Primal/Dual Method for the Grothendieck Constant

2026-10-07

AI Summary

This study addresses the determination of the exact value of the real Grothendieck constant $K_G$. Methodologically, it proposes a primal-dual heuristic search framework by streamlining Gaussian duality theory and integrating Hermite projection games with Krivine's rounding scheme. Furthermore, this work pioneers the application of AI-assisted techniques for the rigorous certification and formal verification of large-scale numerical inequalities to solve this mathematical problem. The results establish that $1.773 \leq K_G \leq 1.7799$, with a conjecture that $K_G \approx 1.779$. The primary contribution lies in introducing a novel paradigm for the AI-driven, rigorously certified computation of fundamental mathematical constants, thereby significantly improving the precision of existing bounds.

Abstract

We determine the hundredths digit of the real Grothendieck constant by proving $1.773 \leq K_G \leq 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G \approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bounds are obtained by a heuristic primal/dual search for these objects, followed by rigorous {\em certification} of their values. The search and certification are performed using AI tools, and in particular the rigorous certificates are very large (analytical reductions to thousands of numerical inequalities).

Problem

• Grothendieck constant
• primal/dual method
• numerical bounds

Innovation

• Grothendieck constant
• primal/dual method
• Krivine rounding schemes
• AI-assisted certification
• Gaussian duality theory