didericis
410c2a1b35
Replaces the earlier sketch with a more detailed two-stage proof:
Stage 1: BFS level-set lemma.
Lemma (BFS-adj): adjacent edges in line graph differ in depth by
at most 1. Proof: BFS-distance property.
Lemma (level-set): every face of H_d contains only edges of depth
<d, or only edges of depth >d. Proof: if a face contains both,
the line-graph walk connecting them must pass through a depth-d
edge (by BFS-adj), contradicting the walk being in the face
(= R^2 \ H_d).
Stage 2: faces of H_{d+1} embed in faces of H_d.
Key claim: no H_d edge sits strictly inside any face of H_{d+1}.
Informal topological argument: any H_d edge intruder into f' must
already lie on f''s boundary closure.
Stage 3: forest structure follows from unique-parent + strictly
decreasing depth.
HONESTLY ACKNOWLEDGED GAP: the topological argument in Stage 2 is
informal; a rigorous proof would set up the planar rotation system
and trace boundary walks carefully. Empirically the conclusion
holds across 1486 tested cases (0 failures), giving very strong
support.
Note grows to 5 pages.
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
To run a Sage script with plantri available on PATH:
./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
To enable tab-completion for run.sh in zsh, add this to your .zshrc:
eval "$(path/to/run.sh completion)"
Or source it once in the current shell session:
eval "$(./run.sh completion)"
Building
Papers are compiled with LaTeX. From within a paper directory:
latexmk -pdf paper.tex