Fix turn-liveness registry across Next dev module instances (globalThis)

In Next dev (Turbopack), each route compiles as its own entry — a
module-level Map in turn-liveness.ts was a DIFFERENT instance in the page
render, GET /api/chats/[id], and POST /api/chat. The chat route held the
live turn (hence the 409), but the page render read an empty registry:
initiallyInFlight=false → mount continuation fired a send → 409 → the
'Something went wrong' banner on returning mid-turn. The registry now
lives on globalThis (the standard Next singleton channel) — one Map per
process regardless of module instances. Semantics unchanged (TTL leak
guard, begin/end round trip).

Browser-verified end-to-end (headless tester): answer quiz → navigate
away → return mid-turn → graded card (no re-prompt), 'Thinking…' shimmer,
composer gated, no error banner; teacher response auto-delivered when the
background turn finished. Polling GET /api/chats/[id] observed working.

src/server/turn-liveness.ts

@@ -5,14 +5,31 @@
  * can tell "the teacher is mid-turn — don't resurrect, don't start anything"
  * apart from "the record IS current".
  *
- * In-memory BY DESIGN: a process restart clears it, so there are no stale
- * flags to clean up — a crashed turn recovers via the write-ahead record +
- * mount continuation on the next visit. The TTL is a leak guard for the
- * pathological case where onFinish never fires (it runs even on aborts);
+ * Lives on globalThis, NOT a module-level Map: Next dev compiles each route
+ * as its own entry, so a module-level singleton is a DIFFERENT instance in
+ * the page render, GET /api/chats/[id], and POST /api/chat — the page would
+ * read an empty registry while the chat route holds the live turn (the bug:
+ * returning mid-turn showed the 409 banner instead of the busy state).
+ * globalThis spans module instances within the process; it is the standard
+ * Next.js singleton channel (same reason Prisma clients live there).
+ *
+ * Still ephemeral BY DESIGN: a process restart clears it, so there are no
+ * stale flags to clean up — a crashed turn recovers via the write-ahead
+ * record + mount continuation on the next visit. The TTL is a leak guard for
+ * the pathological case where onFinish never fires (it runs even on aborts);
  * chat turns are minutes at most, so 30 minutes is far on the safe side.
  */
 
-const inFlight = new Map<string, number>(); // chatId → startedAt (epoch ms)
+interface LhGlobal {
+	__lhTurnRegistry?: Map<string, number>;
+}
+
+function registry(): Map<string, number> {
+	const g = globalThis as typeof globalThis & LhGlobal;
+	return (g.__lhTurnRegistry ??= new Map<string, number>());
+}
+
+const inFlight = registry();
 const MAX_TURN_MS = 30 * 60_000;
 
 /**