Centralize seat capacity accounting and the payment provider registry.

Replace manualProviders.ts with a paymentProviders.ts registry
(automatic vs manual settlement) and move all seat counting into
capacity.ts as the single source of truth: only paid/checked-in tickets
and pending_approval payments hold a seat, so abandoned checkouts never
block sales. Admins can now knowingly approve a payment over capacity
(allowOverCapacity), with the booking and admin UIs updated to match.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

frontend/src/app/(public)/events/[id]/EventDetailClient.tsx

@@ -5,7 +5,7 @@ import Link from 'next/link';
 import Image from 'next/image';
 import { useLanguage } from '@/context/LanguageContext';
 import { eventsApi, Event } from '@/lib/api';
-import { formatPrice, formatDateLong, formatTime } from '@/lib/utils';
+import { formatPrice, formatDateLong, formatTime, eventSpotsLeft, isEventSoldOut } from '@/lib/utils';
 import Card from '@/components/ui/Card';
 import Button from '@/components/ui/Button';
 import ShareButtons from '@/components/ShareButtons';
@@ -43,9 +43,10 @@ export default function EventDetailClient({ eventId, initialEvent }: EventDetail
       .catch(console.error);
   }, [eventId]);
 
-  // Spots left: never negative; sold out when confirmed >= capacity
-  const spotsLeft = Math.max(0, event.capacity - (event.bookedCount ?? 0));
-  const isSoldOut = (event.bookedCount ?? 0) >= event.capacity;
+  // Server-authoritative availability (paid + claimed seats count; abandoned
+  // pending bookings don't) — matches the booking API's sold-out check exactly.
+  const spotsLeft = eventSpotsLeft(event);
+  const isSoldOut = isEventSoldOut(event);
   const maxTickets = isSoldOut ? 0 : Math.min(MAX_TICKETS_PER_PERSON, Math.max(1, spotsLeft));
 
   useEffect(() => {