 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[False.] Take $\alpha=\omega^2+1$, which satisfies the proposed lower bound. Associativity of <ordinal addition> and $1+\omega^2=\omega^2$ give
$$
\boxed{(\omega^2+1)+\omega^2=\omega^2\cdot2,\qquad
\omega^2+(\omega^2+1)=\omega^2\cdot2+1.}
$$
The final successor survives only in the second expression.

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