didericis
e29836c78a
Adds a 'General method' paragraph to menagerie §6 describing how to
compute P_e(G, k) for any G = C_n + (matching of non-crossing chords):
P_e(G, k) = Σ_{(c_1,...,c_r)} N(C_n; forbidden(c_1,...,c_r), k)
where the sum is over chord-color assignments and N counts proper
k-edge-colorings of C_n subject to per-edge forbidden colors (= the
chord colors at adjacent chord endpoints). For each chord-color
choice the inner count is a transfer-matrix product on the polygon,
computed in O(n k^2) time, so the full polynomial is computable in
O(n k^{r+2}) time.
The method specializes to the closed form for θ(1, p, q) (r = 1) and
generalizes to any number of non-crossing chords. Verified against
Sage's chromatic polynomial of the line graph on:
- θ(1, 3, 3): 30
- C_8 + {(0,2), (3,7), (4,6)}: 6
- C_10 + {(0,2), (3,5), (6,8)}: 18
Note grows by ~1/2 page; still 5 pages total.
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