Theorem: Holonomy reduction

Fix a realized marked spinless cone patch with punctured-disk topology, its packet, adapted charts, overlap components, based-loop package, and a presentation as in A:def:presentation. After checking endpoints, developed face closure, reverse maps, and prescribed transitions in (E1), the holonomy obligations are equivalent to:

1. identity products at regular relators;

2. one primitive positive peripheral product equal to $H_{\Pi }$;

3. a collar primitive product equal to the same framed $H_{\Pi }$.

Other based words follow from winding and transport. Particle/worldline matching additionally requires the supplied worldline, rest-vector, and slice-intersection data and the algebraic conditions of A:lem:spinless.

These word checks do not replace realization, spacelikeness, positivity, primitive-winding evidence, or admission to the spinless dictionary.

Proof. The finite-relator lemma gives every based product from (a) and (b). Condition (c) verifies the particular collar identification in the common frame. The spinless lemma determines the required pointwise-axis map from the supplied packet. Chart transport verifies the representations of the worldline and rest vector; the slice intersection is separately checked.

Conversely, the complete packet includes the specified regular, peripheral, and collar words. None of these matrix products controls a face gradient or proves embedding, so realization remains an independent prerequisite.