Human-AI Collaboration Tightens Both Bounds on the Grothendieck Constant
Rahul Saha and six coauthors report AI-assisted lower and upper bounds of $6\pi/11 \leq K_G \leq \pi/(2\log(1+\sqrt{2}))-10^{-4}$ for the real Grothendieck constant, determining its tenths digit while leaving its exact value open; the authors say they independently verified the theorem-level results and exclude stronger machine-only claims from those theorems.
In two preprints submitted on August 11, 2026, Rahul Saha, Alan Li, Anton Xue, Swarat Chaudhuri, Adam Klivans, Pravesh K. Kothari, and Raghu Meka report new lower and upper bounds on the real Grothendieck constant: . The authors state that these bounds determine the previously unknown tenths digit of to be , while its exact value remains open.
The Grothendieck constant measures the worst-case gap between a hard bilinear optimization problem over signs and its efficiently solvable vector relaxation. The authors describe an asynchronously human-steered research system whose reasoning model was OpenAI GPT-5.5-Pro and later GPT-5.6-Sol, while Anthropic Claude Code, first running Claude Opus and later Claude Fable 5, handled coding, experiments, literature retrieval, and session orchestration. They report that central steps of the lower-bound argument originated with this system, while the upper-bound construction predates it and arose through conversations with GPT-5.5-Pro.
The companion preprint contains the complete mathematical proofs, and the case study says the authors independently verified every result stated as a theorem. The same system produced additional stronger machine-verified numerical claims, but the authors explicitly separate those from the theorems because they had not yet checked those certificates themselves; those provisional values are not reported here.