didericis
030ca67afb
Adds orbit_decomposition note + script answering: do structurally
different (T_1, T_2) pairs share the same canonical orbits in
S_1 ∩ S_2?
Findings across the 23 step-2 pairs:
1. EVERY intersection is closed under S_3 color permutation
(structural sanity check, follows from color-symmetry of
proper edge 3-coloring).
2. EVERY S_3-orbit has size exactly 6, with one exception: the
constant orbit {(c,c,c,c) : c ∈ {1,2,3}} of size 3 at γ=4,
T_1=T_2=(4,-,SR). So |S_1 ∩ S_2| = 6 × (# S_3-orbits) almost
always.
3. The RAINBOW orbit (a,b,c,b,c,a) at γ=6 appears in 3 different
(T_1, T_2) pairs -- all with T_1 = (6, (0,3), SP). The two-
chord SP tire (6, (0,2)(3,5), SP) never produces the rainbow
orbit. So rainbow is associated with the antipodal-chord
topology, not the pair as a whole.
4. Other canonical orbits recur across structurally different
pairs. E.g. (1,2,1,3) at γ=4 appears in 7 of 12 tested
γ=4 pairs.
This upgrades the step-2 finding: the intersection isn't just
non-empty -- it has full S_3-symmetric structure, contains at
least one size-6 orbit in essentially all cases, and shares
canonical orbits across varied (T_1, T_2) pairings.
Files:
experiments/orbit_decomposition.py
experiments/orbit_decomposition_data.txt
notes/orbit_decomposition.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