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

Abstract:

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 in the 2-adic setting. 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)·3^(i-1) + (3^L - 2^(L+e))), proof strategy remarks for Conjectures 1–4, and several expository fixes.

v3 (March 2026): Added Proposition 6 (archimedean non-compactness of L, proved by non-equicontinuity); added Baker–Wüstholz proof sketch deriving the constant 25 from Laurent (2008), Corollary 2; added remark clarifying L as projective limit of integer Syracuse maps; added primitivity remark to Theorem 5; moved sup/limsup equality out of Theorem 1(e) proof into a separate remark; noted n = -1/3 exceptional point; softened "closing the Mahler/Amice program entirely" to "in the 2-adic setting"; corrected Baker–Wüstholz growth claim; corrected pair count from 91 to 105; clarified k=1000 density scan as algebraic membership testing; corrected D odd/nonzero justification; corrected Proposition 5 K0 definition; added footnote crediting Davison (1976) as originator of the cycle equation.

Reference corrections (v3): The reference "Siegel, M. Ghost Cycles of the 3x+1 Map. arXiv:2601.12772" in v1 and v2 was incorrect in both author and title. The correct citation is Dhiman, M. and Pandey, R. (2026), 2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles, arXiv:2601.12772. The paper attributed to Siegel does not exist. The author apologizes to Maxwell Siegel and to Madhav Dhiman and Rohan Pandey for the misattribution. Additionally: page range for Amice (1964) corrected to 117–180; title for Kontorovich and Lagarias (2010) corrected to match arXiv preprint.