didericis
702cbcecf7
For each cut tire on G'_1 of Holton-McKay #0 (HM cut: |S|=10, matching 6-cut), brute-force enumerate proper edge 3-colorings, compute the joint (σ_out, σ_in) projection, and check S_3-closure and orbit decomposition. Results (8 cut tires analyzed, 2 too big or trivial): d face |f| out in |E| #col |π| S3-cl orbit sizes 1 0 12 5 0 17 96 93 yes [3, 6^15] 1 1 4 1 0 5 6 3 yes [3] 2 0 7 4 3 14 126 126 yes [6^21] 2 1 7 4 3 14 126 126 yes [6^21] 3 0-2 2 0 0 2 3 1 yes [1] 4 0 4 1 0 5 6 3 yes [3] 4 1 8 2 1 11 24 21 yes [3, 6^3] 5 1 2 0 0 2 3 1 yes [1] 6 0 12 3 2 17 96 93 yes [3, 6^15] 7 0 2 0 0 2 3 1 yes [1] Findings: 1. S_3-closure is universal (structural, expected). 2. Orbit sizes are always 3 (constant) or 6 (generic). 3. Non-trivial cut tires have rich projections (e.g. d=2 has 21 size-6 orbits = 126 elements; d=6 has 16 orbits). Neither conjecture is DIRECTLY testable on this example: - Rainbow conjecture requires antipodal-chord SP face boundary structure. Our cut tires' face boundaries don't naturally have this shape. - König-lift conjecture requires both sides give γ-face partitions on a shared γ. Cut tires at consecutive depths share data via in-spoke ↔ face-boundary-edge bijection, not via γ-face partitions. What CAN be observed: cut tire projections are LARGE and S_3- symmetric (substantially looser than the rainbow case's 36-element prediction). A "loose conjecture" would say π(T) ≥ c · 6 with c depending only on |E(T)|, derivable from Prop 1.13 in paper.tex. Files: experiments/cut_tire_test.py notes/cut_tire_conjecture_tests.tex (3 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