Syracuse descent for odd numbers between 1032607 and 1036607

Let $T$ be the Syracuse map, sending $n$ to the odd part of $3n+1$. For every odd $m$ with $1032607 \le m \le 1036607$, some iterate of $T$ falls strictly below $m$:

This is a descent statement rather than a convergence statement: nothing is claimed about where the orbit eventually goes, only that it dips below its starting point. That is precisely the strength cycle exclusion needs, since the minimum of a nontrivial cycle can never descend.

Descent also keeps the certificate small. Each orbit is followed only until its first drop below $1032607$ — automatically below $m$ — instead of 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 a single step. Only the residues $m \equiv 3 \pmod 4$ require explicit data, here $5976$ step facts.

This lemma is one stage of a ladder raising the cycle-minimum threshold from $860564$ to $1166400$. The stages are mutually independent — none imports another — so they may be checked in any order or concurrently.