Three-phase session status: derived summaries for the picker

- core/phases.ts: Phase3 (probe/plan/teach) + pure derivations —
  phaseOfTree (point-in-time, from the stored plan tree: no nodes → probe,
  no occupied node → plan, occupied → teach) and phaseAt3 (monotonic
  transcript fold for the dividers: failed plan_init never marks a seam;
  phase only advances).
- contracts.ts: ChatSummary gains record-derived phase + pendingQuestion;
  ChatListItem adds liveness-merged busy.
- store/chats.ts listChats computes the record-derived fields (tree +
  pending question scan).
- api/chats route merges busy = turn running || running background job.

Lane crossing granted by master: contracts/store/api (schema arbitration).
Data-only inference: pre-plan planning reads as probe until the plan lands.

app/api/chats/route.ts

@@ -1,7 +1,16 @@
-import { createChat, listChats, titleFromPrompt } from "@/src/server";
+import { createChat, isTurnRunning, listChats, listJobs, titleFromPrompt } from "@/src/server";
+import type { ChatListItem } from "@/src/core";
 
+/** The session picker's list: record-derived summaries (phase, pending
+ *  question) + live liveness — a running turn OR a background job marks
+ *  the session busy (contracts.ts ChatListItem; the store can't see
+ *  liveness, the route is the merge point). */
 export async function GET() {
-	return Response.json({ chats: listChats() });
+	const chats: ChatListItem[] = listChats().map((c) => ({
+		...c,
+		busy: isTurnRunning(c.id) || listJobs(c.id).some((j) => j.status === "running"),
+	}));
+	return Response.json({ chats });
 }
 
 export async function POST(req: Request) {