didericis
1ac80aa5cf
Pins Π_G at the start to be an embedding placing S on the outer face;
such an embedding exists for any single-vertex source. This collapses
the two-embedding split in the previous proof (one for Lemma 2.6,
another for the topological analysis of R_{C'}) into a single
embedding throughout, and removes the "in either order" ambiguity for
B_out and B_in:
- B_out = G[V_{C'} ∩ L_d]: the boundary of R_{C'} closer to S.
- B_in = G[V_{C'} ∩ L_{d+1}]: the boundary farther from S.
The outerplanarity step now cites Lemma 2.6 of [bauerfeld-pds]
directly (no embedding switch). The "tire structure" step pins the
orientation by S's position on the outer face.
Remark 1.9 (degenerate cases) updated: the orientation ambiguity is
gone, so we state the d=0 case has degenerate B_out and the d=D_max
case has degenerate B_in.
(R1) and (R2) remain — they are graph-theoretic and unaffected by
embedding choice (for 3-connected planar graphs the embedding is
essentially unique by Whitney's theorem, so changing the outer face
cannot untangle pinches or merge multi-hole topology).
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