Theorem: Principal persistence theorem

For every $\mu \in I$, formulas (eq:coefficients), (eq:rays), and (eq:family) construct a spinless conical slicing family with the following properties:

1. all affine spatial branch gradients have norm at most ${\lambda }_{\ast }<1$, uniformly in $\mu$;

2. graphs descend through rays and the seam, are embedded, and are globally acausal;

3. regular-face induced metrics are at least $1/9$ times the corresponding spatial metrics;

4. the family is strictly ordered by (eq:ordering);

5. exactly one event of the declared cellulation occurs, at $s=0$, changing the particle link from three to six;

6. every regular slice, the common limiting slice, and both fixed-deficit incidence germs have complete certificates (E1)–(E5).

The particle intersection is (eq:particle-time). Primitive core, collar, and outgoing-star maps are $J(\mu )$ in the reference framing and $AJ(\mu )A^{-1}$ in the declared framing.

At $\mu ={\mu }_0$, all displayed data specialize to the reference formulas in Appendix app:reference.

Proof. Lemmas lem:gradients, lem:bounds, and lem:embedding establish descent, uniform gradients, localization, embedding, the metric lower bound, acausality, ordering, and particle times. Lemmas lem:decomposition, lem:lengths, and lem:event establish the realized cellulation, nondegeneracy, and unique event.

Lemma lem:holonomy verifies endpoints, segments, closure, canonical transitions, regular local words, marked ordered words, and primitive core/collar agreement. Lemma lem:spinless verifies spinless packet matching and pointwise worldline fixation. The graph supplies the slice intersection, and the rest vector is the declared transported $\mathbf{t}$. These are all five checks.

A common $A\in {\mathrm{ISO}}^{+}(2,1)$ transports the spacetime, faces, rest splitting, and affine words coherently. Spacelikeness, embedding, and acausality are preserved. The metric comparison is with the transported spatial metric; it is not an assertion about an unrelated fixed Cartesian time splitting after a boost.

At the event, outgoing collapsed slots are germ metadata, while the limiting slice uses the active incoming triangles. The core meridian supplies its noncollapsing peripheral witness. Finally, substituting $\alpha =\pi /2$ proves the appendix identities.