backend, web: support for resuming streams

docs/api/rest.md

@@ -39,6 +39,22 @@ Chat upload limits:
 ```
 - OpenAI model lists are filtered to models that are expected to work with the backend's Responses API implementation.
 
+## Active Runs
+
+### `GET /v1/active-runs`
+- Response:
+```json
+{
+  "chats": ["chat-id-with-active-stream"],
+  "searches": ["search-id-with-active-stream"]
+}
+```
+
+Behavior notes:
+- Lists in-memory chat/search streams that are still running on this server process.
+- Clients should use this after app start or page refresh to restore per-row generating indicators.
+- The lists are not durable across server restarts.
+
 ## Chats
 
 ### `GET /v1/chats`
@@ -260,6 +276,32 @@ Search run notes:
 - Persists answer text/citations + ranked results.
 - If both search and answer fail, endpoint returns an error.
 
+### `POST /v1/searches/:searchId/run/stream`
+- Body: same as `POST /v1/searches/:searchId/run`
+- Response: `text/event-stream`
+
+Events:
+- `search_results`: `{ "requestId": string|null, "results": SearchResultItem[] }`
+- `search_error`: `{ "error": string }`
+- `answer`: `{ "answerText": string|null, "answerRequestId": string|null, "answerCitations": SearchDetail["answerCitations"] }`
+- `answer_error`: `{ "error": string }`
+- terminal `done`: `{ "search": SearchDetail }`
+- terminal `error`: `{ "message": string }`
+
+Behavior notes:
+- The stream is owned by the backend after it starts. If the original HTTP client disconnects, the backend keeps running and persists the final search state.
+- While a search stream is active, `GET /v1/active-runs` includes the `searchId`.
+- If a stream is already active for the same `searchId`, this endpoint attaches to the existing stream instead of starting a second run.
+
+### `POST /v1/searches/:searchId/run/stream/attach`
+- Body: none
+- Response: `text/event-stream` with the same event names as `POST /v1/searches/:searchId/run/stream`
+- Not found: `404 { "message": "active search stream not found" }`
+
+Behavior notes:
+- Replays buffered events for the active in-memory stream, then emits new events until `done` or `error`.
+- Intended for clients that discovered a pending search via `GET /v1/active-runs`, such as after browser refresh.
+
 ## Type Shapes
 
 `ChatSummary`