didericis
25cb109ab7
experiments/check_v_neighbour_degrees.py reports the cyclic degree
sequence of v's 5 neighbours in G across all (G, v, i) triples
underlying chord-apex+Kempe colourings (|V(G)| ≤ 20).
Result:
- 100.00% of (G, v) pairs have at least one neighbour of v of
degree 5. So the partial proof of the deciding-face conjecture
(Theorem deciding-face-partial, which requires n_k ∈ {5, 6} for
some k) handles all (G, v) pairs IF we can pick any k freely.
- 99.70% of (G, v, i) triples have min(n_i, n_{i+1}) ≤ 6, so the
partial proof's flank-face argument applies to the specific i.
- 24 / 7,930 (0.30%) triples have BOTH n_i ≥ 7 AND n_{i+1} ≥ 7
(the "bad" case where the partial proof's flank-face doesn't
work). These occur at n_G = 19, 20 for triangulations with
cyclic neighbour-degree sequences like (5,7,7,5,5).
For these 24 bad triples, the remaining 3 neighbours typically have
degree 5, so F^♭_outer (length n_{i+2} + n_{i+4} - 3 = 5+5-3 = 7,
not divisible by 3) becomes the candidate deciding face. Extending
the structural proof to cover F^♭_outer is the next step.
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.
Creating a New Paper
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./run.sh init_paper "Your Paper Title"
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Setup
The Python library code in lib/ requires SageMath. Run setup once per machine:
./run.sh setup <sage_python_path> <sage_site_packages> [system_name]
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Setup also compiles the plantri submodule via make.
Running Sage
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./run.sh sage <script.py> [args...]
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./run.sh sage
Linting
./run.sh lint
Runs pyright and pylint on lib/ using the SageMath Python interpreter.
Shell Completion
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eval "$(path/to/run.sh completion)"
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eval "$(./run.sh completion)"
Building
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latexmk -pdf paper.tex