The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

Abstract: The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

• The paper proves that the Grothendieck constant is strictly larger than Davie-Reeds' bound by 10^-12 using a perturbative analysis.
• It introduces a novel Hermite expansion method, adding degree-3 contributions that refine the integrality gap estimation.
• The results have implications in quantum information theory and combinatorial optimization, signaling a non-saturated theoretical lower limit.

Precise Lower Bound Improvement for the Grothendieck Constant

Overview

The paper "The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound" (2603.30039) delivers a rigorous advancement in estimating the real Grothendieck constant ($K_G$), a quantity central to functional analysis with deep implications in quantum information theory, combinatorial optimization, and the geometry of Banach spaces. The authors establish that the best-known lower bound, ascribed to the works of Davie and Reeds, is not tight; specifically, they show $K_G \geq K_{DR} + 10^{-12}$, marking the first increase in the lower bound since the 1980s.

Theoretical Context

$K_G$ quantifies the maximal discrepancy between quantum and classical values for 2-player XOR games, and is interpreted as the integrality gap of the associated semidefinite programming relaxation in combinatorial optimization. The precise value of $K_G$ remains unknown, with numerical estimates and analytical bounds covering the interval $1.6769 \leq K_G \leq 1.7823$. The upper bound is due to Krivine, while the lower bound—$K_{DR} \approx 1.6769$—resulted from the constructions by Davie and Reeds. Recent breakthroughs, such as Braverman et al., have shown the upper bound can be lowered infinitesimally, but the lower bound remained unchanged until this result.

Methodological Innovations

The paper's principal technical innovation is a perturbative analysis of the Davie-Reeds operator, situated in the landscape of Hermite projection games. This class of instances is essential, as Raghavendra and Steurer established their completeness for hard instances of the Grothendieck problem.

The Davie-Reeds operator takes the form $A_{DR} = \bm{\Pi}_1 - \lambda^* \mathbf{I}$, where $\bm{\Pi}_1$ projects onto degree-1 Hermite polynomials and $\lambda^*$ is chosen optimally. The authors demonstrate that every near-extremizer for this problem must carry $\Omega(1)$ weight on degree-3 Hermite coefficients. Thus, the integrality gap can be increased by introducing a small cubic ($K_G \geq K_{DR} + 10^{-12}$0) Hermite perturbation:

$K_G \geq K_{DR} + 10^{-12}$1

for a suitable $K_G \geq K_{DR} + 10^{-12}$2.

To substantiate this intuition, the authors provide detailed operator norm estimates, stability arguments, and careful analysis of optimal (and near-optimal) strategies for the associated games. The improvement is quantitatively small ($K_G \geq K_{DR} + 10^{-12}$3), but it is positive and strictly exceeds the previously established lower bound.

Strong Numerical Results and Claims

The central numerical result is:

$K_G \geq K_{DR} + 10^{-12}$4

where $K_G \geq K_{DR} + 10^{-12}$5. The proof combines spectral norm evaluations (from Hermite projection games) and a stability framework which bounds the deviation between nearly-optimal functions and true optimizers for the Davie-Reeds instance. The argument is robust enough to withstand perturbations and still yield a nontrivial increment to the lower bound.

The paper also emphasizes that the improvement is not merely theoretical: the analysis aligns with concurrent research by Heilman, who achieved $K_G \geq K_{DR} + 10^{-12}$6 using related techniques.

Implications and Future Directions

The result signals two main implications:

• Theoretical: The lower bound for $K_G \geq K_{DR} + 10^{-12}$7 is not saturated by the Davie-Reeds construction. This suggests that the Grothendieck constant's actual value may still be well above this threshold, and provides concrete guidance for constructing future hard instances using Hermite expansions with alternating low-degree coefficients.
• Quantum/Classical Gap: In quantum information theory, this increases the guaranteed quantum advantage in Bell-type XOR games and underscores the non-optimality of certain classical/SDP approximations.
• Combinatorial Optimization: While the numerical gain is minuscule, the methodological advance invites the systematic inclusion of higher-degree Hermite terms in integrality gap constructions, with potential relevance for the Unique Games Conjecture and regularity lemmas.

Speculatively, one expects further refinements by leveraging richer Hermite expansions and more sophisticated perturbation strategies, possibly leading to larger increments in the lower bound. The approach may inspire analogous tactics for other integrality gap problems and approximation resistance phenomena in theoretical CS.

Conclusion

This work rigorously demonstrates that the Davie-Reeds lower bound for the Grothendieck constant is not optimal. Through precise perturbative analysis, the authors obtain a strictly improved lower bound, confirming that every candidate for extremality requires non-negligible degree-3 Hermite contributions. While the numerical progress is incremental, the theoretical insight is significant, and lays the groundwork for continued exploration of integrality gaps and quantum-classical discrepancies across functional analysis, quantum information, and combinatorial optimization.