Lemma: Complete local matching and primitive words

The transitions (eq:transitions) match every shared developed segment, satisfy (eq:ambient-matching), and have inverse reverse transitions. Face polygons close and regular local vertex words are affine identities.

The loops $\rho =29/10$ and $\rho =15/4$ are primitive positive meridians, lying respectively in the core and resting collar. Their products, transported to the reference frame, are both $J(\mu )$. For $s>0$, the six-face particle star has the same primitive peripheral product.

Proof. An ordinary overlap shares one developed segment. On a seam overlap, $J(\mu )$ identifies the upper segment with the lower segment, including its height by descent. Applying the two face maps gives (eq:transitions) and (eq:ambient-matching). Reversing the overlap gives the inverse map.

Every polygon is realized by the displayed endpoint coordinates. Its oriented endpoint displacements telescope to zero. Shared endpoints coincide, including seam copies. In an ordered word, consecutive chart factors cancel, leaving (eq:word). A sufficiently small loop about a regular vertex has winding zero, so its word is the identity.

The bounds

place the core loop outside all inner regions and inside the cap, and place the collar loop in the resting region. Write ${\mathcal{A}}_i={\mathcal{A}}_{V_i}$. The core's chronological crossings give

Transport gives $J$. The collar charts are identities, so its chronological crossings are $I,I,J$, with product $J$. Both loops traverse all three sectors once and have angular lift increment $3\alpha =2\pi -\mu$, proving primitivity independently of matrix values.

For the outgoing star use $\rho =s$, where $s>0$. Since $a<1$,

so this loop lies strictly in the inner region. In angular order set

Its six-factor word is

It has one positive seam crossing, winding one, and transported product $J$. At $s=0$ the nonvanishing core meridian certifies the common peripheral signature of both germs.