Carry Dynamics of the Syracuse Map: Exact Identities, Rank-One Transitions, and Mersenne Phase Transitions

We develop the carry arithmetic of the Syracuse map \(S(x) = (3x+1)/2^{v_2(3x+1)}\) from first principles, establishing exact identities that govern popcount evolution under iteration. For each odd \(x \ge 3\), a single Syracuse step satisfies \(\operatorname{pc}(S(x)) = 2\operatorname{pc}(x) + 1 - C(x)\) where \(C(x)\) is the carry count in the binary addition \(x + (2x+1)\), and \(C(x) \ge \operatorname{trail}(x) + 1\). We introduce the combined return decomposition and prove that the composition of two returns obeys \(P_1 P_2 z = R_1 R_2 x + R_2 C_1 + P_1 C_2\), with the carry envelope composing inductively. Using this framework we prove three structural results. First, the transition matrix of combined returns restricted to the resolved event (total bit consumption \(\le K - 2\)) is exactly rank-one: the output depth distribution is independent of the input depth for all input values. Second, this rank-one property yields an algebraic spectral gap \(1 - |\lambda_2| \ge 1 - M \cdot 2^{-\alpha(K-2)}\) that approaches \(1\) as the resolution parameter \(K\) grows. Third, Mersenne numbers \(x = 2^n - 1\) exhibit a resolution phase transition: the first combined return expands bit-length by a factor \(\le 1.6\) but creates \(K \approx 0.585n\) bits of resolution from \(K = 0\), after which the orbit enters the spectral gap regime immediately. We give a constructive \(O(n)\)-return descent certificate for all Mersenne starters. All results are verified computationally for \(n\) up to \(64\) and seeds up to \(2^{60}\).

Keywords: Collatz conjecture, Syracuse map, carry arithmetic, popcount dynamics, combined return, spectral gap, Mersenne numbers.

MSC 2020: 11B83, 37P05, 37B10.