Minimal local facet-rule description for the three-dimensional certified marking

Determine the minimal number of distinct facet labels needed to realize the certified three-dimensional frame-marking rule, and determine whether the 14 admissible relative poses that never occur in hierarchical tilings can be excluded by a local matching rule.

Background

The certified three-dimensional rule contains 135 facet-contact triples on the 24 panels of C_3 and admits 44 relative touching poses. Of these, 30 occur in hierarchical tilings, while 14 are admissible under the closed rule but never occur in hierarchical tilings.

The authors leave unresolved whether the full frame-derived rule can be compressed to fewer distinct facet labels and whether a purely local rule can eliminate those 14 non-occurring poses without changing the desired hierarchical behavior.

References

The 135 facet triples of $F*$ are what the full-frame marking produces; the minimal number of distinct facet labels (and whether the 14 admissible-but-never-occurring poses of Appendix A can be excluded by a local rule) is open.

— Towards Strongly Aperiodic Monotiles in Higher Dimensions  (2610.00916 - Kamenetsky, 1 Oct 2026) in Section 11, “Future work,” item 4