didericis
c2961b1833
Adds Proposition 1.7 (Source-side simple-cycle property): in any
maximal planar graph G with single-vertex source v_0, for any vertex
v at level d and any connected component C' of G'_d incident to v,
the depth-d faces of F_{C'} at v form a single contiguous arc in v's
rotation in Pi_G. Equivalently: the source-side boundary of R_{C'}
is always a simple cycle in L_d, with no cut-vertices at level d.
Proof: contradiction via Jordan curve theorem. If two arcs of
depth-d faces at v exist, pick level-(d-1) neighbours p, q of v in
the two gaps. The BFS ball G[L_{<d}] is connected so admits a simple
path P from p to q. The closed walk v -> p -> P -> q -> v is a
simple cycle W (since v is the only vertex at level >= d on it). W
separates the plane into two regions; the two arcs at v lie on
opposite sides. Any dual path of depth-d faces from one arc to the
other must avoid v on its intermediate faces, but consecutive
intermediate faces share edges entirely in L_{>= d}, so the dual
path stays on one side of W. This contradicts the endpoints being
on opposite sides.
Lemma 1.8 (the tire-component lemma) now cites Proposition 1.7 to
justify that B_out is always a simple cycle. Level-(d+1) pinches
(cut-vertices of O) remain allowed and are accommodated by the
relaxed Definition 1.5.
The empirical search (n in [7, 12], 47k + 276k = 323k components,
13k+ pinches all level-(d+1) cut-vertices of O, zero level-d
pinches) is now subsumed by the structural proof.
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