Approach: at each vertex of a simple closed cycle C in a 3-regular
planar graph, define turn-sign(v) = +1 if the third edge (off-cycle)
is in C's bounded region (interior), -1 if exterior. Compute
Σ_v turn-sign(v).
Empirical check on standard graphs (K_4, Q_3, dodecahedron, 3-prism):
For a FACE boundary, Σ = -L_face (all third edges outside the face).
For a NON-face cycle, Σ can range from -L to +L.
Plan: under Lemma 5.2's alternation hypothesis (constancy on V(K_b)
forces third edges to alternate sides along K_b), the signs alternate
+,-,+,-,... yielding Σ = 0 for K_b of even length.
This shows K_b is NOT a face boundary (= it bounds a region containing
other vertices/edges), which is true but not a contradiction.
A simple closed planar curve can have Σ = 0; that just means equal
numbers of off-cycle edges are inside vs outside.
So the winding-number approach (option 4) does not yield a direct
contradiction under the chord-apex+Kempe + constancy hypothesis.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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