didericis
db245eecea
- Promote Prop 3.1 (outerplanarity of level subgraphs) to Theorem 3.1 with a proof by contradiction via a BFS-path argument; drop the $n \leq 10$ caveat and the now-resolved open question. - Add Section 5 "An edge-flip resolution algorithm": apex classification of $L_k$-edges, bridge lemma, cross-level flip pass, definition of tricky-everywhere odd cycles and facial depth (seeded from inner faces with $\geq 2$ outer-face edges), and the depth-guided flip procedure. Observation 5.5 records empirical termination at $n = 9, 10, 11$; Question 5.6 asks if it holds in general. - Add experiments/depth_monovariant_check.py (sanity check over triangulation iso-classes, confirms the count-of-tricky-faces monovariant strictly decreases per flip on all 1400 tricky configs at $n \leq 11$), viz_cycling.py and debug_cycling.py, and cycling_visualization.png illustrating the depth-definition fix. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Level Resolution Experiments
Computational investigation of a structural proof strategy for the four color theorem via level resolutions of maximal planar graphs.
See paper.tex for full definitions, conjectures, and findings.
Files
Core library
level_cycles.py— levels, level subgraphs, level cycles, resolution enumeration (used by old-definition coverage).triangulation_gen.py— vertex-insertion + flip closure (good to n=10).triangulation_gen_fast.py— WL-hash pre-filter for n ≥ 11.balanced_layout.py— Tutte-init random-search planar layout.four_color.py— level 4-coloring via parity 2-coloring of L_k.
Experiments
coverage_new_def.py— coverage under the cleaner definition: G' is a level resolution of G via S iff its parity subgraphs are bipartite. Reachability reduces to "G' admits a bipartite 2-partition with cardinality matching some BFS-realizable parity split."coverage.py,coverage_fast.py,coverage_chunked.py— coverage under the OLD (stricter) definition involving specific edge flips on level cycles.face_counting.py— per-target preimage counts (N_iso, N_paths) under the old definition.orbit_check.py— orbit-counting with k-flip reverse-preimages (used for old-definition icosahedron analysis).
Visualizations
plot_oct.py,n7_examples.py,four_color_viz.py.
Summary under the new definition
| n | iso-classes | reachable | md4 reachable |
|---|---|---|---|
| 6 | 2 | 2 | 1/1 |
| 7 | 5 | 5 | 1/1 |
| 8 | 14 | 14 | 2/2 |
| 9 | 50 | 50 | 5/5 |
| 10 | 233 | 233 | 12/12 |
| 11 | 1249 | 1249 | 34/34 |
| 12 | icosahedron | reachable | yes |
Every iso-class is reachable at every tested size. The previously "uncovered" classes T1 (n=7) and T6 (n=8) under the old definition are both reachable under the cleaner definition.
The new definition makes coverage equivalent to 4CT plus a BFS-realizable partition cardinality constraint, raising the question of what additional structure on the preimage G would make the framework non-circular.
Dependencies
pip install networkx matplotlib numpy scipy