Proposition: Framed affine peripheral certificate

On every regular slice and on both one-sided incidence germs:

1. all regular local vertex words are affine identities;

2. core and collar meridians are primitive positive loops with reference-frame product $J_0$;

3. the outgoing six-face star has the same transported primitive product;

4. the packet is

   $$(\mathsf{p},\pi /2,{\mathcal{W}}_0,e_0,\mathrm{id},J_0),$$

   with consistent worldline and rest-vector transport.

Together with the realization checks, these witnesses give the complete matching certificate.

Proof. Every small regular vertex loop has winding zero, including a loop whose written path crosses the seam in opposite directions. Its affine word is identity by A:lem:word.

Write ${\mathcal{A}}_i={\mathcal{A}}_{V_i}$. The core word starting in $V_0$ is

The no-seam connecting path has transport ${\mathcal{A}}_0$, because all intermediate adapted charts cancel. Reference transport gives

The collar product is directly $J_0$.

Both meridians have angular lift increment $3\pi /2$. This proves positive primitive winding independently of the matrix product.

For the outgoing star, the chronological face order is

The five ordinary crossings and one seam crossing telescope to ${\mathcal{A}}_{U_0^{-}}{J}_0{{\mathcal{A}}_{{\mathcal{U}}_0^{-}}}^{-1}$. The chosen no-seam connecting path has transport ${\mathcal{A}}_{U_0^{-}}$, so the reference word is again $J_0$. Its angular increment is $3\pi /2$.

At $s=0$, the nonvanishing core meridian avoids collapsed slots and is common to both limiting descriptions. Outgoing star data at the event mean the certified one-sided family, not a regular loop on collapsed faces.

Finally, $J_0$ fixes ${\mathcal{W}}_0$ pointwise and fixes $e_0$ in its Lorentz part. In chart $f$, the worldline and rest vector are ${\mathcal{A}}_f({\mathcal{W}}_0)$ and $L_f{e}_0$, where $L_f$ is its linear part. Canonical transitions transport them consistently, because $J_0$ fixes the reference objects. The slice intersections are A:eq:particle. Segment matching and face closure follow from A:lem:charts; the finite word presentation was specified above. These verify (E1)–(E5).