Lower the depth of a ternary SUBLEQ step below eleven

Determine whether a ternary threshold network implementing one step of the SUBLEQ machine can have depth strictly below 11, beyond the separate lower bounds established for the read multiplexer and the branch predicate.

Background

The paper constructs a ternary threshold network for SUBLEQ whose standard form has depth 13 and whose folded form has depth 11. The folded construction combines address bits directly into several threshold units, reducing the depth of fetches and operand reads. The authors prove that a read multiplexer and the branch predicate each require more than one ternary threshold level in the relevant formulations, but these results do not establish a lower bound of 11 for the complete SUBLEQ step. The unresolved question is therefore whether a different ternary construction can reduce the overall depth below 11.

References

Other questions are open: whether a ternary step of SUBLEQ can have depth below $11$, since Lemma~\ref{lem:muxdepth} and Remark~\ref{rem:ternarydepth} bound only a read and the branch; how few records a self-reproducing stored network needs; and how much noise a reproducing run tolerates when the perturbations are correlated or drift, which the independence assumed in Section~\ref{sec:robust} excludes.

— Universal Construction and Exact Self-Reproduction in Ternary McCulloch-Pitts Networks  (2610.12251 - Norton, 8 Oct 2026) in Section 9, Conclusion (following the discussion of the fixed network's non-reproduction)