Lemma: Collar realization

The collar module is embedded and nondegenerate on its full window, has no incidence event, and preserves the installed packet and reference-frame signature. Its embedded slice and particle intersection are unchanged.

Proof. By lem:bounds,

Thus $R_{\mathrm{col}}-r>1/8$ and $4-R_{\mathrm{col}}\ge 1/8$. Ring lengths are positive because $0<\alpha <\pi$. The horizontal areas are

both positive. The two refined faces partition exactly the original horizontal face, so their union has not moved.

All new faces have the identity reference chart. Ordinary transitions are identities and seam transitions remain $J(\mu )$. Shared segments, regular words, and primitive core–collar matching are therefore unchanged. Since $s_a$ is fixed, so is $P_{s_a}$. These checks give the complete matching output.