Carry Absorption, Boundary-Pair Reduction, and Exact Anti-Persistence for the Accelerated Odd Map

This paper studies the accelerated odd map F (m) = (3m+1) / 2^nu₂ (3m+1) on the positive odd integers. The paper proves an exact arithmetic decomposition of the odd step into a carry-free quotient over F₂x and an exact carry identity in Zx. The carry-free quotient is shown to be unconditionally contractive in at most d (d+1) /2 steps on degree-d data. The paper then proves an exact boundary-pair reduction D (F (n) ) - D (n) = tau (n) - b (n), where b (n) is determined by the first equal adjacent pair in the binary expansion of n, and tau (n) by the last equal adjacent pair. From this decomposition the paper derives exact shell statistics for the accelerated odd map, including a closed-form shell mean drift formula with limit -1/3, and an exact anti-persistence theorem. If Bd denotes the bad shell state, then F (n) in B₃+₁ if and only if n in Bd, n = 7 mod 8, and n >= Bdˢtar, and the exact bad-to-bad conditional probability tends to 1/6. These are shell-counting theorems, not orbit theorems. The paper also proves a Carry Firewall Creation Theorem for the carry transducer on finite input words. As a conditional application to the classical Collatz conjecture, the paper isolates two residual hypotheses: the Orbit Tail Mixing Hypothesis on the divergent side, and the Cycle Elimination Hypothesis on the cyclic side. The paper does not claim a proof of the Collatz conjecture.