Local reversibility of reversible cellular automata

Definitions and mathematical statement

For a finite alphabet $A$ and integer $d\geq1$, a classical cellular automaton on $A^{{\mathbb{Z}}^d}$ is a map $F$ given by a finite neighborhood $N\subset{\mathbb{Z}}^d$ and a local rule $\varphi:A^N\to A$: \[ F(x)_v=\varphi((x_{v+u})_{u\in N})\qquad(v\in{\mathbb{Z}}^d). \] It is reversible when it is bijective and its inverse is a cellular automaton; on the full finite-alphabet shift, bijectivity yields this inverse-local-rule property. This standard reversibility notion is distinct from mere surjectivity.

Question. For every dimension $d\geq3$, is every reversible cellular automaton locally reversible?

The source does not define "locally reversible." If it meant only that the inverse has a finite neighborhood rule, it would repeat ordinary reversibility and would not explain the dimension-specific question. Possible stronger meanings involve decomposing the evolution into invertible local block operations, but such a definition would need to specify allowed block partitions, finite compositions or time regrouping, extra states/ancillas, and spatial recoding. None is supplied, and no particular circuit or block convention is silently selected.

The exact universal target is therefore clear in dimension and automaton scope, but its crucial conclusion cannot be uniquely formalized until the source definition is recovered. No claim of a proof in all dimensions follows from the standard inverse-rule fact above.

The supplied research summary labels this OPEN-TRIAGE, says it could not verify the $d\geq3$ conjecture with an authoritative citation, mentions alleged two-dimensional counterexamples, and names Richardson (1972) and Hedlund. It also equates injectivity, surjectivity, and invertibility in dimension one without sufficient qualifications; that equivalence is not adopted here. No definition or primary reference is supplied linking the purported two-dimensional phenomenon to the intended local-reversibility concept.

Short English statement

Is every reversible cellular automaton in every dimension $d\geq3$ locally reversible? Ordinary reversibility already gives a finite-rule inverse, so the unspecified word locally must mean a stronger property. The source supplies neither that definition nor an authoritative citation for the conjecture or lower-dimensional claims.

Sources

• Local UnsolvedMath registry, permanent ID 4900020, source code catalogue entry, "Local reversibility of reversible cellular automata"; no proposer supplied. Status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Wikipedia, List of unsolved problems in mathematics, Dynamical systems, current bullet 20 as extracted, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Dynamical_systems. Reported HTML extraction and access: 2026-07-29. The record says Wikipedia lists the item as unsolved but is not an authoritative resolution source. Rights: CC BY-SA 4.0; attribution and share-alike compatibility require release review.
• The summary names Richardson (1972) and Hedlund and reports two-dimensional examples without complete references or a definition of local reversibility. It expressly says authoritative confirmation was not obtained.