feat: implement Gmail metadata synchronization to backend

- Added a new utility function `syncGmailMetadataToBackend` to emit Gmail profile and email summaries to the backend via the `integration:metadata-sync` socket event.
- Updated `SkillProvider` to call the new synchronization function, ensuring Gmail skill state is sent to the backend when updated.

src/lib/skills/gmailMetadataSync.ts

@@ -0,0 +1,62 @@
+/**
+ * Send Gmail profile (and optionally emails) to the backend via the
+ * `integration:metadata-sync` socket event so the server can merge them
+ * into the user's Google OAuth integration metadata.
+ */
+import { emitViaRustSocket } from '../../utils/tauriSocket';
+
+const INTEGRATION_METADATA_SYNC_EVENT = 'integration:metadata-sync';
+const PROVIDER_GOOGLE = 'google';
+
+/** Gmail profile shape from skill state (snake_case). */
+interface GmailProfileLike {
+  email_address: string;
+  messages_total: number;
+  threads_total: number;
+  history_id: string;
+}
+
+/** Single email summary from skill state. */
+interface GmailEmailSummaryLike {
+  id: string;
+  threadId: string;
+  snippet?: string;
+  subject?: string;
+  from?: string;
+  date?: string;
+}
+
+/** Gmail skill state slice we care about for metadata sync. */
+export interface GmailStateForSync {
+  profile?: GmailProfileLike | null;
+  emails?: GmailEmailSummaryLike[] | null;
+}
+
+/**
+ * Emit `integration:metadata-sync` with Gmail profile and emails so the
+ * backend can merge them into the user's Google OAuth integration.
+ * No-op when profile is missing or not in Tauri.
+ */
+export function syncGmailMetadataToBackend(gmailState: GmailStateForSync | undefined): void {
+  if (!gmailState?.profile || typeof gmailState.profile !== 'object') return;
+
+  const profile = gmailState.profile as GmailProfileLike;
+  const metadata: Record<string, unknown> = {
+    email_address: profile.email_address,
+    messages_total: profile.messages_total,
+    threads_total: profile.threads_total,
+    history_id: profile.history_id,
+  };
+
+  if (Array.isArray(gmailState.emails) && gmailState.emails.length > 0) {
+    metadata.emails = gmailState.emails as GmailEmailSummaryLike[];
+  }
+
+  const payload = {
+    requestId: crypto.randomUUID(),
+    provider: PROVIDER_GOOGLE,
+    metadata,
+  };
+
+  void emitViaRustSocket(INTEGRATION_METADATA_SYNC_EVENT, payload);
+}

src/providers/SkillProvider.tsx

@@ -19,6 +19,10 @@ import {
   type GmailProfile,
 } from '../store/gmailSlice';
 import { setSkillError, setSkillState, setSkillStatus } from '../store/skillsSlice';
+import {
+  syncGmailMetadataToBackend,
+  type GmailStateForSync,
+} from '../lib/skills/gmailMetadataSync';
 import { DEV_AUTO_LOAD_SKILL, IS_DEV } from '../utils/config';
 
 // ---------------------------------------------------------------------------
@@ -71,7 +75,7 @@ function parseSkillStatePayload(
   return { skillId, state };
 }
 
-/** Sync profile and emails from gmail skill state into gmailSlice. */
+/** Sync profile and emails from gmail skill state into gmailSlice and send to backend via socket. */
 function syncGmailStateToSlice(
   gmailState: Record<string, unknown> | undefined,
   dispatch: ReturnType<typeof useAppDispatch>
@@ -89,6 +93,7 @@ function syncGmailStateToSlice(
       Array.isArray(gmailState.emails) ? (gmailState.emails as GmailEmailSummary[]) : []
     )
   );
+  syncGmailMetadataToBackend(gmailState as GmailStateForSync);
 }
 
 export default function SkillProvider({ children }: { children: ReactNode }) {