The classification conjecture for number-conserving binary cellular automata

The conjecture asserts that binary number-conserving cellular automata with the von Neumann neighborhood are limited to the identity, shifts, and traffic rules, totaling $4d+1$. The earlier work established this only through dimension $3$ and left higher dimensions conjectural.

Known results

• Dimension $1$: the identity, shifts, and traffic rules give the predicted count.
• Dimension $2$: exactly $9$ rules.
• Dimension $3$: exactly $13$ rules, as established in earlier classification work.
• General dimension: necessary and sufficient conservation conditions were known, but the full count was not initially derived.

2019 general classification theorem

A later paper proves that, for every dimension $d$, there are exactly $4d+1$ such automata and that all are intrinsically one-dimensional. The result is independently restated in the PubMed record and a later survey, with no reported counterexample, proof gap, or retraction.

Current status (as of August 2026): The classification is settled for every dimension $d$ by the general theorem giving exactly $4d+1$ rules; no unresolved mathematical objection was found.