The Carry Ratchet: Why the Collatz Map Always Descends

We identify the mechanism by which the Collatz map drives every trajectory to 1. The map T(n) = 3n+1 (odd) or n/2 (even) has a single active operation: 3n+1, which simultaneously escalates magnitude and injects harmony. Escalation is uniform (log_2(3) bits per step). Harmony injection is variable: the +1 fires a carry chain through trailing bits, creating trailing zeros whose count depends on the binary geometry of n. We prove the Carry Ratchet: when the carry hits a 0-bit, it stops and the loss is bounded at exactly -1; when it hits a 1-bit, it converts it to 0 and continues — gains cascade without bound. Losses are fixed while gains compound. Working modulo 8, we identify four states: two that escalate…