docs(ai-chat): clarify lastEventId is sessionId-keyed across run boundaries

Clearing lastEventId on chat.endRun() looks intuitive (Run ended, cursor must be stale) but breaks the next-message subscription. The cursor is sessionId-keyed, not runId-keyed. Clearing it forces the next sendMessages to subscribe from seq_num=0, where it may hit the prior turn's still-durable turn-complete record and close the SSE empty before the new Run's chunks arrive.

Spells out the invariant in the frontend transport persistence table and adds a Warning in the chat.endRun() reference.

Author: ericallam

docs/ai-chat/backend.mdx

@@ -585,6 +585,10 @@ The current turn streams through normally, `onBeforeTurnComplete` / `onTurnCompl
 
 Use this when the agent knows its work is done (budget exhausted, goal achieved, one-shot response) rather than relying on the idle timeout. Unlike `chat.requestUpgrade()`, no `upgrade-required` signal is sent to the client, so there's no version-migration semantics.
 
+<Warning>
+  If you persist `lastEventId` to your own storage for cross-page-load resume, **don't clear it on `chat.endRun()`**. The cursor is sessionId-keyed and stays valid across Run boundaries — clearing it forces the next `sendMessages` to subscribe from `seq_num=0`, where it may hit the prior turn's stale `turn-complete` record and close the stream empty before the new Run's chunks arrive.
+</Warning>
+
 ### Runtime configuration
 
 #### chat.setTurnTimeout()

docs/ai-chat/frontend.mdx

@@ -113,7 +113,7 @@ Every chat is backed by a durable Session — the row that owns the chat's runs,
 | Field | Type | Notes |
 | --- | --- | --- |
 | `publicAccessToken` | `string` | Session-scoped JWT (`read:sessions:{chatId} + write:sessions:{chatId}`). Refreshed automatically on 401/403 via `accessToken`. |
-| `lastEventId` | `string \| undefined` | Last SSE event received on `.out`. Used to resume mid-stream after a reload. |
+| `lastEventId` | `string \| undefined` | Last SSE event received on `.out`. **Valid for the lifetime of the Session** — keep it across `endRun` / `requestUpgrade` / continuation-run boundaries; only clear when the Session itself closes. The cursor lets the next subscription open past the prior turn's stale `turn-complete` record. |
 | `isStreaming` | `boolean \| undefined` | **Optional.** The transport sets it internally, but you don't have to persist it — the server decides "nothing is streaming" via the session's [`X-Session-Settled`](/ai-chat/client-protocol#x-session-settled-fast-close-on-idle-reconnects) signal on reconnect. If you do persist it, the transport keeps the fast-path short-circuit. If you drop it, reconnects open the SSE and close fast on settled sessions. |
 
 ### Session cleanup (frontend)