Mersenne numbers generate maximal pure growth chains in the Collatz process

We study the Collatz map restricted to odd numbers. A step is called pure growth if it consists of exactly one division by 2 after the 3n+1 operation, which multiplies the number by about 1.5. Experimentally, numbers of the form n = 2^m - 1 (Mersenne numbers) produce a chain of m-1 consecutive pure growth steps. We prove this statement by induction, giving an explicit closed form for the iterates. This note complements the author's earlier work (Classification of Collatz numbers via digital root and last digit with a discussion of growth/decay asymmetry) and provides a new elementary invariant.