didericis
34141322ce
Adds the concrete construction (40 vertices, 60 edges, cubic + planar
+ proper 3-edge-coloured) on which h_φ is simultaneously constant on
two Kempe cycles sharing an edge:
- K_{red, blue} = 8-cycle (the outer frame): all h_φ = -1
- K_{red, green} = 12-cycle (outer frame + upper-left ladder side):
all h_φ = -1
- They share the colour-red edge (0, 7) (and others).
The graph is drawn in TikZiT and stored as
papers/face_monochromatic_pairs/constant_heawood_counterexample.tikz
The Sage transcription + Heawood/Kempe verification + PNG renderer is
papers/face_monochromatic_pairs/experiments/counterexample_conj_5_5.py
Rendered PNG (with the four bent outer-face / trapezoid arcs matching
the tikz drawing) is at
papers/face_monochromatic_pairs/figures/no-two-constant-kempe-counterexample.png
Globally h_φ has 16 vertices at +1 and 24 at -1; the +1 vertices are
concentrated in the inner "tilted ladder" region, leaving the outer
and the K_{red,green}-extension all at -1. This is the structural
reason both Kempe cycles can be constant.
Also includes the TikZiT styles file default.tikzstyles defining the
red/blue/green edge styles used by the .tikz file.
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