Add walk-distance labelling to source-dual cut

Label each dual face by its distance, within the source-dual cut, from
the first entry's cap down tooth. Regenerate the seed 1 and seed 2 full
walk figures and metadata with the new labelling.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_cuts/experiments/full_walk/full_medial_tire_cut_walk_1.md

@@ -5,13 +5,24 @@
 - deep-embedded edges: 84
 - graph seed: 59
 - deep-embedded minimum degree: 3
+- chosen face: (8, 9, 19)
 - chosen source cap vertex: 24
+- root entry tooth: e2 (apex medial vertex = level-1 edge (19, 8))
 - recognised treads: 11
-- skipped treads: [(0, 'only 0 up teeth')]
+- skipped treads: [((0, 0), 'only 0 up teeth')]
 - removed source-dual edges: 29
 - annular/cap cuts: 12
 - up-apex cuts: 17
 
+- **source-dual cut is a tree: True** (56 dual faces, 55 edges, 1 component(s), acyclic=True)
+
+## Walk-distance labelling of the source-dual cut
+
+Each dual face (vertex of the source-dual cut) is labelled by its distance, within the cut, from the **cap down tooth of the first entry**: the triangular face `(8, 24, 19)` = `{source 24, edge (19, 8)}` (dual node 34).
+
+- maximum distance: 21
+- distance histogram (faces by distance): `{0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 2, 6: 3, 7: 3, 8: 4, 9: 4, 10: 4, 11: 4, 12: 3, 13: 3, 14: 2, 15: 2, 16: 1, 17: 2, 18: 3, 19: 4, 20: 2, 21: 1}`
+
 - dual cut figure: `full_medial_tire_cut_walk_1_dual.png`
 - tire cut grid: `full_medial_tire_cut_walk_1_tires.png`
 - combined PDF: `full_medial_tire_cut_walk_1.pdf`

papers/medial_tire_cuts/experiments/full_walk/full_medial_tire_cut_walk_2.md

@@ -7,7 +7,7 @@
 - deep-embedded minimum degree: 3
 - chosen face: (4, 12, 11)
 - chosen source cap vertex: 20
-- root entry tooth: e3
+- root entry tooth: e3 (apex medial vertex = level-1 edge (11, 4))
 - recognised treads: 3
 - skipped treads: [((0, 0), 'only 0 up teeth')]
 - removed source-dual edges: 20
@@ -16,6 +16,13 @@
 
 - **source-dual cut is a tree: True** (38 dual faces, 37 edges, 1 component(s), acyclic=True)
 
+## Walk-distance labelling of the source-dual cut
+
+Each dual face (vertex of the source-dual cut) is labelled by its distance, within the cut, from the **cap down tooth of the first entry**: the triangular face `(4, 20, 11)` = `{source 20, edge (11, 4)}` (dual node 15).
+
+- maximum distance: 17
+- distance histogram (faces by distance): `{0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 3, 7: 3, 8: 3, 9: 3, 10: 4, 11: 3, 12: 2, 13: 1, 14: 1, 15: 1, 16: 1, 17: 1}`
+
 - dual cut figure: `full_medial_tire_cut_walk_2_dual.png`
 - tire cut grid: `full_medial_tire_cut_walk_2_tires.png`
 - combined PDF: `full_medial_tire_cut_walk_2.pdf`