Exclude nontrivial Syracuse cycles of least period at least ninety-four

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. Suppose $m$ is a positive integer whose least positive return time under $T$ is $p$: that is, $T^p(m) = m$ while $T^k(m) \neq m$ for all $0 < k < p$. Under the additional assumption $p \ge 94$, prove that $m = 1$.

Since $1$ has least period $1$, this would rule out such a cycle altogether.

This is the remaining open cycle obligation once least periods $1$ through $93$ have been excluded, and it supersedes the earlier frontiers at periods $8$ and $17$. The elementary margin method stops here. It requires $2^{K_a}/3^a$ (with $K_a$ least such that $3^a < 2^{K_a}$) to exceed $\bigl((3B+1)/B\bigr)^a/3^a$ for the largest verified small-value threshold $B$, and at $a = 94$ this fails for $B = 1193$. Enlarging $B$ buys only finitely much more ground: verifying every odd value up to $3342$ would move the first gap to $147$, up to $6724$ to $200$, and up to $27113$ to $306$ — and $306$, a denominator of a convergent of $\log_2 3$, is where the approach is defeated outright, because $2^{485}/3^{306}$ is within $10^{-5}$ of $1$.

A treatment of all large periods is therefore a statement about linear forms in the logarithms of $2$ and $3$, not something the counting argument can reach. This is not a claim that the full Collatz conjecture has been proved.