Pursue the paper's medial pigeonhole programme (R_T restriction relation,
chain-pigeonhole conjecture) at the data level.
Findings: R_T (outer<->inner boundary necklace, one Kempe-balanced colouring)
is genuinely coupled, not a product of its projections. A uniform per-size
boundary-state family threading every tile EXISTS at n=9 (unique per size, the
balanced-block necklaces 0011/000011/012/00012 -- not monochromatic), but FAILS
at n=12: size-7 seams admit no universal state (|D[7]|=0; near-universal 0001112
realised on 210/211 boundaries, blocked by one tile). So the uniform "same state
everywhere" shortcut breaks once large odd seams appear and universals vanish as
the tile population grows; the per-interface pigeonhole choice is genuinely
needed. Pairwise gluability still holds, so this locates the conjecture's
difficulty rather than obstructing gluing.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis
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