Sessions keep running across navigation: turn liveness + write-ahead persist

The bug: answering a quiz and navigating away mid-turn, then returning,
re-prompted the same question. Root causes: (1) the answer was durable only
at onFinish — every mid-turn view read the stale pre-answer record;
(2) the server had no notion of a turn in flight; (3) remount treated the
stale record as current, with no re-attach path. Worse, the stale dock let
the quiz be answered again, racing two turns into a last-writer-wins
saveMessages clobber.

Fix (four pieces):
- Write-ahead persist in chat-handler: the request body IS the client's
  canonical state — saved to disk BEFORE the stream starts. The answer is
  durable the instant the request arrives; crashes recover via the existing
  mount continuation.
- Turn liveness registry (src/server/turn-liveness.ts): in-memory chatId →
  in-flight map with a TTL leak guard; beginTurn rejects concurrent turns
  (409) so one turn ever owns a chat. endTurn runs in onFinish AFTER
  persistTurn, so pollers that see not-in-flight always read a finished
  record; a turn-start failure clears the registry (no lockout).
- GET /api/chats/[id] reports turnInFlight; the chat page passes
  initiallyInFlight to the client.
- Client re-attach (components/chat.tsx): while in flight, busy shimmer +
  composer gated + mount continuation suppressed; poll /api/chats/[id] until
  the turn finishes, then syncFromServer adopts the completed transcript.
  Stop stays local-stream only (a server turn has no client request to
  abort).

Verified by scripted-model tests: concurrent turn 409 + registry lifecycle,
write-ahead durability mid-turn, failed-turn lockout guard.

app/chat/[id]/page.tsx

@@ -1,7 +1,7 @@
 import { notFound } from "next/navigation";
 import { Chat } from "@/components/chat";
 import type { TeacherUIMessage } from "@/src/core";
-import { getChat } from "@/src/server";
+import { getChat, isTurnRunning } from "@/src/server";
 
 export const dynamic = "force-dynamic";
 
@@ -22,6 +22,9 @@ export default async function ChatPage({
 			title={chat.title}
 			initialMessages={chat.messages as TeacherUIMessage[]}
 			initialPrompt={prompt}
+			// A teacher turn that outlived its client: the remounted view must
+			// attach (busy shimmer + poll), never resurrect the stale record.
+			initiallyInFlight={isTurnRunning(chat.id)}
 		/>
 	);
 }