Every odd number above one and below 583288 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 $1 < m < 583288$, some iterate of $T$ falls strictly below $m$.

This is a descent statement, deliberately weaker than convergence: nothing is asserted about where the orbit eventually goes. It is nonetheless exactly the strength cycle exclusion requires, since the minimum of a nontrivial cycle can never descend.

Below $330750$ the result is inherited from the previous descent bound. Above it, the statement comes from a ladder of $65$ 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.