Concrete empirical example added to boundary_cut_tire.tex (page 2):
HM_0 cut #1 side 1, d=2:
- H_2 has 3 faces (lengths 4, 4, 12).
- H_1 has 3 faces (lengths 4, 4, 12).
- The length-12 H_2 face is low-side (contains pendants + H_1
edges in its interior).
- Adjacent H_1 edges come from ALL THREE H_1 faces:
H_1 face 0: edge (15,19)
H_1 face 1: edge (17,21)
H_1 face 2: edges (23,27), (28,33), (24,29), (28,34)
- No single H_1 face contains all of them → no unique parent.
This is a genuine empirical case, not a schematic. The figure
(uniqueness_break_example.pdf) shows the planar embedding from
Sage with:
- Orange = H_2 face 2 boundary (12 edges)
- Green / purple / blue = H_1 edges grouped by their H_1 face
- Gray = pendants (d=0) and depth-3+ edges
- Red dots = pendant vertices
Two new scripts:
- find_uniqueness_break.py: searches for empirical cases
- draw_uniqueness_break.py: renders the figure using Sage's
planar embedding
Note grows to 6 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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