§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
This page was written with AI assistance (Claude).