fix(gateway): warn when a live gateway's heartbeat goes stale instead of printing running

The reporter's case in #113372 is "not a crash": the process stays alive,
`gateway_state.json` keeps saying `running`, and housekeeping, cron and the
kanban dispatcher are frozen. Nothing wrote `updated_at` periodically, so the
file was a stored status, not a heartbeat, and both `hermes gateway status`
and `/api/status` rendered a wedged gateway as healthy (the existing stale arm
only fires when the recorded PID is gone).

Make the housekeeping tick re-stamp `gateway_state.json` first thing every
tick (60 s), so `updated_at` is a heartbeat that stops when the thread — or a
chore blocked on the loop — wedges. Readers then warn on `running`/`starting`
+ stale stamp + live PID: `hermes gateway status` prints
`⚠ Gateway heartbeat stale: housekeeping has not refreshed gateway_state.json
for N s (event loop or housekeeping wedged; pid X alive)`, `/api/status`
carries `gateway_heartbeat_stale_s` (null when healthy) and the sidebar strip
shows "Heartbeat stale". Liveness (`gateway_running`, busy/drainable) still
keys off the PID, never the stamp. Draining is excluded: shutdown drains the
housekeeping thread before the process exits, so its stamp legitimately ages.

Invariant tests: the CLI line names the age and the live PID and is silent on
a fresh stamp; /api/status sets/clears `gateway_heartbeat_stale_s` the same way.

web/src/components/SidebarStatusStrip.tsx

@@ -65,6 +65,11 @@ export function gatewayLine(
     },
     stopped: { label: g.stopped, tone: "text-muted-foreground" },
   };
+  // Alive but housekeeping stopped stamping the heartbeat: 'Running' would be the lie the
+  // reporter saw (loop/housekeeping wedged while gateway_state.json still said running).
+  if (status.gateway_heartbeat_stale_s != null) {
+    return { label: g.heartbeatStale, tone: "text-destructive" };
+  }
   if (status.gateway_state && byState[status.gateway_state]) {
     return byState[status.gateway_state];
   }

web/src/i18n/en.ts

@@ -67,6 +67,7 @@ export const en: Translations = {
     gatewayStrip: {
       degraded: "Degraded",
       failed: "Start failed",
+      heartbeatStale: "Heartbeat stale",
       off: "Off",
       running: "Running",
       starting: "Starting",

web/src/i18n/types.ts

@@ -86,6 +86,7 @@ export interface Translations {
     gatewayStrip: {
       degraded: string;
       failed: string;
+      heartbeatStale: string;
       off: string;
       running: string;
       starting: string;

web/src/lib/api.ts

@@ -1927,6 +1927,10 @@ export interface StatusResponse {
   env_path: string;
   gateway_exit_reason: string | null;
   gateway_health_url: string | null;
+  /** Seconds since the gateway's housekeeping last stamped gateway_state.json, set only when the
+   * process is alive but the stamp is past the freshness TTL (loop/housekeeping wedged).
+   * null when healthy; absent on older backends. */
+  gateway_heartbeat_stale_s?: number | null;
   gateway_pid: number | null;
   gateway_platforms: Record<string, PlatformStatus>;
   gateway_running: boolean;