didericis
8f0245aa3d
User observation: the cut tires can at most have a tree structure.
This is correct: each face of H_{d+1} lies inside exactly one face
of H_d in the planar embedding, giving a parent-child relation that
is a forest (rooted at depth-1 cut tires).
PROPOSITION: parent(T_{d+1}^{(f')}) = T_d^{(f)} where f is the
unique face of H_d containing f' in its interior. Well-defined and
unique because H_d's faces partition the plane minus H_d's edges.
CONSEQUENCE FOR CHAIN HALF: chain pigeonhole reduces to a tree-DP
problem. Process tires bottom-up from leaves; at each node, combine
with children via the in-spoke ↔ face-boundary-edge bijection;
at the root, R_i is the projection. Tree DP is well-understood;
counterexamples (if any) must come from tree-DP failures, which is
much narrower than general-graph compatibility.
EMPIRICAL CHECK on G'_1 of HM#0:
Root (1, 0): |f|=12, no children (outer shell).
Root (1, 1): |f|=4, deep substructure all the way to depth 7
with single chain of children.
EMPIRICAL CHECK on G'_0:
Root (1, 0): |f|=9, one depth-2 child.
Root (1, 1): |f|=9, no children.
In both cases the structure is a tree (= 2-root forest).
CAVEATS:
- The empirical parent test used a vertex-sharing heuristic that
gives ambiguous candidates in some cases (8 ambiguous in G'_1).
A rigorous test would use point-in-region containment via the
planar embedding's face structure.
- The proposition itself is uncontested; the ambiguity is just an
artifact of the empirical detection.
NEXT STEPS:
1. Prove the proposition rigorously via point-in-region.
2. Implement tree DP on the cut tire forest.
3. Bound |R_i| as a function of tree size.
Note: cut_tire_tree_structure.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"
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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