Eccoci.

For some positive integers $k$, there exists an integer partition of the form $4^t = 2^t + k$,
where $4^t$, $2^t$, and $k$ are all positive integers and $t$ is a real number.

The first two such partitions are $4^1 = 2^1 + 2$ and $4^{1.5849625\cdots} = 2^{1.5849625\cdots} + 6$.

Partitions where $t$ is also an integer are called perfect.
For any $m \ge 1$ let $P(m)$ be the proportion of such partitions that are perfect with $k \le m$.
Thus $P(6) = 1/2$.

In the following table are listed some values of $P(m)$.

\begin{align} P(5) &= 1/1\\ P(10) &= 1/2\\ P(15) &= 2/3\\ P(20) &= 1/2\\ P(25) &= 1/2\\ P(30) &= 2/5\\ \cdots &\\ P(180) &= 1/4\\ P(185) &= 3/13 \end{align}

Find the smallest $m$ for which $P(m) \lt 1/12345$.