New Lower and Upper Bounds for the Grothendieck Constant
Rahul Saha, Alan Li, Anton Xue, Swarat Chaudhuri, Adam Klivans, Pravesh K Kothari, Raghu Meka
cs.CC, cs.DS
2026-08-12
A long-horizon AI research system working with humans pushed the Grothendieck constant's bounds to 1.7135 and 1.7818, fixing its tenths digit at 7 for the first time.
The Grothendieck constant KG dates to a 1953 inequality: relax a quadratic optimization over ±1 signs to one over real vectors, and the vector optimum can exceed the sign optimum by at most a factor KG. Existence has been known since Grothendieck; the exact value has resisted 70 years of attacks. The constant is not a curiosity. It sets approximation ratios for combinatorial problems such as matrix cut norm, and it surfaces across Banach space geometry, operator spaces, harmonic analysis and quantum information.
Both sides of the bound were stuck. On the lower side, 1.6769 came from high-dimensional Gaussian constructions by Davie (1984) and Reeds (1991) and stood for four decades; two 2026 works moved it by 10⁻²⁶ and 10⁻¹² respectively. On the upper side, Krivine's 1977 bound of 1.7822 came with his conjecture that it was tight. Braverman, Makarychev, Makarychev and Naor disproved that conjecture in 2011, but explicit numerical improvements since have been on the order of 10⁻⁵.
Both bounds revolve around Krivine rounding schemes: randomized procedures that map the relaxed vector solution back to ±1 signs, built from an infinite-dimensional preprocessing step and a random hyperplane step.
For the upper bound, the paper introduces limiting Krivine schemes, an analytic framework for studying families of rounding schemes of arbitrarily high dimension, and constructs within it a cubic-quintic scheme whose coordinate-wise correlation function takes the form (t − s₃²t³ + s₅²t⁵)/(1 + s₃² + s₅²). All previous work stayed in one or two dimensions, so this is the first genuinely asymptotic construction, and it affirmatively answers a 2011 question of Braverman et al.: raising the dimension does improve the bound. The proof splits into an analytic step, which reduces controlling an inverse series to a lower bound on the linear coefficient plus an upper bound on the total nonlinear mass, and a computational step that certifies a finite list of inequalities in outward-rounded interval arithmetic. The certificates are on GitHub.
The lower bound inverts the usual strategy. Instead of constructing gap instances, the paper shows that Krivine schemes themselves have a ceiling: the correlation function of any rounding scheme obeys the affine constraint b₃ ≥ 2b₁ − 11/6 on its Taylor coefficients. Affine constraints survive mixtures and coefficientwise limits, so this bounds the whole asymptotic family. Combined with the Naor-Regev theorem that mixed Krivine schemes are asymptotically optimal, the ceiling translates into KG ≥ 6π/11.
| Quantity | Previous best | This paper |
| Upper bound | 1.7822 (Krivine 1977; recent gains 10⁻⁵) | π/(2log(1+√2)) − 3.47×10⁻⁴ ≈ 1.7818 |
| Lower bound | 1.6769 (Davie-Reeds, unmoved for 40+ years) | 6π/11 ≈ 1.7135 |
| Tenths digit | unknown | 7 |
The upper-bound gain of 3.47×10⁻⁴ is more than an order of magnitude larger than the best previous improvement, and the lower bound jumps by 0.037, its first substantive move since 1984. Together they pin KG between 1.7135 and 1.7818.
For theory readers this is the largest substantive movement on the constant in decades, and the first evidence that higher-dimensional rounding schemes actually help. The research process is at least as notable: the bounds came out of a long-running collaboration between humans and a long-horizon AI research system, with a companion paper dissecting how the collaboration worked. For anyone tracking whether AI systems can sustain months-long mathematical research, this is a concrete data point.
The exact value of KG remains open; the 0.068 gap between 1.7135 and 1.7818 leaves the hundredths digit unsettled. The lower-bound argument depends on the Naor-Regev optimality theorem, so the ceiling binds Krivine-type schemes, and a rounding procedure outside that framework could still push the upper bound down. The paper itself says little about what the AI system actually contributed and how reliable it was; that account lives in the companion paper.