The only positive Syracuse periodic point with return time seven is 1

Let T(n) be the odd part of 3n+1. For every positive integer m, if T iterated seven times returns m to itself, then m=1. Seven is a return time, not necessarily the least positive period. This is a symbolic statement for all positive inputs, not a finite trajectory check. It supplies the seven-step case in the Collatz mission's reduction to exclusion of nontrivial Syracuse cycles.