New Lower and Upper Bounds for the Grothendieck Constant
2026-08-11 • Computational Complexity
Computational ComplexityData Structures and Algorithms
AI summaryⓘ
The authors found new, tighter limits for an important mathematical number called the Grothendieck constant, showing it lies between about 1.71 and just under 1.78. Instead of the usual methods, they used a new approach for the lower limit and created the first large-scale method for the upper limit. Their results clarify a previously uncertain detail about this constant's value. This work was achieved through a long-term team effort of both people and a specialized AI system.
Grothendieck constantKrivine schemesrounding schemesasymptotic analysismathematical boundscollaborative AIgap instanceslogarithmπ (pi)optimization
Authors
Rahul Saha, Alan Li, Anton Xue, Swarat Chaudhuri, Adam Klivans, Pravesh K Kothari, Raghu Meka
Abstract
We establish new bounds on the Grothendieck constant $K_G$: \[ \frac{6π}{11} \le K_G \le \fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.