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Theorem · cell-complex classification

CRD-RW-3a

The five-leaf rotation complex has nine native faces

Exact statement

The five-leaf binary-tree rotation complex K5K_5 has 14 vertices, 21 edges, six contextual pentagons, and three residual-interchange squares. Its nine faces are indexed by the proper contiguous clusters of the five leaves.

StatusProved for the schedule-quotient five-leaf complex
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

At five leaves the first independent rotations appear, so the native coherence geometry contains both pentagonal and square faces.

Definitions

  • K5K_5 is the rotation complex of planar full binary trees on five leaves after quotienting disjoint contraction schedules.
  • A proper cluster is a contiguous leaf interval of size two, three, or four.

Hypotheses and scope

  • Contextual associativity rotations and residual interchange defined by disjoint active-constructor occurrences.

Proof or evidence

There are C4=14C_4=14 trees and 21 rotation edges. A cluster of size rr supports Cr−1C5−rC_{r-1}C_{5-r} vertices: five for r=2r=2 or 44, and four for r=3r=3. The six size-two and size-four intervals give pentagons; the three size-three intervals give squares. The active-constructor classification shows that these exhaust the native elementary faces.

Face enumeration and completeness proof →

Verification notes

The Catalan product count and active-constructor completeness argument were reconstructed; exhaustive enumeration was used only as a finite cross-check.

Limitations

  • This is the tree-level schedule quotient, not the raw 24-history complex.
  • Leaf-interval disjointness alone is not the correct general interchange test.
  • No all-arity face-classification theorem is claimed.

Open work

Generalize the cluster-indexing argument without assuming the five-leaf enumeration.