Every odd number above one and below 1883432 has a strictly smaller Syracuse iterate

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. For every odd $m$ with $1 < m < 1883432$, some iterate of $T$ falls strictly below $m$.

This is a descent statement, deliberately weaker than convergence, and exactly the strength cycle exclusion requires: the minimum of a nontrivial cycle can never descend.

Below $1505449$ the result is inherited from the previous descent bound. Above it, the statement comes from a ladder of $195$ mutually independent range lemmas, each following orbits only until their first drop below that range's own lower bound. Half of every range needs no orbit data at all: if $m \equiv 1 \pmod 4$ then $4 \mid 3m+1$, so $T(m) \le (3m+1)/4 < m$ in a single step.