The Cylinder Horizon of the Accelerated 3x + 1 Map A Number-Theoretic Case Study in Descriptive Boundaries and Projectability Limits

We record an elementary, self-contained observation about the accelerated Collatz map
T (m) = (3m + 1)/2ν2(3m+1) on odd integers. Let Dk = ν2(3mk + 1) be the discharge (valuation)
sequence of an orbit. We show that a positive integer can reproduce a fixed eventually-periodic
low-discharge valuation word—one whose mean discharge satisfies ¯d < log2 3—for at most
¯d−1 log2 m0 + O(1) steps.