didericis
b27a4d20a3
Attempts to prove the antipodal-chord rainbow conjecture; result is
nuanced:
1. Upper bound (proven cleanly): pi_D ⊆ {σ : (σ_0..σ_{m/2-1}) and
(σ_{m/2}..σ_{m-1}) are both perms of {1,2,3}}. This follows
from the proper-coloring constraint at the two O-face dual
vertices, each of degree m/2 in T'_{f'}.
2. Lower bound at m_1 ≥ m - 1 (constructive): every σ in the
above set extends to a proper edge 3-coloring of T'_{f'}.
Explicit construction at m=6, m_1=6. In particular rainbow
⊂ pi_D.
3. Counterexample at m_1 ≤ m - 2 (refutes original conjecture):
at m=6, m_1=4, the rainbow σ = (1,2,3,2,3,1) is NOT in pi_D.
Explicit forcing-propagation contradiction: two length-1
inter-D-position gaps on T'_ann force conflicting cycle-edge
colors at a U-position. Empirically |pi_D| = 18 (half the
full set) at m=6, m_1 ∈ {3, 4}.
REVISED conjecture: pi_D equals the full "perm-per-face" set
(containing the rainbow orbit) iff m_1 ≥ m - 1. The threshold
m_1 ≥ m - 1 is sharp. Verified for m=4 (all m_1 ≥ 3) and m=6
(m_1 ≥ 5).
Consequence: chain-pigeonhole at γ length m reduces to a smaller
overlap condition under m_1 ≥ m - 1. The case m_1 < m - 1
remains open -- pi_D still nonempty but the rainbow orbit is
missing; structural characterization of the surviving 18-element
support not addressed.
Note: rainbow_proof_sketch.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"
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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