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 Z₂ is shown to generate a full shift on the countable alphabet of step sizes 1, 2, 3,. . .: for every s >= 1, T (Pₛ) = Zₒdd. With the Haar-referenced potential psigamma (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-hatₘ on 1, 3,. . . , 2ᵐ - 1 is not a projection of the Z₂-dynamics: the Syracuse step is not well-defined on residue classes modulo 2ᵐ. 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-hatₘ 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 Z₂-cycles of T whose fixed points x* = C/ (2Sigma - 3L) are odd 2-adic rationals outside N, e. g. -817/601 and -2123/1675. The infinite family xₛ = 1/ (2ˢ - 3) already shows that Z₂ 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 Z₂ (full shift, with abundant rational cycles) and N (the actual conjecture) remains the fundamental open problem.