Grothendieck Constant Updates

README.md

@@ -29,7 +29,7 @@ Bounds for which the level of available verification is currently at minimal lev
 | [7b](https://teorth.github.io/optimizationproblems/constants/7b.html) | Irrationality measure of $\Gamma(1/4)$ | 2 | $10^{143}$ |
 | [8](https://teorth.github.io/optimizationproblems/constants/8a.html) | Classical zero-free region constant | 0.755106 | 5.558691 |
 | [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.2578 | 3.3177 |
-| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.782214 |
+| [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $1.67696 + 10^{-26}$ | 1.78215358819137 |
 | [10b](https://teorth.github.io/optimizationproblems/constants/10b.html) | The complex Grothendieck constant | 1.338 | 1.40491 |
 | [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.414214 | 3.674235 (3.65*) |
 | [11a](https://teorth.github.io/optimizationproblems/constants/11a.html) | $L^1$ Poincaré constant on the Hamming cube | $\sqrt{\pi/2} \approx 1.2533$ | $\pi/2 - 0.00013 \approx 1.5707$ |

constants/10a.md

@@ -28,6 +28,8 @@ $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$).
 | $2.261$ | [R1974] | Improvement of the original upper bound |
 | $\dfrac{\pi}{2\ln(1+\sqrt{2})} \approx 1.782214$ | [K1979] | Krivine’s bound; best known **explicit** numerical upper bound |
 | $< \dfrac{\pi}{2\ln(1+\sqrt{2})}$ | [BMMN2011] | Strict improvement over Krivine’s bound (no widely cited explicit numerical gap) |
+| $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 10^{-5}$ | [Hei26b] | $10^{-5}$ improvement over Krivine's bound |
+| $< \dfrac{\pi}{2\ln(1+\sqrt{2})} - 6.039*10^{-5}$ | [LSXKKMC26] | $6.039*10^{-5}$ improvement over Krivine's bound |
 
 ## Known lower bounds
 
@@ -46,15 +48,18 @@ $u_{i},v_{j} \in S^{d-1}$ for some sufficiently large finite dimension $d$).
 - [Wikipedia page on the Grothendieck inequality](https://en.wikipedia.org/wiki/Grothendieck_inequality)
 - [JM26] proves that the Davie–Reeds lower bound is not optimal, using a perturbative analysis of the Davie–Reeds operator to establish $K_{G}^{\mathbb R} \ge K_{DR} + 10^{-12}$; the paper also notes concurrent work [Hei26] proving the weaker bound $K_{G}^{\mathbb R} \ge K_{DR} + 10^{-26}$.
 - The perturbative strategy of [Hei26] and [JM26] could possibly be used to improve the complex Grothendieck constant lower bound.
+- [Hei26b] uses thresholding by degree 3 Hermite polynomials.  [LSXKKMC26] upgrades this to a degree 9 thresholding mixed with hyerplane thresholding.
 
 ## References
 
 - [BMMN2011] Braverman, Mark; Makarychev, Konstantin; Makarychev, Yury; Naor, Assaf. *The Grothendieck constant is strictly smaller than Krivine's bound.* Forum of Mathematics, Pi, Volume 1, 2013, e4. [arXiv:1103.6161](https://arxiv.org/abs/1103.6161)
 - [Dav1984] Davie, A. M. *Lower bound for $K_{G}$.* Unpublished note (1984).
 - [G1953] Grothendieck, Alexandre. *Résumé de la théorie métrique des produits tensoriels topologiques.* Bol. Soc. Mat. São Paulo **8** (1953), 1–79.
 - [Hei26] Heilman, Steven. *A lower bound for Grothendieck's constant.* (2026) [arXiv:2603.22616](https://arxiv.org/abs/2603.22616)
+- [Hei26b] Heilman, Steven. *An Upper Bound on Grothendieck's constant.* (2026) [arXiv:2606.00247](https://arxiv.org/abs/2606.00247)
 - [JM26] Jones, Chris; Malavolta, Giulio. *The Grothendieck constant is strictly larger than Davie-Reeds' bound.* (2026) [arXiv:2603.30039](https://arxiv.org/abs/2603.30039)
 - [K1979] Krivine, Jean-Louis. *Constantes de Grothendieck et fonctions de type positif sur les sphères.* Advances in Mathematics **31** (1979), 16–30.
+- [LSXKKMC26], Alan Li, Rahul Saha, Anton Xue, Adam Klivans, Pravesh K Kothari, Raghu Meka, Swarat Chaudhury. *The Grothendieck Constant is Less Than $\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-5}$* [arXiv:2606.03991](https://arxiv.org/abs/2606.03991)
 - [Pis2012] Pisier, Gilles. *Grothendieck’s theorem, past and present.* Bull. Amer. Math. Soc. (N.S.) **49** (2012), 237–323. [arXiv:1101.4195](https://arxiv.org/abs/1101.4195)
 - [Ree1991] Reeds, James A. *A new lower bound on the real Grothendieck constant.* Unpublished manuscript (1991).
 - [R1974] Rietz, Ronald E. *A proof of the Grothendieck inequality.* Israel J. Math. **19** (1974), 271–276.