didericis
e6880371ff
Two more diagnostics on chord-apex+Kempe colourings (n <= 18,
13,800 colourings) probing how thin the non-constancy obstacle on
V(K_b) is:
1. check_min_flip_structure.py
- Flip count on K_b drops as low as 2 (at n = 18, 12 colourings):
these have a single minority Heawood vertex on K_b. So the
structural obstacle has NO slack: proving "at least 1 minority
vertex on V(K_b)" is the bar.
- All n=14 colourings (216) have flip count = 8 exactly. At
larger n the distribution spreads.
2. check_minority_location.py
- For colourings with K_b flip count <= 4, identify the minority
Heawood vertices and tally where they sit:
v_n : 12.86%
A_{i+1} : 10.82%
A_{i+2} : 8.98%
A_i : 7.76%
A_{i+4} : 5.31%
A_{i+3} : 5.10%
"other" : 49.18%
- About half the minority vertices live on non-named vertices in
the rest of G'. No single named vertex is *always* the
minority. The obstruction is genuinely diffuse / global, not
anchored to a specific structural location.
These together imply that the structural proof of "h_phi non-constant
on V(K_b)" must be global (no local "this vertex must flip"
argument suffices) and handle the edge case where only one minority
vertex exists. Likely requires a topological / homological / global
counting argument.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
math-research
Personal mathematics research repository by Eric Bauerfeld. Papers are written in AMS-LaTeX using the amsart document class.
Papers
kempe_style_search_for_smaller_contradiction
Humans Suffice: A Novel Proof of the Four Color Theorem
An in-progress proof of the Four Color Theorem via a minimal counterexample argument. The paper builds on Kempe's 1879 strategy — establishing valid cases for vertices of degree ≤ 4, then extending the argument to the degree-5 case using properties of non-adjacent degree-5 vertices, merged subgraphs, and locked colorings.
plane_depth_labelling
Plane Depth Labelling
Early-stage paper. Title and author information set; content in progress.
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