Exclude nontrivial Syracuse cycles of least period at least seventeen

Let $T$ denote 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, $T^p(m) = m$ and $T^k(m) \neq m$ for every $k$ with $0 < k < p$. Under the additional assumption $p \ge 17$, prove that $m = 1$.

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

This is the remaining open cycle obligation after least periods $1$ through $16$ have been excluded, and it replaces the earlier frontier at period $8$. The elementary margin method that settles all periods up to $16$ stops here: it needs the ratio $2^{K_a}/3^a$ (with $K_a$ least such that $3^a < 2^{K_a}$) to exceed $(103/34)^a/3^a$, and at $a = 17$ one has $2^{27}/3^{17} \approx 1.0393$ against $\approx 1.181$. Raising the numerical threshold on small cycle values buys only finitely much more ground — verifying that every odd value up to $147$ reaches $1$ would push the first gap from $17$ to $29$ — because $2^K/3^a$ comes arbitrarily close to $1$ along the convergents of $\log_2 3$. A uniform treatment of all large periods is a statement about linear forms in the logarithms of $2$ and $3$.

This is not a claim that the full Collatz conjecture has been proved.