Push notification dedupe: one message = one notification — an offline message was notified twice (FCM push, then again when the app synced the outbox and mirrored the replayed frames). Fix: the gateway records the highest outbox cursor delivered per device via push (devices.last_pushed_cursor, advanced only on successful send) and returns it in hello.ack; sync-replayed frames carry their outbox cursor in the envelope; the app skips system notifications for replayed frames at/below the watermark (live frames never suppressed — that is the case where no push fired). Also: 5s per-chat push coalescing so a cron delivery (notification frame + message frame) pushes once, and the FCM handler no longer posts a redundant notification (skips when WS is connected or FCM already displayed the notification payload; data-only messages are the exception). Docs: frames.schema.json, 04-wire-protocol.md, 08-push.md §8.8

gateway-plugin/ws_server.py

@@ -290,6 +290,9 @@ class WsServer:
             server_caps=self._adapter.server_caps(),
             sync_cursor=self._adapter._outbox.latest_cursor(),
             channels=self._adapter.channel_list(),
+            # M5: lets the app dedupe sync-replayed notifications that
+            # already woke this device via push (docs/08 §8.7).
+            last_pushed_cursor=self._adapter._devices.last_pushed_cursor(device_id),
         )
         try:
             await ws.send(ack.to_json())