fix: recurring event display and AI query improvements

- Use occurrenceStart instead of startTime in getEventsInRange so
  recurring events show their actual occurrence date to the AI
- Add lazy CalDAV sync in ChatService (syncOnce before first DB access)
- Add CaldavService.sync() with internal config check (silent no-op)
- Show German recurrence description (e.g. "Jede Woche") instead of
  generic "Wiederkehrend" in EventCardBase via formatRecurrenceRule()
- Move RepeatType and REPEAT_TYPE_LABELS from editEvent to shared
- Separate calendar overlay useFocusEffect from event loading

packages/shared/src/utils/rruleHelpers.ts

@@ -11,6 +11,24 @@ const REPEAT_TYPE_TO_FREQ: Record<RepeatType, string> = {
   Jahr: "YEARLY",
 };
 
+const FREQ_TO_REPEAT_TYPE: Record<string, RepeatType> = Object.fromEntries(
+  Object.entries(REPEAT_TYPE_TO_FREQ).map(([k, v]) => [v, k as RepeatType]),
+);
+
+export const REPEAT_TYPE_LABELS: Record<RepeatType, string> = {
+  Tag: "Tage",
+  Woche: "Wochen",
+  Monat: "Monate",
+  Jahr: "Jahre",
+};
+
+const REPEAT_TYPE_SINGULAR: Record<RepeatType, string> = {
+  Tag: "Jeden Tag",
+  Woche: "Jede Woche",
+  Monat: "Jeden Monat",
+  Jahr: "Jedes Jahr",
+};
+
 /**
  * Build an RRULE string from repeat count and type.
  *
@@ -27,3 +45,30 @@ export function buildRRule(repeatType: RepeatType, interval: number = 1): string
 
   return `FREQ=${freq};INTERVAL=${interval}`;
 }
+
+/**
+ * Format an RRULE string into a human-readable German description.
+ *
+ * @param rrule - RRULE string like "FREQ=WEEKLY;INTERVAL=2"
+ * @returns German description like "Alle 2 Wochen" or "Wöchentlich"
+ */
+export function formatRecurrenceRule(rrule: string): string {
+  const rule = rrule.replace(/^RRULE:/, "");
+  const parts = Object.fromEntries(
+    rule.split(";").map((p) => p.split("=") as [string, string]),
+  );
+
+  const freq = parts.FREQ;
+  const interval = parts.INTERVAL ? parseInt(parts.INTERVAL, 10) : 1;
+  const repeatType = FREQ_TO_REPEAT_TYPE[freq];
+
+  if (!repeatType) {
+    return "Wiederkehrend";
+  }
+
+  if (interval <= 1) {
+    return REPEAT_TYPE_SINGULAR[repeatType];
+  }
+
+  return `Alle ${interval} ${REPEAT_TYPE_LABELS[repeatType]}`;
+}