didericis
d659bf40d5
For each chord-apex+Kempe colouring, record:
- |V(K_b)|, |V(K_c)|, |V(K_b) cap V(K_c)|, |V(K_b) cup V(K_c)|
- "Flip count" on each cycle: #consecutive pairs whose third-colour
edges lie on opposite local sides (= #same-Heawood pairs by Lemma A).
Results (n in [12, 18], 13,800 colourings):
- |V(K_b) cap V(K_c)| is NEVER 2 -- always >= 6. The two Kempe cycles
through merged share many vertices.
- Distributions of flip_Kb and flip_Kc are identical multisets
(consistent with b <-> c symmetry of the construction).
- But per-colouring, flip_Kb == flip_Kc only 39.65% of the time --
the symmetry is statistical, not pointwise.
- Max observed flip count is 20, never the maximum possible |V(K)|.
Consistent with h_phi never being constant on V(K_b) U V(K_c).
The substantial overlap of K_b and K_c (>= 6 shared vertices) means
the constancy hypothesis would impose simultaneous alternation
constraints from both cycles at every shared vertex -- the topological
"trap" the proof needs to exploit.
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
Use run.sh to scaffold a new paper from the AMS-LaTeX template:
./run.sh init_paper "Your Paper Title"
This creates a new directory (name derived from the title) containing a paper.tex pre-filled with the title and author.
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]
sage_python_path— path to the SageMath Python interpreter (e.g./opt/sage/local/bin/python3)sage_site_packages— path to SageMath's site-packages directorysystem_name— optional label for this machine (defaults tohostname -s); used to store per-machine env files as.env.<system_name>
On subsequent runs the paths default to whatever was saved in .env, so ./run.sh setup alone re-runs setup with the existing configuration.
Setup also compiles the plantri submodule via make.
Running Sage
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./run.sh sage <script.py> [args...]
Or to open an interactive Sage session:
./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 "$(path/to/run.sh completion)"
Or source it once in the current shell session:
eval "$(./run.sh completion)"
Building
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latexmk -pdf paper.tex