didericis
902db37b50
Adopts the k>=2 refinement of the loose chain pigeonhole conjecture
(per loose_conjecture_counterexamples.tex) and runs a broader sweep:
- All 6 Holton-McKay non-Hamiltonian 38-vertex cubic plane graphs.
- 3 candidate matching 6-edge cuts per graph (greedy search,
preferring matching cuts then balance).
- Both sides of each cut.
- All depths d >= 1.
- Brute-force enumerate proper edge 3-colorings (skipping cut
tires with > 14 edges due to runtime).
Results:
- 287 total cut tires examined.
- 154 with k >= 2 in/out spokes.
- 107 verifiable (≤ 14 edges).
- ALL 107 passed: |π(T)| >= 6 with at least one full S_3-orbit.
- 0 counterexamples found.
This is strong empirical support for the k>=2 form of the loose
conjecture's per-tire half.
The cut_depth_label note (now 7 pages) is updated with:
- k >= 2 restriction in the conjecture statement.
- Restriction rationale (k=1 trivially excluded).
- Status: empirical sweep + provable spoke-only case.
Files:
experiments/loose_conjecture_sweep.py
experiments/loose_conjecture_sweep_data.txt
notes/cut_depth_label.tex (updated)
Next step: the per-tire half is essentially provable for spoke-only
cut tires via Prop 1.13. The chain half remains the genuinely open
piece.
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