Conjectured multidimensional and non-interval neighborhood criterion for non-idempotency and infinite order

Prove that for any dimension d ≥ 1 and any finite neighborhood S ⊂ Z^d containing the origin, a cellular automaton τ: A^{Z^d} → A^{Z^d} with a unique active transition p ∈ A^S is not idempotent if and only if there exists t ∈ S such that, with U = S ∩ (t + S), the restriction of p to U \ {t} equals the restriction of the translate of p by −t to U \ {0}. Additionally, show that τ is not idempotent if and only if τ has infinite order.

Background

The main theorem of the paper provides a complete idempotency criterion for one-dimensional cellular automata with a unique active transition when the neighborhood S is an interval containing 0, yielding a dichotomy: idempotent or strictly almost equicontinuous. Extending this characterization beyond interval neighborhoods and to higher dimensions remains a natural next step.

The authors propose a conjecture that generalizes their one-dimensional interval-based criterion to arbitrary finite neighborhoods in any dimension via a self-agreement condition on overlaps U = S ∩ (t + S). They also conjecture the dynamical consequence that non-idempotency is equivalent to infinite order. Examples in the paper indicate that the subshift with a single forbidden pattern is insufficient alone, motivating a structural approach based on overlaps of pattern translates.