Adds a Motivation section to paper.tex explaining that the
quadrilateral sequencing is intended to support an inductive 4-coloring
of the underlying maximal planar graph, with ring completion as the
suspected obstacle.
Adds commentary.tex recording (a) why a pure pigeonhole argument is
unlikely to close the conjecture, (b) the observation that under any
strictly local online rule every G'-edge constraint is enforced when
its second endpoint is colored (so ring completions cannot fail at the
moment they fire), and (c) the empirical finding that pure greedy
fails at non-ring-completion moves on every 3-connected triangulation
of order 5-7.
Adds quad_sequence_coloring_check.py, an enumeration check over small
triangulations via Sage's planar_graphs that runs greedy online
4-coloring under the canonical sequence and classifies failures.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
Extend the deep embedding to include the outer face, decompose into
quadrilaterals via level-edge pairing on the sphere, and define a
deterministic sequence built from four moves (anchor drop, level add,
join, ring completion) with a recursive lex-smallest tiebreak on the
initial quadrilateral. Attempt the termination theorem and the per-move
case analyses.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis