Spectral Phenomena in Finite Approximations of the Syracuse Full Shift: Phase Transitions, Fake Cycles, and the Eigenvalue Ratio 1/√2

The Syracuse map 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) = Z₂^odd, meaning all transitions between step sizes are admissible. Sarig's thermodynamic formalism applies directly, yielding a spectral gap, unique Gibbs measure, and exponential mixing for the RPF operator with potential ψ_γ = -γ log|S|, for all γ ∈ (0, 1). However, finite approximations mod 2ᵐ exhibit rich spectral phenomena absent in the infinite system: fake Markov dependencies, fake cycles, spectral gap collapse at γ=0, and a first-order phase transition with gap (γ, m) ≈ C (m) ·γ. Computer-assisted certification using Bauer-Fike perturbation bounds with 50-digit arithmetic confirms the spectral gap rigorously for m=8, 10, 12 with safety margins exceeding 10⁵. The eigenvalue ratio λ₂/λ₁ converges to 1/√2 as m increases, suggesting deep connections to 2-adic structure. This paper does NOT claim to prove the Collatz conjecture. The measure-zero barrier between Z₂ and the natural numbers N remains the fundamental open problem. Computational code for all results is included.