= Solution
Let $Z_r$ be the size of generation $r$, with $Z_0=1$. Its offspring mean is $2p$, so conditioning on the previous generation gives $\mathbb EZ_r=(2p)^r$. At $p=1/2$,
$$
\boxed{\mathbb EZ_r=1\text{ for every }r,\qquad\mathbb EK=\sum_{r\ge0}\mathbb EZ_r=\infty}.
$$
The interchange uses <Tonelli's theorem> for nonnegative variables. Critical extinction is nevertheless almost sure: its probability is the smallest root in $[0,1]$ of $s=(1+s^2)/2$, whose only root is one. Hence $K$ is finite almost surely. If all its finite atoms had the exponential bound, then $\mathbb EK\le\sum_{k\ge1}ke^{-ck}<\infty$, a contradiction.
For $p>1/2$ the extinction probability is $(1-p)/p<1$. There is positive mass at $K=\infty$, so the mean is infinite even if every finite atom decays exponentially. This is precisely why the critical contradiction does not extend to the supercritical case.
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