Theorem: Complete matching for the explicit family

Every regular member and both one-sided incidence germs of Section sec:family have an accepted certificate.

For a closed ordered chart word $\gamma$, based in $f_0$,

Regular small vertex words have $N=0$. Primitive positive core, collar, and outgoing-star meridians have $N=1$, and their reference-frame peripheral map is exactly $J_{\mu }$.

Proof. The graph and cell decomposition already establish geometric realization. Each polygon boundary closes by telescoping its vertex differences. Across ordinary overlaps the ambient map is the identity before face adaptation; across the marked seam it is $J_{\mu }$. The descended graph values ensure that endpoints and shared segments match under these maps. Conjugating by the specified face charts gives (eq:transition), including inverse reverse maps.

Multiplication of successive transitions cancels all intermediate chart factors, proving (eq:holonomy). An angular lift of a regular small loop has zero net seam crossing. A loop crossing the three sectors once has angular increment $3\vartheta =2\pi -\mu$ and winding one.

For a noncollapsing core loop choose constant

which lies between every inner dominance region and the cap. For the collar choose ${\rho }_{\mathrm{collar}}=15/4$. Both wind once, and both lie in regions with the explicitly given charts. For $s>0$, a sufficiently small loop around $P$ meets the six inner faces once in cyclic order and also has one positive seam crossing. At zero the core loop remains noncollapsing and witnesses the common peripheral signature.

Transporting the products back by the base chart gives $J_{\mu }$ for each primitive meridian. The product lift preserves the axial coordinate. Lemma lem:spinless, the declared rest data, and ${\tau }_P(s)$ prove (E5). The shared core–collar boundary is the same realized crest. These checks establish all five components.