didericis
c34c754c7e
Tested 10 closed PDS chains under SR (degenerate-inner T_1, varied
middle tires, outer-triangle T_n with m_n=3). In all cases:
- Forward state grows in the middle (widest tires), shrinks toward outer.
- Final state at outer-triangle L_n has size EXACTLY 6.
- Those 6 elements are EXACTLY the permutations of {1,2,3}.
- Outer-face dual-vertex constraint (degree-3, distinct colors) is
satisfied in every chain.
This is strong empirical evidence that under SR+PDS, the entire
chain-pigeonhole story closes for 4CT:
step 1 (saturation): proven
step 2 (pairwise): automatic from step 1
step 3 (chain consistency, open): always works
step 4 (closed with outer-triangle constraint): always works,
with the 6 outer-permutations as a clean attractor.
If this holds for all internally 6-connected G under SR (likely from
Birkhoff degree ≥ 5), it's a structural proof path for 4CT for
PDS-decomposable triangulations.
Remaining: prove SR holds for all internally 6-connected G; verify
exhaustively across more chains; find symbolic proof of the
"final state = exactly 6 permutations" attractor behavior.
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