didericis
626a837a06
Case A: K_1 visits the four shared vertices {p, p', q, q'} in the same
cyclic order as K_0. The K_0-arc A_1 from p' to q and the K_1-arc B_1
from p' to q share both endpoints, so they bound a "lune" face Φ* of
K_0 ∪ K_1 with two (b, c)-corners.
- Planarity: B_1's interior is non-shared, so B_1 lies on one side of
K_0; hence c-edges at p' and q (endpoints of B_1) point to the
SAME side of K_0.
- Lemma 5.2 along A_1 (m odd): c-edges at p' and q alternate, so
they point to OPPOSITE sides of K_0.
→ Contradiction.
This is clean and uses only Lemma 5.2 + planarity (does not require
the mod 3 face-sum from Step 3).
Case B: K_1 visits the shared vertices in opposite order
{p, p', q', q}. Then K_0 ∪ K_1's faces are 3-corner triangles instead
of lunes; both Lemma 5.2 alternations and the mod 3 face-sum hold
consistently. Listed as the explicit open case.
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