Empirical evidence (from check_r2.py + classify_r1_pinches.py over
maximal planar graphs in n ∈ [7, 11]):
Total components: 47,253
Total claimed (R1) pinches: 1,319
All 1,319 pinches are at level-(d+1) vertices that are cut-vertices
of O = G[V_{C'} ∩ L_{d+1}].
ZERO level-d pinches (which would have been a problem for B_out).
Under the relaxed Definition 1.5 (where B_in is the outer-face
boundary closed walk of O, not necessarily a simple cycle), cut-
vertices of O are naturally accommodated: the closed walk visits the
cut-vertex multiple times. So what I previously called "(R1)
violations" are not obstructions at all — they're just structural
features of O that the closed-walk B_in captures.
Changes:
- Lemma 1.7: dropped (R1) hypothesis. Lemma is now unconditional
(modulo the BFS-on-the-outer-face embedding choice already in the
setup).
- Proof: rewritten boundary-structure paragraph to describe the
cut-vertex case naturally instead of citing (R1).
- Definition 1.5: removed the "2-manifold" assertion (since R is not
a manifold at cut-vertices of O); added an explicit note that R may
fail to be a 2-manifold at cut-vertices and that the closed walk
B_in visits them multiple times.
- Remark 1.9 (was rem:R1-when): rewritten as "no extra hypotheses
needed", documenting that both cut-vertex / multi-arc structures
and multi-hole topology are already accommodated by Definition 1.5.
Adds experiments/check_r1_concrete.py and
experiments/classify_r1_pinches.py for verification, plus
experiments/draw_r1_candidate.py and r1_candidate.png showing the
n=9 bowtie example concretely.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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