feat(web): add same-origin SPA authentication

apps/web/src/lib/auth-client.ts

@@ -1,8 +1,9 @@
 import { createAuthClient } from 'better-auth/react';
 import { adminClient, genericOAuthClient } from 'better-auth/client/plugins';
 
+// The gateway and BetterAuth client both use /api/auth. Omitting baseURL keeps
+// every browser request on the current origin in development and production.
 export const authClient = createAuthClient({
-  baseURL: process.env['NEXT_PUBLIC_GATEWAY_URL'] ?? 'http://localhost:14242',
   plugins: [adminClient(), genericOAuthClient()],
 });