Calendar reminders as push notifications

A job every minute finds the event reminders that fell due — repeating events
and moved occurrences expanded as the calendar view does — and pushes each
once to every device with notifications on (sdk.push, new: plugins can notify
through the new-mail push delivery). A reminder missed while the worker was down
is still sent within 10 minutes; cancelled events and declined invitations stay
quiet. Title, then "In 15 minutes, 14:00 · Room 2" in the user's zone and clock
format; it stays until dismissed and shows even with the app open; clicking it
opens that day (/p/calendar?date=…, new deep link).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

backend/app/plugins/sdk.py

@@ -179,6 +179,12 @@ class PluginSdk:
         CalDAV/CardDAV/ICS. Returns status + headers + body."""
         return PluginHttp()
 
+    # --- notifications ------------------------------------------------------
+    @property
+    def push(self) -> PluginPush:
+        """Notifications on the user's devices (Web Push) — calendar reminders."""
+        return PluginPush(self.name)
+
     # --- secret vault -------------------------------------------------------
     @property
     def vault(self) -> PluginVault:
@@ -187,6 +193,39 @@ class PluginSdk:
         return PluginVault()
 
 
+class PluginPush:
+    """``sdk.push``: show a notification on every device where the user turned
+    notifications on (Settings → Accounts → Notifications)."""
+
+    def __init__(self, plugin: str) -> None:
+        self._plugin = plugin
+
+    async def notify(
+        self,
+        user_id: uuid.UUID,
+        *,
+        title: str,
+        body: str = "",
+        url: str = "/mail",
+        tag: str | None = None,
+        sticky: bool = False,
+    ) -> int:
+        """Send one notification; returns how many devices it went to. ``tag``
+        replaces an earlier notification with the same tag (it is prefixed with
+        the plugin's name); ``sticky`` keeps it on screen until dismissed."""
+        from app.push.service import deliver
+
+        payload: dict[str, object] = {
+            "type": f"plugin.{self._plugin}",
+            "title": title[:200],
+            "body": body[:500],
+            "url": url if url.startswith("/") else "/mail",
+            "tag": f"{self._plugin}-{tag}" if tag else self._plugin,
+            "sticky": sticky,
+        }
+        return await deliver(user_id, [payload])
+
+
 def sdk(name: str) -> PluginSdk:
     """Return the SDK facade bound to plugin ``name`` (SPEC §11)."""
     return PluginSdk(name)