didericis
56ebf49b48
Replaces the earlier proof sketch with a clearer attempt that:
1. Corrects the sharp threshold: m_1 >= m - 1 (not m_1 >= m).
Empirically verified for m ∈ {4, 6} across m_1 ≥ m - 1.
2. Proves the ⊆ direction cleanly: each O-face dual vertex has
degree m/2 ≤ 3 in T'_{f'}, so its incident spokes must be
pairwise distinct, putting σ in the "perms-per-half" set P_m.
3. Reduces the ⊇ direction to a cyclic 2-SAT solvability claim
(Conjecture 2sat): for each σ ∈ P_m, find an "orientation"
o ∈ {0,1}^m at the m D-positions such that each length-1 gap
has R_j = L_{j+1} and each length-2 gap has R_j ≠ L_{j+1}.
4. Acknowledges the gap: a naive "all-zero orientation" fails
(e.g. rainbow at m_1 = 6 has the all-zero attempt fail at
gap (4,5)). A satisfying assignment exists in every tested
case (6-18 per σ at m=6, m_1=6) but a clean general proof
awaits. Two routes outlined: S_3-equivariant case analysis,
or global implication-graph analysis.
5. Confirms sharpness with explicit forcing-propagation
counterexample at m=6, m_1=4 (rainbow not in π_D).
6. States provisional corollary: π_D = P_m at m_1 ≥ m - 1
(conditional on Conjecture 2sat); chain pigeonhole reduces
to π_U meeting P_m.
Honest about what's proven (the ⊆ half, the 2-SAT reduction,
the sharpness counterexample) and what's left (the 2-SAT
solvability proof).
Note: rainbow_proof.tex (4 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