- Update abstract and coverage table: computational evidence now includes n=12
(previously n=6..11). All iso-classes remain reachable.
- Correct conjecture statement: minimum degree ≥5 (not ≥4).
- Add graphicx package (for potential figure support).
- Add exploratory experiment files for exception characterization, preimage
search, and visualization (directed toward understanding the full orbit
of the T*_9 case and related structural questions).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Section 5.6 sketches an inductive route to the simple-resolution md4
surjectivity conjecture:
- Lemma 5.8 (good contraction): every md4 triangulation on n >= 7
vertices has a degree-4 vertex with an md4-preserving diagonal
contraction. Empirically true at n=7..11; proof obligation called
out.
- Lemma 5.9 (lift): given a labelled preimage of the contracted
triangulation, reinserting the contracted vertex at the
diagonal-bounded quadrilateral yields a preimage of the original
triangulation. Proof obligation called out.
- Inductive scheme paragraph chains the two lemmas with the octahedron
at n=6 as the base case, citing the n=7 hand-verification (already
scripted in experiments/inductive_lift_check.py).
Lemmas are stated without proof; the three remaining proof
obligations are explicit.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- Generalize Phase 1 to include even interior faces as optional flip
candidates and allow the source-triangle break in $L_0$ to be skipped;
generalize Phase 2 so even outer-incident cycles may have at most one
outer-face edge flipped (odd cycles still must have one).
- Define "simple level resolution" as a triangulation $G'$ obtained from
some $(G, S)$ via the algorithm with bipartite parity subgraphs
(Definition 5.4).
- Add Conjecture 5.7 (simple-resolution md4 surjectivity) and
Observation 5.6: every minimum-degree-4 plane triangulation iso-class
on $n \in \{6, ..., 11\}$ vertices is reached as a simple level
resolution. Counts: 1, 1, 2, 5, 12, 34. The md4 restriction is
necessary -- specific non-md4 iso-classes (iso 5 at n=8; iso 25, 183
at n=10) are not reachable.
- Add experiments/simple_level_resolution_coverage.py implementing the
branched algorithm and coverage check, plus supporting scripts for
Phase 1 cycling debugging, Phase 2 gap diagnosis, inductive-lift
scaffolding (inductive_lift_check.py for the route-1 proof strategy),
and visualizations of the unreached n=10 iso-classes and the original
Phase 2 gap example.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
- 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>
didericis