Lemma 52 ([The switch graph seen by a long window]). There is a uniform depth such that the following holds for every centered monotone window W whose two endpoints are at least that far from its sampling center and whose pointwise …

First justify the passage from a finite window to bilateral witnesses. A local state consists of a centered event, or a pair of events with centers at distance at most one, together with a fixed-size path neighborhood large enough for phase uniqueness. There are finitely many normalized states. More generally, fixing an actual regular horizontal marking, there are finitely many carried extensions of any prescribed finite depth: the relevant paths have finitely many orbits and their cores have finite edge quotients, while the marking has trivial stabilizer. Thus the extension sets form a finitely branching inverse system. If a given state occurs in arbitrarily deep window cores, choose compatible extensions successively. The resulting bilateral path carries its specified event or pair in every finite subwindow. Local finiteness of the horizontal tree preserves the full lines carrying those arcs. States that do not persist in this way have a maximum possible depth. Taking a maximum over the finitely many short states gives a uniform depth beyond which every state in question has a bilateral witness. The witness is initially required to agree only on the fixed-size neighborhood recorded in the state. Lemma 47 ([Phase uniqueness]), applied to the witness and the finite window, then forces its expanding ray to agree throughout the corresponding side of W . This last step is what permits use of a fixed-size state for an arbitrarily long W . Every vertex in the finite-window core admits a centered event. Assign its expanding color using a bilateral witness. Phase uniqueness makes the assignment unambiguous and equivariant under (W) . Adjacent same-sign labels agree: use a common bilateral witness for the pair. Adjacent opposite-sign labels give a switch through W . Connectedness of the core therefore connects all recorded colors in the switch graph. A recorded color either participates in a switch or is the unique color and is preserved by the noncyclic window group. In the switch case, periodicity and Lemma 51 ([Periodic switches and noncyclic active tails]) imply W⊂Y_z . In the unique-color case, the ray to that end extends W monotonically; the window group fixes that end and W , and hence its entire continuing tail. The tail stabilizer is noncyclic, so again W⊂Y_z . Conversely, suppose W⊂Y_z . Take events far along an expanding trajectory for z , and use periods and Lemma 48 ([Changing the sampling offset]) to bring them to the required levels. By the transitivity in Lemma 50 ([Branch states and tail groups]), match the starting edge on the side away from z . The path towards z from that edge is unique. Taking the original event far enough along its expanding part ensures that its comparison subwindow lies on this ray in Y_z ; it therefore realizes the desired event in the central core of W . The same argument applied to switch paths shows that a locally isolated occurrence has no global switch incidence. It remains to exclude a cycle. At the center, use the convex cores of the tail groups M_ z,e for the colors just obtained, with one fixed thickening. Nearby paired events are uniformly close to the corresponding tail cores. To verify uniformity, normalize their marked fixed-window data; there are finitely many types, each determining the actual end, and the tail group acts coboundedly on its core. Every switch through W has such nearby paired events by periodicity and Lemma 48 ([Changing the sampling offset]). Thus every switch edge gives an intersection of the two thickened cores. For sufficiently deep W , the thickenings of distinct same-sign cores are disjoint. Otherwise normalize a pair at bounded distance near one horizontal origin. There are only finitely many such actual lifts: use finite color orbits, edge transitivity on Y_z , and coboundedness of each tail core. Increasing depths then give arbitrarily long agreement of the corresponding rays, forcing equality of their ends by phase uniqueness. This contradicts distinctness. Finally, the intersection graph of subtrees of a tree is chordal, by convexity and the Helly property. With no same-sign intersections it is bipartite, hence a forest. The switch graph under consideration is a subgraph of this intersection graph; extra opposite-sign intersections cause no problem. It is connected by the preceding argument, and therefore is a tree.