I am currently reading "Combinatorial Game Theory" by Aaron Siegel. In the section about temperature and cooling, there seem to be some issues in the proof that cooling of games is a group homomorphism (Theorem 5.14).
Recall of the definitions: For $t \geq -1$ and a game $G$, Siegel defines $$\tilde G_t = \{G^L_t - t \mid G^R_t + t\}.$$ If $G$ is equal to an integer $n$, then the cooled game $G_t$ is defined as $n$. If $\tilde G_{t'}$ is infinitesimally close to a number $x(t')$ for some $t' < t$, then $G_t$ is defined as $x(t')$ for the smallest such $t'$. In all other cases, $G_t = \tilde G_t$. The temperature $t(G)$ is defined as the smallest $t \geq -1$ such that $t(G)$ is numberish.
In the proof of $(G + H)_t = G_t + H_t$, Siegel wants to prove $(G+H)^\sim_t = G_t + H_t$ given that $t(H) < t \leq t(G)$. The proof goes as follows:
$G_t$ is not equal to number, so there is a left option such that $L(G_t) = R((G_t)^L) = R(G_t^L) - t$. First Question: Why is $G_t$ not a number? If $t = t(G)$ then $G_t$ is numberish. Why can't the infinitesimal part vanish here?
$$R(G^L_t + H_t) = R(G^L_t) + H_t = L(G_t) + t + H_t = L(G_t + H_t) + t > R(G_t + H^L_t)$$ Second Question: Where does the last inequality come from?
The last step is realising that the option $G_t + H^L_t - t$ of $(G + H)^\sim_t$ is dominated by $G^L_t + H_t - t$. Third question: How does that follow?
My guess about questions two and three is, that one might even prove $H_t + t > H_t^L$. That would prove the inequality in the second question even with $L(G_t + H^L_t)$. The third question is solved easily then.