Eden's conjecture asserts that the maximal local Lyapunov dimension on a compact global attractor is attained at an equilibrium point or a periodic orbit (Kuznetsov and Leonov 2016). The unrestricted formulation does not require the attractor to have a dense orbit.
Colbrook (2026) reported a counterexample given by a polynomial system on 
, identified with 
. For 
and 
, the system is
 |  | ![[q(|z_1|^2)+i]z_1](/images/equations/EdensConjecture/Inline7.svg) |
(1)
|
 |  | ![[q(|z_2|^2)+isqrt(2)]z_2](/images/equations/EdensConjecture/Inline10.svg) |
(2)
|
 |  |  |
(3)
|
Its compact global attractor is 
.
The reported maximal local dimension is
 |
(4)
|
attained precisely on the invariant torus 
. Here 
denotes the Kaplan-Yorke
dimension obtained from the ordered Lyapunov
characteristic exponents. The angular velocities on 
are 1 and 
, so 
contains no periodic orbit.
The maximum over all equilibrium points and
periodic orbits is instead 3.
The same maxima and maximizing sets occur for the finite-time local dimension at every positive time. These dimensions use derivatives in the ambient space 
. The Hausdorff dimensions
of 
and 
are 4 and 2, respectively. The attractor is not transitive,
so the construction does not refute formulations restricted to transitive attractors.
Colbrook (2026) credits ChatGPT and Codex with obtaining the counterexample and assisting its Lean formalization. The author reports checking the proof and its formulation. As of Oct. 25, 2026, independent specialist review had not been reported.
