didericis
1513922dec
Attempts to prove item 1 (non-emptiness of state at L_n in closed
SR+PDS chains ending at outer triangle). Results:
PROVEN:
- S_3-closure preserved by chain propagation.
- State at L_n is either empty OR equals all 6 permutations of {1,2,3}
(the only non-empty S_3-closed subset of permutations).
- Non-emptiness propagates through intermediate tires under outward
PDS via step-1 saturation.
REMAINING GAP (conjecture, empirically true): state at L_{n-1}
intersects the "perm-paired" subset of T_n's σ_D-projection (the
σ_D values that pair with permutation σ_U). At the final step T_n
has m_n=3 < k_n, so saturation fails — chain state at L_{n-1} could
in principle lie entirely in the (non-perm-paired) parity-matching
σ_D's, but empirically doesn't.
KEY STRUCTURAL FINDING: for T=(3, k), the σ_D's paired with a
permutation σ_U equal exactly the (parity-matching σ_D's) ∩ (T's
σ_D-projection). Verified for k=3..10.
HONEST OBSERVATION: a structural proof of the remaining conjecture
(without invoking 4CT) would constitute a new proof of 4CT under
the SR+PDS modelling assumption. The chain-pigeonhole framework
reduces to this single reachability question.
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"
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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>
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Setup also compiles the plantri submodule via make.
Running Sage
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./run.sh sage <script.py> [args...]
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./run.sh sage
Linting
./run.sh lint
Runs pyright and pylint on lib/ using the SageMath Python interpreter.
Shell Completion
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eval "$(path/to/run.sh completion)"
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eval "$(./run.sh completion)"
Building
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latexmk -pdf paper.tex