Before attempting to prove the loose chain pigeonhole conjecture
("|π(T)| ≥ 6" for every non-trivial cut tire), looked for
counterexamples and found TWO in the existing empirical data:
(d, face) = (1, 1): 1 out spoke, |π(T)| = 3, orbit size [3].
(d, face) = (4, 0): 1 out spoke, |π(T)| = 3, orbit size [3].
Reason: a cut tire with exactly k = 1 in/out spoke has σ ∈ {1,2,3}.
S_3 acts with stabilizer of size 2 on any single-color σ, so the
orbit has size 6/2 = 3, never 6. The "|π(T)| ≥ 6" claim is
automatically false for k = 1 tires.
For k ≥ 2: σ can use 1 color (size-3 orbit) or ≥ 2 colors
(size-6 orbit). |π(T)| ≥ 6 requires at least one multi-color σ to
extend, which is not automatic but typically holds.
Three refined conjectures proposed:
1. Restrict to k ≥ 2 spokes (avoids the trivial counterexample).
2. Weaken to "non-empty and S_3-closed" (very weak; needs the
chain composition to preserve non-emptiness).
3. Just describe orbit sizes 3 or 6 (no useful claim).
The two found counterexamples are at "side" faces in the chain;
they don't break the bottom-line chain pigeonhole because the main
chain runs through larger faces.
To find harder counterexamples: look for k ≥ 2 cut tires whose face
boundary forces all spoke colors equal (= |π(T)| = 3 with k ≥ 2).
Such examples might exist but weren't found in the current data.
Recommended next step: restrict the conjecture to k ≥ 2 and re-run
the empirical sweep.
Note: loose_conjecture_counterexamples.tex (3 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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