didericis
401dc86853
NEW: Theorem 1.5 (Tire treads partition the bounded faces).
For a maximal planar graph G with level source S on the outer face,
the family of tire treads { R_{C'} : d ≥ 0, C' a connected component
of G'_d } supplied by the tire-component lemma partitions the
bounded part of |Π_G|:
(i) every bounded face of G lies in exactly one tread R_{C'};
(ii) distinct treads have disjoint interiors.
Proof: each bounded face has a unique dual depth d, hence its dual
vertex lies in G'_d alone, and within G'_d in a unique component C'.
By the tire-component lemma, that C' carries the unique tread
containing the face.
This is the first step toward a chain pigeonhole argument that
colorings extend across the nested tire treads induced by a level
source.
Paper grows to 10 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.
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