didericis
415c33cfc3
Replaces the informal Stage 2 argument with a rigorous one,
achieved by refining the proposition to high-side faces only.
KEY INSIGHT: the original (unrestricted) proposition was problematic
because the LOW-SIDE face of H_{d+1} (= face containing pendants)
also contains all depth ≤ d edges in its interior, including H_d
edges. Hence low-side H_{d+1} faces span multiple H_d faces.
The fix: restrict to HIGH-SIDE faces only.
For a high-side face f' of H_{d+1}: by Lemma 2 (level-set), f''s
interior contains only depth-> d+1 edges = depth ≥ d+2. Since
depth-d edges are NOT in this range, no H_d edge sits inside f'.
Therefore f' is contained in a unique H_d face (by partition).
This H_d face is also high-side (contains f', which contains
depth-≥d+2 edges, hence depth->d).
Result: high-side cut tires form a forest, rigorously. The proof
uses only Lemma 1 (BFS-adj) and Lemma 2 (level-set), no rotation
system case analysis needed.
Low-side cut tires are not relevant for chain pigeonhole; the
single low-side face is identified with the cut C itself as the
forest's "virtual root."
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