Reduced C prime(1/6) diagrams satisfy the standard combinatorial curvature count

Interpret $D$ with the toolkit conventions. For a fixed prefix $p$, a literal word has a unique suffix after $p$, so the distinct pairs $(r,u)$ and $(s,v)$ in the original piece definition are distinct exactly when $r\mathrm{\ne }s$. Thus the piece conditions agree. Reversing one face orientation across a shared edge also identifies the two definitions of a cancellable face pair. The diagrams and their outer walks are identical, not replacements by minimum-area diagrams. If the symmetrised set contains the empty word, discard it: it labels no polygonal face and contains no nonempty piece, so this changes none of the hypotheses on $D$.

Choose an end disc block using [F5]. If the block tree has just one block, it is a disc because $D$ has a face. Otherwise a finite tree has at least two ends, as seen from the endpoints of a longest simple path. An end bridge gives a spur, whose incoming and outgoing boundary occurrences are inverse. Freely reducedness excludes such a pair unless it straddles the chosen start of the literal boundary word. Only the spur at that starting vertex can do this, so there is at most one end bridge. Some other end is therefore a disc block. This argument permits a whisker at the starting vertex; freely reduced does not mean cyclically reduced.

Apply [F3] and [F4] to this block, keeping the original word lengths. The block has the same face words and cancellation condition as in $D$, hence the same strict piece bounds. A one-face block is a zero-shell. For a multi-face block, [F4] supplies a shell with at most three internal arcs whose exterior arc avoids its attachment vertex internally. If the block is all of $D$, mark the boundary word's starting vertex instead and use the same two-shell argument. No minimum-area replacement is made.

Expand the suppressed vertices. The selected face remains the same face of $D$; its exterior arc is contiguous in the full outer walk by the attachment qualification in [F4]. Other blocks attach only at the selected block's attachment vertex, not along an edge, so its internal arcs remain internal arcs of $D$. That vertex is not internal to any such arc. Expansion therefore gives at most three maximal internal arcs of $D$, as required. In the zero-shell case the full face circuit is the contiguous excursion at the attachment.