chat: persistent Agent SDK session per chat — decouple worker from turn (Phase 1)

Root fix for orphaned background tasks: the platform drove Claude Code as a
one-shot `claude -p` per turn (stdin ignored, process exits at turn end), so
run_in_background/Monitor work — and its task_notification — had no live harness
to return to. Now each chat session runs ONE long-lived Agent SDK query() with
streaming input; turns are user messages pushed onto it, and the session stays
warm between turns.

- claude-manager: persistent `query({ prompt: AsyncIterable, options })` per
  sessionKey (bypassPermissions, --resume, mcp via extraArgs, CLAUDECODE stripped).
  Single consumer loop maps every SDK message → ChatEvent, incl. post-turn
  task_started / task_notification. interrupt() = stop-turn; abort() = kill-session;
  30-min idle GC.
- stream-parser: processMessage() (object-level, reused by the SDK loop) + task
  message handling. ChatEvent/ServerMessage gain task:started / task:notification.
- API: the sidecar event subscription is now SESSION-scoped (no longer unsubscribes
  on 'result'), so background events after turn-end still reach the client. First
  turn opens the session; later turns push onto it. handleStop → interrupt (keeps
  session warm); disconnect/deleteSession → kill.
- protocol/sidecar-registry/user-instance: claude:interrupt command + interruptClaude.
- client: render task:started / task:notification in the transcript.

Verified end-to-end through the real chat WS: a run_in_background task's completion
arrives ~6s AFTER the turn's result; multi-turn on one warm session works.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

src/servers/sidecar/claude/stream-parser.ts

@@ -118,6 +118,62 @@ function handleResult(
   callbacks.onEvent({ type: 'result', cost });
 }
 
+function handleSystem(msg: Record<string, unknown>, callbacks: StreamParserCallbacks): void {
+  const subtype = msg.subtype as string | undefined;
+  if (subtype === 'init') {
+    const sessionId = msg.session_id as string | undefined;
+    if (sessionId) callbacks.onSessionId(sessionId);
+  } else if (subtype === 'task_started') {
+    callbacks.onEvent({
+      type: 'task:started',
+      taskId: (msg.task_id as string) ?? '',
+      description: (msg.description as string) ?? '',
+      taskType: msg.task_type as string | undefined,
+    });
+  } else if (subtype === 'task_notification') {
+    callbacks.onEvent({
+      type: 'task:notification',
+      taskId: (msg.task_id as string) ?? '',
+      status: (msg.status as 'completed' | 'failed' | 'stopped') ?? 'completed',
+      summary: (msg.summary as string) ?? '',
+    });
+  }
+}
+
+/**
+ * Map a single already-parsed message object (a Claude Code stream-json message, or the equivalent
+ * Agent SDK message — same shapes) to ChatEvents. Returns false if the message type was ignored.
+ */
+export function processMessage(
+  msg: Record<string, unknown>,
+  state: ParseState,
+  callbacks: StreamParserCallbacks,
+): boolean {
+  const type = msg.type as string;
+
+  switch (type) {
+    case 'stream_event':
+      handleStreamEvent(msg, state, callbacks.onEvent);
+      break;
+    case 'assistant':
+      handleAssistant(msg, state, callbacks.onEvent);
+      break;
+    case 'user':
+      handleUser(msg, callbacks.onEvent);
+      break;
+    case 'system':
+      handleSystem(msg, callbacks);
+      break;
+    case 'result':
+      handleResult(msg, state, callbacks);
+      break;
+    default:
+      return false;
+  }
+
+  return true;
+}
+
 /**
  * Process a single NDJSON line from Claude Code's stream output.
  * Returns false if the line was skipped (empty or malformed), true otherwise.
@@ -136,32 +192,7 @@ export function processLine(
     return false;
   }
 
-  const type = msg.type as string;
-
-  switch (type) {
-    case 'stream_event':
-      handleStreamEvent(msg, state, callbacks.onEvent);
-      break;
-    case 'assistant':
-      handleAssistant(msg, state, callbacks.onEvent);
-      break;
-    case 'user':
-      handleUser(msg, callbacks.onEvent);
-      break;
-    case 'system':
-      if (msg.subtype === 'init') {
-        const sessionId = msg.session_id as string | undefined;
-        if (sessionId) callbacks.onSessionId(sessionId);
-      }
-      break;
-    case 'result':
-      handleResult(msg, state, callbacks);
-      break;
-    default:
-      return false;
-  }
-
-  return true;
+  return processMessage(msg, state, callbacks);
 }
 
 /**