Surface the event-bus WebSocket state in the UI

SDK (sdk-event-bus amended on the fork): a new `WsState` enum
(`Stopped`/`Connecting`/`Connected`/`Reconnecting`) is broadcast through a
`watch::Sender` inside `EventBusClient`, exposed via `state()` for
observers. Transitions are emitted at every step of the reconnect loop,
plus a `Drop` guard guarantees a final `Stopped` even on a panic in the
caller's task tree. Two new SDK tests cover the initial value and
multi-subscriber broadcast.

Bridge: capture `bus_client.state()` at start, mirror it into a serde
`WsStatus` field on `BridgeStats`, and clear on stop.

UI: add a third stat card "Realtime" with a colored dot (green Connected,
orange pulsing Connecting/Reconnecting, gray Off) reading
`stats.ws_status`. Stats grid switches from 2 to 3 columns.

Submodule pointer bumped to the rebuilt `tutabridge-integration` which
cherry-picks all six SDK branches on `upstream/master`; the move-mails
fixup (use `make_test_facade` so its test compiles alongside the blob
branch) is re-applied. 341/341 SDK lib tests pass, 154/154 bridge tests
pass.

ui/src/App.css

@@ -230,10 +230,33 @@
 
 .stats-grid {
   display: grid;
-  grid-template-columns: 1fr 1fr;
+  grid-template-columns: 1fr 1fr 1fr;
   gap: 12px;
 }
 
+.ws-badge {
+  display: inline-flex;
+  align-items: center;
+  gap: 8px;
+  font-family: var(--brand);
+  font-size: 14px;
+  font-weight: 500;
+  color: var(--on-surface);
+  line-height: 1;
+}
+
+.ws-dot {
+  width: 10px;
+  height: 10px;
+  border-radius: 50%;
+  display: inline-block;
+  flex-shrink: 0;
+}
+
+.ws-dot.pulsing {
+  animation: breathe 2s ease-in-out infinite;
+}
+
 .stat-card {
   display: flex;
   flex-direction: column;

ui/src/components/Dashboard.tsx

@@ -1,7 +1,21 @@
 import { useState } from "react";
-import type { BridgeStatus, BridgeStats } from "../types";
+import type { BridgeStatus, BridgeStats, WsStatus } from "../types";
 import { isError } from "../types";
 
+function wsBadge(status: WsStatus): { color: string; label: string; pulsing: boolean } {
+  switch (status) {
+    case "Connected":
+      return { color: "var(--green)", label: "Connected", pulsing: false };
+    case "Connecting":
+      return { color: "var(--orange)", label: "Connecting…", pulsing: true };
+    case "Reconnecting":
+      return { color: "var(--orange)", label: "Reconnecting…", pulsing: true };
+    case "Stopped":
+    default:
+      return { color: "var(--gray)", label: "Off", pulsing: false };
+  }
+}
+
 interface Props {
   status: BridgeStatus | null;
   stats: BridgeStats;
@@ -122,6 +136,23 @@ export function Dashboard({ status, stats, hasSavedSession, loading, onStart, on
           </span>
           <span className="stat-label">Uptime</span>
         </div>
+        <div className="stat-card">
+          <span className="ws-badge">
+            {(() => {
+              const b = wsBadge(stats.ws_status);
+              return (
+                <>
+                  <span
+                    className={"ws-dot" + (b.pulsing ? " pulsing" : "")}
+                    style={{ background: b.color }}
+                  />
+                  <span className="ws-text">{b.label}</span>
+                </>
+              );
+            })()}
+          </span>
+          <span className="stat-label">Realtime</span>
+        </div>
       </div>
     </div>
   );

ui/src/hooks/useBridge.ts

@@ -9,7 +9,11 @@ const POLL_INTERVAL = 1000;
 export function useBridge() {
   const [config, setConfig] = useState<Config | null>(null);
   const [status, setStatus] = useState<BridgeStatus | null>(null);
-  const [stats, setStats] = useState<BridgeStats>({ uptime_secs: null, mails_synced: 0 });
+  const [stats, setStats] = useState<BridgeStats>({
+    uptime_secs: null,
+    mails_synced: 0,
+    ws_status: "Stopped",
+  });
   const [hasSavedSession, setHasSavedSession] = useState(false);
   const [bridgePassword, setBridgePassword] = useState<string | null>(null);
   const [logs, setLogs] = useState<string[]>([]);

ui/src/types.ts

@@ -9,9 +9,12 @@ export interface Config {
 
 export type BridgeStatus = "Stopped" | "Starting" | "Running" | { Error: string };
 
+export type WsStatus = "Stopped" | "Connecting" | "Connected" | "Reconnecting";
+
 export interface BridgeStats {
   uptime_secs: number | null;
   mails_synced: number;
+  ws_status: WsStatus;
 }
 
 export function statusLabel(status: BridgeStatus): string {