didericis
f0bc82b88d
Adds a side-by-side comparison of the two proof attempts now in
the repo:
Approach 1 (cyclic 2-SAT, in rainbow_proof.tex):
Proves π_D = P_m (perms-per-half) for one antipodal-chord SP
tire when m_1 ≥ m - 1. Open piece: 2-SAT solvability
(Conjecture 1.5).
Approach 2 (König lift, in worst_case_proof_sketch.tex):
Proves |S_1 ∩ S_2| ≥ 6 for two adjacent SP tires sharing γ
when both chords are on γ. Open piece: T_2 induces a
γ-face partition (Conj t2-induces-partition).
Assessment: Approach 2 is more promising because (a) the hard step
is already proven (König's theorem), (b) it proves exactly what we
need (chain-pigeonhole non-emptiness, not the full π_D
characterisation), and (c) it directly explains the empirical
worst-case |S_1 ∩ S_2| = 6 = single S_3-orbit phenomenon.
Approach 1 still has value if we need finer control over π_D's
shape, but for just establishing non-empty overlap Approach 2
suffices.
Both approaches witness the SAME canonical 6-element worst-case
intersection (the rainbow S_3-orbit at γ=6 = the König-lifted
Latin S_3-orbit).
Recommended next move: attack Conj t2-induces-partition. Write
down the candidate induced γ-partition explicitly, verify it
computationally, then prove inclusion via transfer-matrix / fibre
lifting.
Note: two_approaches_comparison.tex (3 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