The bounded-face sum omits the outer face at outer-boundary vertices, so
restrict the gluing identity to interior vertices (where all cluster
interfaces live) and recover a colouring by carrying a single +/-1 label
on the unbounded face f_inf, giving Heawood's identity on the full cubic
dual for the Tait step.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add the two-sided cluster decomposition proposition: a vertex's full
Heawood face-sum splits as exactly one child-cluster contribution plus
one parent-cluster contribution (the at-most-two-clusters bound makes the
pairing binary and complete). Explain why this fails per-tire -- a vertex
on many same-depth tires has only a fragment of its face-star in any one
tire -- and recast the chain-pigeonhole and 4CT conjectures to nested
clusters with a cluster restriction relation.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define a connected tire cluster (union of same-depth tires joined by
shared vertices, transitive closure), prove same-depth tires meet only
in vertices, and prove every vertex lies in at most two clusters (one at
each of two consecutive depths) -- the bounded coarsening of the
unbounded per-vertex tire count.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Define a +/-1 Heawood face-labelling of a tire, its induced boundary
Heawood sequences and restriction relation, and interface compatibility
(0<->0, +1<->-1 = vertex face-sum vanishes mod 3). State the Heawood
chain-pigeonhole conjecture and a tire route to the Four Colour Theorem,
parallel to the medial programme.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Add a new paper stub referencing the nested tire decompositions paper,
with intro, Heawood bibliography entry, and an empty restrictions section.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
didericis