didericis
cb6a79f799
Per user spec: instead of including the actual depth-(d±1) edges
incident to the face boundary, redefine the cut tire as:
- Face boundary walk of f (depth-d edges in H_d).
- For each vertex v on the boundary walk with degree-2 in H_d:
add a fresh vertex n_v and fresh edge {v, n_v}, labelled
"out spoke" if v has an incident depth-(d-1) edge in G'_i,
"in spoke" if v has an incident depth-(d+1) edge.
Result: each cut tire is intrinsically "cycle (or closed walk) +
labelled pendants," structurally isomorphic to the partial tire
dual D(T) from paper.tex. Pendants ↔ D(T)'s leaves, face boundary
↔ T'_ann.
This means propositions about D(T) (chromatic polynomial counts,
S_3-orbit structure, rainbow conjecture, etc.) apply verbatim to
each cut tire.
Updates:
- notes/cut_depth_label.tex: Definition rewritten, structural
remark added, table of spoke counts updated to match new defn.
- experiments/cut_tire.py: cut_tire_at() now computes labelled
pendants instead of incident edges; draw_cut_tire renders
pendant vertices (orange squares for out, green squares for in)
with edges offset toward parent-graph neighbor.
- notes/fig_cut_tire.png: regenerated.
Note grows to 6 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