Lemma: Certificate for every successor

Every successor is an admitted label and advances the selected slice time by $\tau +a\delta >0$. Each marking is embedded and spacelike on the regular product. The same spatial cells and any fixed compact regular collar are used on every label. Original-name cell transport is the identity, permission decreases by one, and the complete peripheral affine word remains $J_{\mu }$.

Proof. The bounds and parity in (eq:labels) are preserved. A horizontal slice has induced metric equal to the spatial metric of (eq:cone). Formula (eq:slice-time) gives the positive increment. Horizontal time translation commutes with $J_{\mu }$; it creates no time or axial shift in the peripheral word. No spatial cell or conical parameter changes.