NEW: chain_dp_joint.py — chain DP tracking full per-tire colorings,
edge-tuple-based parent/child sharing, and ground-truth comparison
against brute-force G' edge-coloring enumeration.
KEY EMPIRICAL FINDING (4th issue in chain_half_analysis):
When H_d is a tree (no internal cycles), the high-side cut tire
forest is EMPTY. The single H_d face is forced (by the level-set
lemma) to be entirely low-side or high-side; for a tree containing
the pendants, it's low-side. Hence high-side forest has 0 tires.
This happens at dodecahedron cut #0 side 0 (|S_0|=4):
- depths {0: 2, 1: 3}, |H|=6, |E(H)|=5
- H_1 is a tree, 1 face of length 6 (= low-side)
- No high-side cut tires
- DP gives R_dp=0, but ground truth R=36
DP correctly produces non-empty output on side 1 (where H_1 has
2 faces, one high-side), but the high-side framework's coverage
is incomplete for thin (small |S_i|) cuts.
This is a STRUCTURAL gap, not a code bug. The path forward
suggested in chain_half_analysis.tex: introduce a "boundary cut
tire" T_0 representing the low-side face + its pendants, so the
chain DP runs from leaves through T_0 to the cut.
Compounding with prior gaps:
(1) cut tires aren't always spoke-only (branched H_d faces)
(2) OUT-only projection loses S_3 orbit
(3) heuristic parent-finding (vertex overlap)
(4) tree H_d → empty high-side forest (this commit)
Net: the loose conjecture's chain half is genuinely open and
requires framework extension before the DP can be tested cleanly.
S_3 equivariance and high-side forest structure are the proven
pieces.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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