Theorem: Particle-supported slicing certificate

For the fixed spinless cone with

the family

defined sectorwise on the three identified quadrants, has the following certificate.

1. The formulas define one continuous single-valued graph on the cone, not three incompatible developments.

2. The slices are embedded and uniformly spacelike with $\lambda =2\sqrt{2}/3$, and strictly ordered with $c=1$.

3. There is exactly one incidence event, at $s=0$. The particle is in the changing incidence structure, and its link changes from three to six incident faces.

4. Vertices, faces, one-sided edge laws, finite edge lengths, positive face areas, and collapsed germ slots are given exactly in Section sec:cells.

5. All regular local affine products are the identity. A declared primitive positively oriented meridian has framed affine product $J$ on both germs and throughout the parameter window. In every changed affine framing, the conjugate holonomy fixes the corresponding developed particle worldline pointwise.

6. The compact patch matches one prescribed exterior through an annular collar disjoint from the worldline. The collar and the moving core have the same peripheral affine holonomy.

7. The particle packet is preserved. Particle realization transport and matter-name cell transport are separately defined.

8. The event guard and geometric output satisfy spectator-blind control. No occupancy test, coupling, or measurement is used.

9. The finite decoder reconstructs exact dyadic family members from symbolic data. Reports are auxiliary, and the event guard tests the stored integer $j=0$.

10. For every fixed finite original-name list, the forward event has an exact reversible implementation with retained history, uncomputed work, and identity action on internal matter.

Consequently this is a particle-supported $G$-rule certificate for Contract contract:G and Definition def:objects.

Proof. Item (1) follows from the matching values of $\rho ,h,k$ on all identified rays. Item (2) is Proposition prop:spacelike. Items (3) and (4) are Theorem thm:event and Propositions prop:lengths–prop:areas.

Item (5) follows from Lemma lem:product, Corollary cor:regular, Proposition prop:holonomy, the explicit six-face star product, and (eq:affine-conjugacy). Item (6) follows from $F_s=2+s$ for $\rho \ge 3$ and the common peripheral calculation.

Item (7) is the construction in Section sec:rule. Item (8) is Proposition prop:blind. Item (9) is the decoder construction and Proposition prop:decoder. Item (10) is Theorem thm:implementation. No new geometric assertion is required in this assembly step.