Lemma: Local relators and one generator suffice

In a presentation satisfying A:def:presentation, suppose every regular two-cell boundary has identity ordered product and the marked primitive positive peripheral word has product $H$ in the common frame. Then every based loop of winding $w$ has product $H^w$. Conversely, this loop formula implies those local and peripheral conditions.

Proof. Deleting an edge followed by its reverse preserves the product. The identity boundary relators permit replacement of one boundary path of a regular two-cell by its complementary path without changing the product. The declared presentation makes these moves the required edge-path homotopies.

The punctured disk, or its annular truncation, has fundamental group $\mathbb{Z}$, generated by the marked positive meridian. A winding-$w$ loop is based-homotopic to its $w$-fold power, with negative powers represented by reversed loops. Product invariance gives $H^w$.

Conversely, regular boundaries have winding zero and the marked meridian has winding one.