No nontrivial Syracuse cycle contains an element $\le 33$

Let $T(n) = (3n+1)/2^{\,v_2(3n+1)}$ be the Syracuse map. No nontrivial $T$-cycle contains a small element:

Mathematical role. The length $a$ is completely unconstrained: this excludes small cycle elements at every period simultaneously, not one period at a time. It is the combination of two facts — that the odd numbers up to $33$ all reach $1$ under $T$, and that a periodic point reaching $1$ must equal $1$ — and it is the form in which those facts are actually used.

Its purpose is to discharge the residual half of every individual cycle-length exclusion. Each such argument bounds a hypothetical cycle below by an explicit threshold $B_a$, obtained by comparing $2^{K}$ with $3^{a}$, and is then left with a finite check on the values beneath it. For the first twelve lengths those thresholds are

all at most $33$. So for every one of those lengths the check is now immediate, and no separate small-case analysis is needed. Transferred through the correspondence between classical and accelerated cycles, the same statement constrains cycles of the original Collatz map.

What remains. This supplies only the residual half. The other half, the lower bound $B_a$ on a cycle's elements, is proved separately for each length, and the thresholds grow without bound along the lengths for which $2^{K}$ approaches $3^{a}$ closely — the convergents of $\log_2 3$. So the cutoff $33$ can be raised by further verification, but no fixed cutoff serves all lengths.

Formalization note. No oddness hypothesis appears: a point of period $a \ge 1$ under $T$ is automatically odd, since $T$ returns an odd part. The period is not assumed minimal.