The four-letter history graph is a hexagon

Exact statement

For complete occurrence-labelled reductions of $a^4$ , the graph generated by disjoint and overlap moves is the six-cycle $C_6$ . Contracting its unique disjoint-scheduling edge gives the five-cycle $C_5$ , the rotation graph of the five planar binary trees on four leaves.

StatusProved by complete enumeration and edge classification

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The smallest connected example with a loop already distinguishes concrete histories from their quotient by independent scheduling.

Definitions

The raw graph has complete occurrence-labelled histories as vertices.
The schedule quotient contracts edges that merely exchange independent contractions.

Hypotheses and scope

The fixed source $a^4$ in $R=\{aa\to a\}$ .

Proof or evidence

The six orders of the three original gaps give the six histories. Adjacent transpositions give a Cayley cycle $C_6$ . Exactly one edge represents disjoint scheduling; its contraction identifies the two histories with the same merge tree and leaves the five-tree associahedral cycle $C_5$ .

Six-history enumeration →

Verification notes

All six histories and their edge types were reconstructed from the preserved enumeration. The raw hexagon and quotient pentagon are kept separate.

Limitations

This is a fixed arity-four calculation.
The quotient changes the presentation even though both graphs have one independent loop.

Open work

Retain the raw and schedule-quotient graphs as distinct presentations in later coherence arguments.