One higher cell fills the arity-four loop

Exact statement

The unique loop in the arity-four transformation graph survives backtrack cancellation and residual-interchange squares. In the fixed presentation, attaching one 3-cell along the raw hexagon—or equivalently along the quotient pentagon—is necessary and sufficient to fill it.

StatusProved for the fixed arity-four component

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

Context

Connected histories can still support inequivalent transformations. This result identifies the first missing higher-dimensional relation.

Definitions

A residual-interchange square records two independent transformation moves performed in either order.

Hypotheses and scope

The raw $C_6$ graph, or its schedule quotient $C_5$ , from CRD-RW-2a.

Proof or evidence

The graph has cycle rank $E-V+1=1$ . At four leaves there is no pair of independent rotations and hence no residual-interchange 4-cycle to fill the generator. A single disk attached with degree $\pm1$ along the primitive cycle kills the loop; without a new 2-dimensional filler the loop remains.

Loop and cell-attachment proof →

Verification notes

The cycle-rank argument, the arity bound on independent rotations, and the raw-versus-quotient boundary words were checked against the preserved proof.

Limitations

Necessity and sufficiency are relative to this fixed one-skeleton.
The result does not prove coherence in all arities or uniqueness across presentations.
A one-way orientation of the overlap rule is not itself a higher cell.

Open work

Test whether the arity-four filler participates in a uniform all-arity presentation.