C}}$ itself.