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/lib/utils.ts

@@ -205,3 +205,33 @@ export function getTpagoLink(
   const key = (count <= 1 ? 'tpagoLink' : `tpagoLink${count}`) as keyof TpagoLinkConfig;
   return config[key] || config.tpagoLink || null;
 }
+
+/**
+ * Spots left for an event, trusting the server's `availableSeats` (which uses
+ * the same seat-holding formula the booking API enforces: paid + claimed
+ * seats count, abandoned pending bookings don't). Falls back to deriving it
+ * from the counts for older API responses.
+ */
+export function eventSpotsLeft(event: {
+  capacity: number;
+  bookedCount?: number;
+  claimedCount?: number;
+  availableSeats?: number;
+}): number {
+  if (typeof event.availableSeats === 'number') {
+    return Math.max(0, event.availableSeats);
+  }
+  return Math.max(
+    0,
+    (event.capacity ?? 0) - (event.bookedCount ?? 0) - (event.claimedCount ?? 0)
+  );
+}
+
+export function isEventSoldOut(event: {
+  capacity: number;
+  bookedCount?: number;
+  claimedCount?: number;
+  availableSeats?: number;
+}): boolean {
+  return eventSpotsLeft(event) <= 0;
+}