Open on the site: the label is OPEN, which the commentary attaches to the corrected Statement (page last edited 29 December 2025; no proof claim on its tab; read 2026-10-07), and the standing derived from the claim pages is open, since every claim is partial. The growth of $v(k)$ is fixed at the double-logarithmic scale and not beyond: for all large $k$, $e^{e^{k/600}}\le v(k)$ and $\log\log v(k)=\Theta(k)$ with $\liminf\log\log v(k)/k\ge\log2/257$ and $\limsup\le\log2$, by Corollary 1.3 of the OpenAI mathematics release's manuscript of 25 September 2026, accepted as a partial claim on formalized evidence (its claim page (OpenAI, 2026)). The best upper bound remains $v(k)\le c_0^{(2/5+o(1))2^k}$ with $c_0=1.26408\ldots$ the Vardi constant, from van Doorn and Tang's inequality (1.2) and the Elsholtz–Planitzer count (the paper prints $(1/5+o(1))2^k$; see the Upper bound below); whether the slope tends to $\log2$ and any asymptotic for $v(k)$ are open. Before the release the best lower bound was $v(k)\ge e^{ck^2}$ (van Doorn and Tang, Theorem 1.1, for every $k\ge1$), accepted as a refereed partial claim (its claim page (van Doorn and Tang, 2025)); a claimed intermediate bound, doubly exponential in $\sqrt k$, has its own page (van Doorn and GPT-6 Astra Pro, 16 September 2026).