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We prove L263 in the notation of its statement. The proof is author-recorded by Plasma AI.

Let $\iota_2(v)=\sum_{i=1}^rv_i2^{i-1}$ for $v\in\{-1,0,1\}^r$. If the last nonzero coordinate of $v$ is $v_j=1$, then $\iota_2(v)\ge2^{j-1}-\sum_{i<j}2^{i-1}=1$; and $\iota_2(v)\le2^r-1$. So $\iota_2$ maps $Q_r^+$ into $[1,2^r-1]$, though not injectively. It is linear, so $v+w=u$ gives $\iota_2(v)+\iota_2(w)=\iota_2(u)$.

Let $n=2^r-1$ and let $\gamma$ be a coloring of $[1,n]$ by $H$ colors with no monochromatic $x+y=z$. Then $\gamma\circ\iota_2$ colors $Q_r^+$, and a monochromatic triple $v+w=u$ would give a monochromatic $\iota_2(v)+\iota_2(w)=\iota_2(u)$. Hence $\chi_r\le H$ whenever $S(H)\ge2^r-1$.

Now let $k\ge1$ and $r=\lfloor\log_2(S(k)+1)\rfloor\ge1$, so that $2^r-1\le S(k)<2^{r+1}-1$. By the previous paragraph with $H=k$ and the hypothesis, $cr\le\chi_r\le k$. Hence $S(k)+1<2^{r+1}\le2^{k/c+1}$. $\square$