User observation: the cut tires can at most have a tree structure.
This is correct: each face of H_{d+1} lies inside exactly one face
of H_d in the planar embedding, giving a parent-child relation that
is a forest (rooted at depth-1 cut tires).
PROPOSITION: parent(T_{d+1}^{(f')}) = T_d^{(f)} where f is the
unique face of H_d containing f' in its interior. Well-defined and
unique because H_d's faces partition the plane minus H_d's edges.
CONSEQUENCE FOR CHAIN HALF: chain pigeonhole reduces to a tree-DP
problem. Process tires bottom-up from leaves; at each node, combine
with children via the in-spoke ↔ face-boundary-edge bijection;
at the root, R_i is the projection. Tree DP is well-understood;
counterexamples (if any) must come from tree-DP failures, which is
much narrower than general-graph compatibility.
EMPIRICAL CHECK on G'_1 of HM#0:
Root (1, 0): |f|=12, no children (outer shell).
Root (1, 1): |f|=4, deep substructure all the way to depth 7
with single chain of children.
EMPIRICAL CHECK on G'_0:
Root (1, 0): |f|=9, one depth-2 child.
Root (1, 1): |f|=9, no children.
In both cases the structure is a tree (= 2-root forest).
CAVEATS:
- The empirical parent test used a vertex-sharing heuristic that
gives ambiguous candidates in some cases (8 ambiguous in G'_1).
A rigorous test would use point-in-region containment via the
planar embedding's face structure.
- The proposition itself is uncontested; the ambiguity is just an
artifact of the empirical detection.
NEXT STEPS:
1. Prove the proposition rigorously via point-in-region.
2. Implement tree DP on the cut tire forest.
3. Bound |R_i| as a function of tree size.
Note: cut_tire_tree_structure.tex (4 pages).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
8f0245aa3d