feat(brainstorm-server): 4h configurable idle timeout; close WS on shutdown

The companion shut down after only 30 minutes idle — too short for real
brainstorming, where a single question can sit far longer. And shutdown() never
closed upgraded WebSocket sockets, so an open browser connection could keep the
Node process alive after it was supposed to exit.

- Default idle timeout raised to 4 hours, configurable via BRAINSTORM_IDLE_TIMEOUT_MS
  and start-server.sh --idle-timeout-minutes (validated positive integer).
- Reported as idle_timeout_ms in the server-started JSON / server-info.
- shutdown() now destroys all client sockets so the process exits even with an
  open WebSocket.
- Watchdog check interval is configurable (BRAINSTORM_LIFECYCLE_CHECK_MS, default
  60s) so the lifecycle can be tested without minute-long waits.

Adds lifecycle.test.js (configured timeout reported; idle shutdown exits despite
an open WS — teeth-verified; the start-server flag). Wires ws-protocol,
lifecycle, and stop-server suites into npm test.

Closes #1237
Refs #1689

skills/brainstorming/scripts/start-server.sh

@@ -11,6 +11,7 @@
 #   --host <bind-host>    Host/interface to bind (default: 127.0.0.1).
 #                         Use 0.0.0.0 in remote/containerized environments.
 #   --url-host <host>     Hostname shown in returned URL JSON.
+#   --idle-timeout-minutes <n>  Shut down after n minutes idle (default 240 = 4h).
 #   --foreground          Run server in the current terminal (no backgrounding).
 #   --background          Force background mode (overrides Codex auto-foreground).
 
@@ -22,6 +23,7 @@ FOREGROUND="false"
 FORCE_BACKGROUND="false"
 BIND_HOST="127.0.0.1"
 URL_HOST=""
+IDLE_TIMEOUT_MINUTES=""
 while [[ $# -gt 0 ]]; do
   case "$1" in
     --project-dir)
@@ -36,6 +38,10 @@ while [[ $# -gt 0 ]]; do
       URL_HOST="$2"
       shift 2
       ;;
+    --idle-timeout-minutes)
+      IDLE_TIMEOUT_MINUTES="$2"
+      shift 2
+      ;;
     --foreground|--no-daemon)
       FOREGROUND="true"
       shift
@@ -59,6 +65,14 @@ if [[ -z "$URL_HOST" ]]; then
   fi
 fi
 
+if [[ -n "$IDLE_TIMEOUT_MINUTES" ]]; then
+  if ! [[ "$IDLE_TIMEOUT_MINUTES" =~ ^[0-9]+$ ]] || [[ "$IDLE_TIMEOUT_MINUTES" -lt 1 ]]; then
+    echo "{\"error\": \"--idle-timeout-minutes must be a positive integer\"}"
+    exit 1
+  fi
+  export BRAINSTORM_IDLE_TIMEOUT_MS=$(( IDLE_TIMEOUT_MINUTES * 60 * 1000 ))
+fi
+
 # Some environments reap detached/background processes. Auto-foreground when detected.
 if [[ -n "${CODEX_CI:-}" && "$FOREGROUND" != "true" && "$FORCE_BACKGROUND" != "true" ]]; then
   FOREGROUND="true"