New note cut_tire_chain_pigeonhole.tex (3 pages) walking through the
chain pigeonhole argument applied to the cut-tire framework:
Setup: minimum counterexample G' to 4CT, 6-edge cut splits into
G'_0, G'_1, depth labelling gives chains of cut tires on each side.
Argument shape:
(1) Minimality ⇒ both G'_i have proper 3-edge-colorings.
(2) Restrict to depth-0 pendants → boundary configurations σ_0, σ_1.
(3) Glue iff σ_0 = σ_1; counterexample ⇒ no such matching.
(4) Layered description: each cut tire has inner/outer projection
constraints; adjacent tires share layers (outer of T_d =
inner of T_{d+1}).
(5) Chain pigeonhole: if each π_in is "large enough,"
R_0 ∩ R_1 ≠ ∅, contradiction.
What this needs to be a proof:
(a) Chain well-definedness: each H_d has ≥ 1 face, adjacencies
clean, no degenerate cases.
(b) Quantitative chain pigeonhole at each layer (= the rainbow
conjecture or König-lift conjecture from existing notes).
(c) Cut-tire-specific issues: H_d not cubic, face boundaries may
not be simple cycles, transfer of primal-tire results requires
verification.
Empirical chain on G'_1 of Holton-McKay #0: chain length 7,
irregular face structure across depths. No counterexample yet but
no proof either.
Net assessment: structurally sound reformulation, but inherits all
open conjectures from the existing approaches plus new technical
issues.
Concrete next step: extend tire_fiber_step2-style pairwise compatibility
check to the cut-tire setting and see if R_0 ∩ R_1 = ∅ empirically
on the Holton-McKay graphs.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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