Syracuse descent for odd numbers between 531801 and 535801

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. For every odd $m$ with $531801 \le m \le 535801$, some iterate of $T$ falls strictly below $m$:

This is a descent statement, not a convergence statement: nothing is claimed about where the orbit eventually goes, only that it dips below its starting point. That is exactly the strength cycle exclusion needs, since the minimum of a nontrivial cycle can never descend.

Descent is also what keeps the certificate small. Each orbit is followed only until its first drop below $531801$ — automatically below $m$ — rather than all the way to $1$, and half the 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 one step. Only the residues $m \equiv 3 \pmod 4$ require explicit data, here $6117$ step facts.

This lemma is one stage of a ladder raising the cycle-minimum threshold from $330750$ to $583288$. The stages are mutually independent: none imports another, so they can be checked in any order or concurrently.