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-registry.ts

@@ -326,6 +326,12 @@ export function killClaude(sessionKey: string, email: string): void {
   if (sc) sendFireToSidecar(sc, { type: 'claude:kill', id: nextId(), sessionKey });
 }
 
+// Interrupt the current turn but keep the persistent session warm (the "stop" button).
+export function interruptClaude(sessionKey: string, email: string): void {
+  const sc = findSidecarByName(`claude:${email}`);
+  if (sc) sendFireToSidecar(sc, { type: 'claude:interrupt', id: nextId(), sessionKey });
+}
+
 export function clearClaudeSession(sessionKey: string, email?: string): void {
   if (email) {
     const sc = findSidecarByName(`claude:${email}`);