Proposition: Fixed-presentation transition uniqueness

In a fixed presentation, two distinct proposed values for the same required transition cannot both be accepted. This is not uniqueness of transitions from peripheral holonomy alone.

Proof. Since ${\mathcal{A}}_f$ is invertible, (eq:ambient-matching) requires

Both accepted proposals must equal this map.

For the qualification, changing face coordinates by invertible maps $B_f$ changes transitions to $B_g{t}_{g\leftarrow f}{B_f}^{-1}$ and changes a based peripheral word by conjugation at its base. If that base map is the identity, the peripheral word is unchanged despite changes in individual transitions. Such a change is excluded only after the presentation has been fixed.