@@ -150,7 +150,7 @@ Bounds for which the level of available verification is currently at minimal lev
 - [3d](https://teorth.github.io/optimizationproblems/constants/3d.html) **solved:** $C_{3d} = 2$ by [H. Lin and S. Li](https://arxiv.org/abs/2607.27199), 29 Jul 2026.
 - [3b](https://teorth.github.io/optimizationproblems/constants/3b.html) **improved lower bound:** $C_{3b} \geq 1.77898884$ by Mosaic Intelligence, [entropy certificate](https://doi.org/10.5281/zenodo.20794135) on a 13-point support.
 - [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) **improved upper and lower bounds:** $\frac{6\pi}{11} \leq C_{10a} \leq \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$ by [R. Saha, A. Li, A. Xue, S. Chaudhuri, A. Klivans, P. K. Kothari, R. Meka](https://arxiv.org/abs/2608.11158), 11 Aug 2026 — determines the tenths digit of $C_{10a}$ to be $7$.
-- [2](https://teorth.github.io/optimizationproblems/constants/2a.html) **solved:** $C_2 = 2$ — Crouzeix's conjecture, by [Jin](https://www.preprints.org/manuscript/202607.1919) (July 2026); a second proof followed shortly after in [E. Lorist and F. L. Schwenninger](https://arxiv.org/abs/2608.03841), 4 Aug 2026, which acknowledges Jin.
+- [2](https://teorth.github.io/optimizationproblems/constants/2a.html) **solved:** $C_2 = 2$ — Crouzeix's conjecture, by [S. Jin](https://www.preprints.org/manuscript/202607.1919), 27 Jul 2026; an independent proof by a different route followed in [E. Lorist and F. L. Schwenninger](https://arxiv.org/abs/2608.03841), 4 Aug 2026.
 - [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) **improved lower bound (limit value):** $C_{3a} \geq 1.19519192*$ by [L. Kleinwaks](https://github.com/kleinwaks/masked-digit-sum-difference-bound), 14 Aug 2026.
 - [15a](https://teorth.github.io/optimizationproblems/constants/15a.html) **improved upper bound:** $C_{15a} \leq 2.371177$ by [E. Dupont, M. Eisenberger, B. Kozlovskii, A. Mehrabian, F. J. R. Ruiz, A. See, R. Zhou, J. Alman, V. Vassilevska Williams, M. Balog](https://arxiv.org/abs/2608.16884), 17 Aug 2026.
 
 
@@ -32,8 +32,8 @@ $$
 | ----- | --------- | -------- |
 | $11.08$ | [C2007] | First dimension-free bound. Also holds in the completely bounded (matrix-valued) setting. |
 | $1+\sqrt{2} \approx 2.41421$ | [CP2017] | Best universal upper bound prior to the [Jin2026] / [LS2026] pair. Also holds in the completely bounded setting. |
-| $2$ | [Jin2026] | Jin: claimed proof of Crouzeix's conjecture, posted on preprints.org in July 2026. |
-| $2$ | [LS2026] | Lorist–Schwenninger: proof of Crouzeix's conjecture, combining previous tools for weaker estimates with a "simple perturbation lemma for 2-dilations" applied to the iterates $f^n$ in the double-layer potential representation.  The paper acknowledges [Jin2026]. |
+| $2$ | [Jin2026] | Jin: proof of Crouzeix's conjecture, first posted on preprints.org on 27 July 2026.  The current version 4 is marked there as not peer-reviewed. |
+| $2$ | [LS2026] | Lorist–Schwenninger: independent proof by a different route, combining previous tools for weaker estimates with a "simple perturbation lemma for 2-dilations" applied to the iterates $f^n$ in the double-layer potential representation.  See the note on priority below. |
 
 ## Known lower bounds
 
@@ -44,7 +44,8 @@ $$
 
 ## Additional comments and links
 
-- Crouzeix conjectured (in [C2004]) that $C_{2}=2$.  The lower bound $2$ matches the upper bounds of [Jin2026] and [LS2026], so **the conjecture is (claimed to be) settled**: $C_{2} = 2$.  Jin's preprint appeared in July 2026 and Lorist–Schwenninger's followed in August 2026, acknowledging Jin's.
+- Crouzeix conjectured (in [C2004]) that $C_{2}=2$.  The lower bound $2$ matches the upper bounds of [Jin2026] and [LS2026], so **the conjecture is (claimed to be) settled**: $C_{2} = 2$.
+- **Priority and independence.**  [Jin2026] was posted first, on 27 July 2026, and so has priority of public announcement; [LS2026] appeared on arXiv on 4 August 2026.  The two proofs are nonetheless independent, and take different routes.  [LS2026, §1] has said so since its first version: "a proof of Crouzeix's conjecture appeared independently in [Jin2026] using a different approach involving function-theoretic representations of matrix-valued functions.  Our proof instead uses the operator-theoretic Lemma 1."  The AI-disclosure statement in [LS2026, v2] dates the interactions that produced its key observation to 25 and 26 July 2026, and records in a footnote that "the authors became aware of [Jin2026] on July 28, 2026, after the interactions described above."  Between 12 and 27 August 2026 this page instead described [LS2026] as merely acknowledging [Jin2026]; that read a citation of prior public posting as a statement of dependence, and was incorrect.
 - The conjectured constant $2$ is known to hold in a number of special cases; see, for instance, [Cho2013], [GKL2018], [CGL2018].
 - Numerical experiments strongly support the conjecture; see [GO2018].
 - [Wikipedia page on Crouzeix’s conjecture](https://en.wikipedia.org/wiki/Crouzeix%27s_conjecture)
@@ -61,10 +62,10 @@ $$
 - [GKL2018] Glader, Christer; Kurula, Mikael; Lindström, Mikael. *Crouzeix’s conjecture holds for tridiagonal $3\times 3$ matrices with elliptic numerical range centered at an eigenvalue.* SIAM J. Matrix Anal. Appl. **39** (2018), no. 1, 346–364. DOI: 10.1137/17M1110663. [arXiv:1701.01365](https://arxiv.org/abs/1701.01365).
 - [CGL2018] Caldwell, Trevor; Greenbaum, Anne; Li, Kenan. *Some Extensions of the Crouzeix–Palencia Result.* SIAM J. Matrix Anal. Appl. **39** (2018). DOI: 10.1137/17M1140832. [arXiv:1707.08603](https://arxiv.org/abs/1707.08603).
 - [RS2018] Ransford, Thomas; Schwenninger, Felix L. *Remarks on the Crouzeix–Palencia proof that the numerical range is a $(1+\sqrt2)$-spectral set.* SIAM J. Matrix Anal. Appl. **39** (2018), no. 1, 342–345. DOI: 10.1137/17M1143757. [arXiv:1708.08633](https://arxiv.org/abs/1708.08633).
-- [Jin2026] Jin. *A proof of Crouzeix's conjecture.* Preprints.org manuscript [202607.1919](https://www.preprints.org/manuscript/202607.1919) (July 2026).
+- [Jin2026] Jin, Shanmu. *The Numerical Range Is a $2$-Spectral Set.* Preprints.org manuscript [202607.1919](https://www.preprints.org/manuscript/202607.1919), first posted 27 July 2026; version 4 posted 7 August 2026, marked there as not peer-reviewed. DOI: [10.20944/preprints202607.1919.v4](https://doi.org/10.20944/preprints202607.1919.v4).
 - [LS2026] Lorist, Emiel; Schwenninger, Felix L. *A solution to Crouzeix's conjecture.* [arXiv:2608.03841](https://arxiv.org/abs/2608.03841) (2026).
 
 
-# Acknowledgements
+## Contribution notes
 
-Prepared with ChatGPT 5.2 Pro.
+Prepared with ChatGPT 5.2 Pro.  The priority note above, and the [Jin2026] title, author and posting dates, were checked against the preprints.org record and against versions 1 and 2 of [LS2026] on 27 August 2026, with assistance from Claude Opus 5.