Geometrically Constrained Tree Search

complete A₂-sliced census

Long runs expose hidden tilers—and narrow hard survivors

We enumerated every connected union of 9 and 10 consecutive-layer A₂ alcoves in the chosen rigid-lattice model, then used exact translational-quotient screens to remove the easy periodic cases. The size-10 pass alone checked 98,537 tiles. A complete determinant-12 campaign then exhausted all 455 eight-copy quotients for eight recovered size-9 leads: two have exact periodic tilings, while six survive every quotient in that screen. A subsequent determinant-15 campaign found a third hidden tiler: size-10 tile 36194 has a replayed nine-copy translational quotient. For recovered size-9 leader 04636, exhaustive substitution screens now also reject every connected five-copy parent at scale 2 in both symmetry models: 17,707 proper and 931,637 reflected metatiles. The same complete reflected screen now closes leader 01085 across 68,758 proper and 1,109,220 reflected five-copy metatiles, with a fresh geometric replay of every obstruction. A ten-million-node continuation also closes most of the hard nine-copy quotient classes for size-10 leaders 36141 and 35323: 81/85 and 77/85 orbit classes are now exact-negative.

This is a search success, not an aperiodicity claim: the survivors are hard benchmark cases whose larger periodic domains—or failure to tile—remain unresolved. Every positive quotient carries a replayed scale-2 quotient-cluster substitution; every reported five-copy rejection has a replayed atomic local obstruction.