New note cut_tire_chain_pigeonhole.tex (4 pages) examining what
chain pigeonhole looks like under the redefined cut tire (face
boundary + labelled pendants).
KEY CHANGE: each cut tire is now structurally isomorphic to a partial
tire dual D(T), so all results from paper.tex / rainbow_proof.tex /
worst_case_proof_sketch.tex / k9_surviving_partitions.tex transfer
directly without re-derivation.
ARGUMENT SHAPE:
Setup: min counterexample G', 6-edge cut, depth labelling, cut
tires at each depth.
Reduction: minimality ⇒ G'_i 3-edge-colorable ⇒ boundary
configurations σ_0, σ_1.
Layered: σ_i = π_out({T_1^{(i, f)}}). Chain compatibility:
out spoke of T_d ↔ face boundary edge of T_{d-1} via the
parent-graph correspondence.
Pigeonhole: if at each layer the projection support contains
S_3-symmetric structure, the chain propagates and forces
R_0 ∩ R_1 ≠ ∅.
WHAT'S NEW UNDER THE REDEFINITION:
1. Direct result transfer: cut tires LITERALLY are partial tire
duals; no translation overhead.
2. Cubicity restored: face-boundary vertices have degree 3 in the
cut tire (degree 2 in H_d + 1 pendant).
3. Combinatorial rigidity: cut tire data = face + degree-2 boundary
vertices + in/out classification.
WHAT STAYS OPEN:
(a) Chain pigeonhole at each layer: same conjectures (rainbow,
König-lift) gate the argument.
(b) Chain well-definedness: trivial H_d faces (length 2),
degree-> 2 boundary vertices not getting pendants.
(c) Depth-by-depth variability: no uniform bound on |π_out|
across depths.
ASSESSMENT: strict improvement over the previous cut tire definition
(no transfer overhead, cubicity restored), but the hard step
remains the same as in the partial-tire-dual framework.
Concrete next step: cut-tire analogue of tire_fiber_step2 — for
each Holton-McKay graph, build cut tire chains on both sides of a
6-cut and check R_0 ∩ R_1 empirically.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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