From d99365f3773b97c64a9565991759e7c76113bb84 Mon Sep 17 00:00:00 2001
From: Sebastian Griego <177274079+sebastian-griego@users.noreply.github.com>
Date: Fri, 29 May 2026 10:40:57 -0700
Subject: [PATCH] Improve C42 upper bound to 0.6906538

---
 constants/42a.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/constants/42a.md b/constants/42a.md
index c5362a7..f483871 100644
--- a/constants/42a.md
+++ b/constants/42a.md
@@ -13,6 +13,7 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
 | 1 | Trivial |  |
 | 5/6 | Biró [Bir00] | |
 | 0.69368 | Harcos [Bir00] | |
+| 0.6906538* | Griego [Gri26] | Proposed asymptotic two-block certificate with exact rational interval verification of the limiting inequality. No explicit finite threshold $N$ is supplied. |
 
 ## Known lower bounds
 
@@ -37,3 +38,8 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
 - [Bir00] Biró, A. *An upper estimate in Tur\'{a}n's pure power sum problem.* Indag. Math. (N.S.) **11** (2000), no. 4, 499-508.
 - [Bir00b] Biró, A. *An improved estimate in a power sum problem of Tur\'{a}n.* Indag. Math. (N.S.) **11** (2000), no. 3, 343-358.
 - [CG96] A. Y. Cheer and D. A. Goldston *Tur\'{a}n's pure power sum problem.* Math. Comp. **65** (1996), no. 215, 1349-1358.
+- [Gri26] Griego, S. *An improved asymptotic certificate for Turan's pure power sum constant $C_{42}$.* GitHub repository, version `v1.0.0`, commit `c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7` (2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0
+
+## Contribution notes
+
+The [Gri26] certificate repository and this update were prepared with AI assistance for formatting, exposition, and verification scripts. The mathematical claim, constants, references, and computations are provided for independent review.