didericis
2f82f6e0bc
For each of the 1,314 chord-apex+Kempe colourings on which Lemma
flank-covering-hex's conclusion empirically fails (the audit-revealed
sub-case (b)(ii) bad cases), classify the actual deciding face.
experiments/check_bad_subcase_deciding_face.py findings:
Deciding-face TYPE distribution (per colouring; multiple deciding
faces possible per colouring):
G-prime-face (= face of G' not modified by reduction): 7,872
outer (F_outer^♭): 1,236
flank-upper: 1,188
merged: 516
Per-colouring coverage:
G-prime-face available: 1,314 / 1,314 = 100.00% ← always
outer: 1,236 / 1,314 = 94.06%
flank-upper: 1,188 / 1,314 = 90.41%
merged: 516 / 1,314 = 39.27%
100% of bad colourings have at least one G'-pentagon (length 5) as a
deciding face -- i.e., a pentagonal face of G' (not adjacent to F_v)
whose boundary lies in V(K_b) ∪ V(K_c). This suggests the missing
piece is a "G'-pentagon fallback" lemma.
Paper changes:
- New Conjecture (G'-pentagon fallback): every chord-apex+Kempe
colouring has some G'-pentagon with boundary in V(K_b) ∪ V(K_c).
- Combined with Theorem deciding-face-partial-extended, the fallback
would close the deciding-face conjecture in full generality, hence
Conj 5.1 (face-monochromatic-pair). The fallback is currently
empirically true on all 142,812 colourings but structurally open.
- Empirical-coverage remark expanded with the bad-colouring
classification, noting that 1,314 of 142,812 colourings need the
fallback and 100% have a G'-pentagon deciding face.
Paper grows from 21 to 22 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"
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