didericis
688b944f95
Adds two explicit hypotheses (R1) and (R2) to Lemma 1.7 (tire-component
lemma) and tightens the proof to use them precisely:
(R1) R_{C'} is a topological 2-manifold with boundary; equivalently,
at every v ∈ V_{C'} the depth-d faces of F_{C'} incident to v form
a single contiguous arc in v's rotation in Π_G.
(R2) R_{C'} has at most two boundary components.
These rule out, respectively, pinch points (where C' wraps around a
vertex via a global path of depth-d faces, producing a non-manifold
region) and multi-hole topology (a "pair of pants" or worse, which can
occur when several disjoint depth->d lobes sit inside one depth-d
component).
The proof is reorganised into labelled steps:
1. Outerplanarity of the two level parts (via Lemma 2.6 of [1]).
2. Layer containment V_{C'} ⊆ L_d ∪ L_{d+1}.
3. Boundary edges are monochromatic in level (full case analysis
on the third vertex of the outside face f', using the
bounded-step property of δ).
4. Boundary components are simple cycles (uses R1: locally at any
boundary point R_{C'} is a half-disk, so the boundary walk
visits each vertex once).
5. Topological type: planarity + R1 + R2 forces disk or annulus
via the classification of compact orientable 2-manifolds with
boundary.
6. Tire structure: identifies the two boundary parts as the level-d
and level-(d+1) induced subgraphs, in either order.
Adds Remark 1.10 documenting when (R1) and (R2) hold or fail:
- (R1) fails iff there is a pinch vertex whose cyclic level sequence
around it enters and leaves {d, d+1} more than once.
- (R2) fails iff R_{C'} encloses two or more disjoint depth->d
sub-regions (the depth-<d region is always a single connected
BFS ball, so the multi-hole obstruction is always on the away side).
- Notes the d=0 single-vertex-source case satisfies both hypotheses
automatically (R_{C'} is the star of v_0).
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