Syracuse Map · 2-Adic Dynamics

Ghost Cycles
of the Syracuse Map

2-Adic Periodic Orbits and the Exceptional Set
k = 9 resolution
256 residues odd integers mod 2k
— ghost cycles
§1 — The Discovery

Extra modular cycles are real 2-adic orbits

Ghost cycles are not transient artifacts of modular reduction.

The Syracuse map S(n) = (3n+1) / 2v(3n+1) acts on odd integers. Its finite approximations — transfer matrices Pk on odd residues mod 2k — occasionally produce extra cycles beyond the expected fixed point {1}. These were long assumed to be artifacts of the truncation.

They are not. Exhaustive search through k = 36 (34 billion residues) and algebraic analysis through k = 200 show that every ghost cycle is the modular projection of a true periodic orbit on the 2-adic integers ℤ2 — with negative rational elements, persisting at arithmetic progressions of levels. The exceptional set E is infinite, with density ≥ 4%.

The animation above cycles through k = 9 (non-exceptional: only the gold fixed point) and k = 10 (exceptional: a 26-node ghost cycle appears in teal).

§2 — Key Results

What we proved

Operator Norm
‖ℒ‖ = ²⁄₃
The transfer operator on C(ℤ2odd) is bounded with norm 2/3, achieved at residues ≡ 2 mod 3.
Spectral Radius
ρ ≤ ½
All eigenvalues satisfy |λ| ≤ 1/2. Lower bound ρ ≥ 2−16/15 ≈ 0.4774 from the L=15 ghost family.
Lasota–Yorke Obstruction
✗ Hölder spaces
ℒ does not preserve any Hölder or Lipschitz space on (ℤ2odd, |·|2). Standard spectral gap methods cannot apply.
Density of E
≥ 10%
The exceptional set has density ≥ 10.0% from the product formula over materializing ghost types. Empirical: ≈ 10.2%.
§3 — Publication

The paper

Ghost Cycles of the Syracuse Map: 2-Adic Periodic Orbits and the Exceptional Set
Adam McKenna · March 2026
DOI 10.5281/zenodo.18949342 github.com/mysticflounder/collatz