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Construct a reversible 1D universally self-replicating cellular automaton

Construct a one-dimensional reversible cellular automaton that belongs to UniversalSelfReplicating_1, or equivalently prove the existence of a reversible 1D CA supporting universal self-replication under the paper’s minimal local self-replicating condition and local universality requirements.

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Background

The paper proves existence of universal self-replicators in 1D (via a modified von Neumann construction compressed into 1D) and discusses reversible CAs’ limitations regarding global universality. While reversible CAs exist that are universally self-replicating in higher dimensions (via embeddings), a true 1D reversible universal self-replicator is conjectured but not known.

Establishing this would deepen the understanding of reversibility constraints on replication and computation in minimal dimensional settings.

References

We conjecture: There is a reversible CA in one dimension which is universally self-replicating, or equivalently \textnormal{\textsf{UniversalSelfReplicating}_1 \cap \textnormal{\textsf{Reversible} is non-empty.

— Self-replication and Computational Universality (2510.08342 - Cotler et al., 9 Oct 2025) in Section 'Open problems and conjectures' (Effect of dimensionality of CAs)