didericis
cebe6e5dbd
Two diagnostic scripts probing the side-classification of c-edges at
K_b-vertices and their relationship to Heawood numbers:
1. check_heawood_side_correlation.py (first attempt)
- Defines "side" as connected component of H \ K_b.
- Result: K_b separates H into 2 components in 0% of cases, so
this notion doesn't capture the planar side. (Negative result --
kept for the record / so we don't redo it.)
2. check_heawood_local_side.py (correct version)
- Defines "side" locally via the planar CW embedding at v: c-edge
is on local RIGHT if, going CW from incoming K-neighbour at v,
we hit the c-neighbour before the outgoing K-neighbour; local
LEFT otherwise.
- Result on 625,200 consecutive K_b-pairs across 13,800
chord-apex+Kempe colourings (n in [12, 18]):
same h, same side: 0
same h, diff side: 372,456 (59.57%)
diff h, same side: 252,744 (40.43%)
diff h, diff side: 0
The empirical biconditional holds perfectly:
h_phi(v_0) == h_phi(v_1) <==> c-edges on opposite sides
This is "Lemma A" -- the corrected version of the proposed
orientation lemma. Equivalently: constant Heawood on a Kempe
cycle K forces the c-edges (off-K) to ALTERNATE inside/outside
of K along the cycle (not all on one side as I initially
conjectured).
This empirical result revises the spiral picture for Path 4: under
the Lemma 5.3 hypothesis of constant h on V(K_b) U V(K_c), the
c-edges alternate sides on K_b (and the b-edges alternate sides on
K_c). K_c must then cross K_b at every K_b-vertex it shares -- a
strong topological constraint we can now 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
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