Spectral Ghosts of Rational Cycles in Bit-Truncated Syracuse Dynamics: A 2-adic Bernoulli Shift and its Finite Approximations

The Syracuse map T on the 2-adic integers Z_2 is shown to generate a full shift on the countable alphabet of step sizes {1,2,3,...}: for every s >= 1, T(P_s) = Z_odd. With the Haar-referenced potential psi_gamma(x) = -S(x) log 2 - gamma log S(x), the associated RPF operator reduces to an i.i.d. Bernoulli shift and admits a spectral gap, unique Gibbs measure, and exponential mixing as immediate consequences of independence (and, formally, as a special case of Sarig's thermodynamic formalism). The bit-truncation of T to a standard-representative map T-hat_m on {1, 3, ..., 2^m - 1} is not a projection of the Z_2-dynamics: the Syracuse step is not well-defined on residue classes modulo 2^m. We study the RPF matrix of this truncation, certify spectral gaps for m in {8, 10, 12} using Bauer-Fike bounds with 50-digit construction precision, and compute the full cycle spectrum exhaustively for 8 <= m <= 18. Non-trivial cycles of T-hat_m appear only at m in {10, 11, 12}, and we show that the four observed cycles (of lengths 6, 7, 25, 26) are spectral ghosts of genuine rational Z_2-cycles of T whose fixed points x* = C/(2^Sigma - 3^L) are odd 2-adic rationals outside N, e.g. -817/601 and -2123/1675. The infinite family x_s = 1/(2^s - 3) already shows that Z_2 contains infinitely many rational periodic orbits of T, so any "no non-trivial cycle" statement must be restricted to N to be meaningful. This paper does not claim to prove the Collatz conjecture. The gap between Z_2 (full shift, with abundant rational cycles) and N (the actual conjecture) remains the fundamental open problem.