didericis
36ed7bac38
Record the partition sweep on the n=24 Fig 2.10 dual. New subsection + experiments/bridge_partition_sweep.py. Findings: - A bridge switch is a constrained diagonal flip; bridge-derived via L means lying in an Even-Level-Graph component of the restricted flip graph. So the question is which flip-components contain an ELG. - Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each face has one within-pair edge), so total parity-subgraph Betti = (c_A+c_B)-2; intertwining trees are the Betti-0 case. - Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3; min is 1 (T not intertwining). All 27 partitions found bridge-derived (depth 2-3) have the minimum Betti 1 -> necessary. - But not sufficient: only 27 of 288 Betti-1 partitions yield a witness; the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase found none. The discriminator is flip-component structure (sharp orbit-size dichotomy), not a numerical invariant. Characterizing which Betti-minimal partitions sit in an ELG component is left open. Co-Authored-By: Claude Opus 4.7 (1M context) <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