NEW NOTE: boundary_cut_tire.tex (3 pages)
NEW SCRIPT: experiments/boundary_cut_tire.py
CONCEPT: T_∂^(i) per side i = the unique low-side face of H_1
(= face containing all pendants) treated as a virtual root tire.
- Cycle = boundary walk of f_∂ (depth-1 edges)
- OUT pendants = depth-0 cut edges in f_∂'s interior
- IN pendants = depth-2 edges at boundary vertices going into
adjacent high-side faces
T_∂ adjoins the high-side forest as a boundary node: not parent
or child geometrically, but shares edges with adjacent high-side
tires (depth-1 boundary edges, depth-2 in-pendants).
The extended chain DP includes T_∂ and uses edge-sharing
compatibility with adjacent high-side tires.
EMPIRICAL RESULTS (vs. ground truth from brute-force enumeration):
Dodecahedron:
- cut #0 side 0 (|S|=4, H_1 = tree): MATCH 9=9 ✓
[previously high-side DP gave 0, framework failed]
- cut #3 side 1 (|S|=4): MATCH 9=9 ✓
- cut #4 side 0 (|S|=4): MATCH 9=9 ✓
- HM_0 cut #0 side 0 (|S|=4): MATCH 9=9 ✓
Thicker sides: |R_dp| < |R_ground| (DP over-restricts).
This is a separate issue (probably heuristic parent-finding
or shared-edge logic when multiple high-side tires interact),
not the coverage gap.
Some cuts have side too large for brute-force enumeration in
T_∂ (n_edges > 18 limit), marked 'bdy too big'.
KEY WIN: the coverage gap is closed for the thin-side case where
H_d is a tree. The boundary cut tire converts these from
"framework gives R=0" to "framework gives R = ground truth."
NOT YET CLOSED:
- Thicker sides where DP under-counts vs ground truth
(different sets, similar cardinality sometimes)
- Branched per-tire half (T_∂'s cycle can traverse edges twice)
- Strong per-tire extendibility conjecture
But the framework now has principled coverage on ALL sides,
not just those with cycles in H_1.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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