didericis
a8af5612c4
Concrete picture demonstrating "low-side faces span multiple
parent faces of H_{d-1}":
- H_{d-1} drawn as the outer blue circle.
- H_d as a smaller nested orange circle inside.
- Face A of H_d (high-side, inside inner cycle): a small disk,
sits entirely inside face X of H_{d-1}. Unique parent. ✓
- Face B of H_d (low-side, outside inner cycle): RED REGION
spanning across the H_{d-1} cycle. It is one connected face
of ℝ² \ H_d, but it intersects BOTH face X (annulus between
cycles) and face Y (exterior of H_{d-1}). Neither X nor Y
contains all of B → no unique parent. ✗
This makes the uniqueness step's failure visible: the forest
proposition's containment argument works for high-side (= face A,
nested inside) but fails for low-side (= face B, spanning across).
The motivation section of the note now has 3 figures:
1. Low-side spans uniqueness failure (this commit, page 1)
2. T_∂ thick-H_1 hexagon (page 2)
3. T_∂ thin-H_1 tree (page 2)
Plus the nested-cut-tires figure on page 3.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>