Margin criterion for Syracuse cycles at a general small-value threshold

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. This is the threshold-parametrised form of the margin criterion for excluding Syracuse cycles of a fixed period.

Fix a period $a \ge 1$ and a threshold $B \ge 1$, and suppose two things are known about $B$:

• (small values) every periodic point below $B$ is $1$; equivalently, no nontrivial cycle contains a value $< B$;
• (margin) every power of two exceeding $3^a$ also exceeds $\bigl((3B+1)/B\bigr)^a$, i.e. $(3B+1)^a < 2^K B^a$ whenever $3^a < 2^K$.

Then every $a$-periodic point of $T$ equals $1$.

The mechanism is a comparison of two inequalities holding around any cycle. Writing $m_0$ for the cycle minimum and $K = \sum_{i<a} v_2\bigl(3T^i(m_0)+1\bigr)$ for the total number of halvings, one has $3^a < 2^K$ and $2^K m_0^{\,a} \le (3m_0+1)^a$. A nontrivial cycle has $m_0 \ge B$ by the first assumption, which gives the linear estimate $B(3m_0+1) \le (3B+1)m_0$; raised to the $a$-th power this contradicts the margin assumption.

Separating the threshold $B$ from the argument is what makes the criterion usable: improving the verified range of small values immediately widens the set of periods that can be excluded, with no change to the proof. With $B = 34$ the margin condition holds exactly for $a \le 16$; with $B = 1193$ it holds exactly for $a \le 93$. It never holds for all $a$, since $2^K/3^a$ returns arbitrarily close to $1$ along the convergents of $\log_2 3$.