didericis
7cd1f0d08d
New note cut_tire_chain_pigeonhole.tex (4 pages) examining what
chain pigeonhole looks like under the redefined cut tire (face
boundary + labelled pendants).
KEY CHANGE: each cut tire is now structurally isomorphic to a partial
tire dual D(T), so all results from paper.tex / rainbow_proof.tex /
worst_case_proof_sketch.tex / k9_surviving_partitions.tex transfer
directly without re-derivation.
ARGUMENT SHAPE:
Setup: min counterexample G', 6-edge cut, depth labelling, cut
tires at each depth.
Reduction: minimality ⇒ G'_i 3-edge-colorable ⇒ boundary
configurations σ_0, σ_1.
Layered: σ_i = π_out({T_1^{(i, f)}}). Chain compatibility:
out spoke of T_d ↔ face boundary edge of T_{d-1} via the
parent-graph correspondence.
Pigeonhole: if at each layer the projection support contains
S_3-symmetric structure, the chain propagates and forces
R_0 ∩ R_1 ≠ ∅.
WHAT'S NEW UNDER THE REDEFINITION:
1. Direct result transfer: cut tires LITERALLY are partial tire
duals; no translation overhead.
2. Cubicity restored: face-boundary vertices have degree 3 in the
cut tire (degree 2 in H_d + 1 pendant).
3. Combinatorial rigidity: cut tire data = face + degree-2 boundary
vertices + in/out classification.
WHAT STAYS OPEN:
(a) Chain pigeonhole at each layer: same conjectures (rainbow,
König-lift) gate the argument.
(b) Chain well-definedness: trivial H_d faces (length 2),
degree-> 2 boundary vertices not getting pendants.
(c) Depth-by-depth variability: no uniform bound on |π_out|
across depths.
ASSESSMENT: strict improvement over the previous cut tire definition
(no transfer overhead, cubicity restored), but the hard step
remains the same as in the partial-tire-dual framework.
Concrete next step: cut-tire analogue of tire_fiber_step2 — for
each Holton-McKay graph, build cut tire chains on both sides of a
6-cut and check R_0 ∩ R_1 empirically.
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