Ghost Cycles of the Syracuse Map: 2-Adic Periodic Orbits and the Exceptional Set

We study the Syracuse map S(n) = (3n+1)/2^v(3n+1) on odd integers through its transfer operator L on C(Z_2^odd) and its finite approximations P_k. We prove ‖L‖ = 2/3 and ρ(L) ≤ 1/2, and establish two independent obstructions to spectral gap methods: L does not preserve any Hölder or Lipschitz space on (Z_2^odd, |·|_2), and L is unbounded on C(Z_2^odd, Q_2) with ‖P_k‖_2-adic = 2^(k+O(1)) → ∞, closing the Mahler/Amice program entirely. Both obstructions trace to a common root cause: the weight 2^(-v) is archimedeanly small but 2-adically large. Our central result is that "ghost cycles" — extra modular cycles beyond the fixed point {1} — are not transient artifacts: exhaustive enumeration through k = 36 and algebraic analysis through k = 200 show they are projections of true 2-adic periodic orbits with negative rational elements in all computed cases. Case-(a) ghosts persist  at arithmetic progressions of levels, making the exceptional set E infinite with density ≥ 4%. A census identifies 88+ materializing ghost types through cycle length L = 12, organized into families by excess e = V − L, with record spectral radius ρ ≥ 2^(-16/15) ≈ 0.4774. We propose four replacement conjectures; notably, Conjecture 4 (negative rationality) asserts that all orbit elements of D < 0 ghost types are negative rationals — verified through L = 12 for 5,996 cases — establishing that the entire high-spectral-radius regime (ρ > 1/3, equivalently D < 0) consists of purely negative 2-adic orbits with no positive-integer elements. All computations are reproducible from the accompanying open-source repository.

v2 (March 2026): Adds Theorem 6 (unconditional proof of negative rationality for concentrated ghost patterns, with closed-form $R_i = 2^{L-i+1}(2^e-1)\cdot 3^{i-1} + (3^L - 2^{L+e})$), proof strategy remarks for Conjectures 1–4, and several expository fixes.