Definition: Matching packet and certificate

For a realized description and proposed transition maps, the evaluated matching packet is

It records the following checks:

1. Vertex and overlap matching. Proposed maps equal A:eq:canonical on the specified components; reverse maps are inverses; paired endpoints and segments match; developed polygonal boundaries close; and regular local vertex words are affine identities.

2. Particle matching. The proposed peripheral map equals $H_{\Pi }$ in the declared framing. The deficit, affine worldline, and future rest direction satisfy A:eq:packet-holonomy.

3. Marked holonomy words. Ordered chart words, overlap components, connecting paths, and their topological classes are supplied. A positive primitive particle meridian has winding one, independently of its matrix value.

4. Collar matching. Core and collar share their prescribed boundary realization and transported reference framing. Their primitive peripheral words equal the same framed $H_{\Pi }$.

5. Worldline data. Every chart describes the transported singular worldline and rest vector consistently. The peripheral map fixes the worldline pointwise, and the declared slice intersection belongs to that worldline.

A complete certificate $\mathsf{Cert}(g)$ contains finite witness tables, exact equalities, winding evidence, and realization verification. It is accepted only when every applicable geometric and matching obligation holds.