feat(auth): hide Google button when OAuth env is missing

- Export enabledSocialProviders from @ketopath/auth: ['google'] when
  GOOGLE_CLIENT_ID/SECRET are both set, [] otherwise
- sign-in and sign-up pages read enabledSocialProviders server-side and pass
  googleEnabled to their forms; the divider and button disappear when false
- No changes to the Better Auth instance — the Google provider is still
  conditionally registered, this just keeps the UI honest about it

docs/runbooks/google-oauth-setup.md walks through the Google Cloud Console
flow end-to-end (project, consent screen, credentials, redirect URIs, env
injection, verification, troubleshooting, prod considerations).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

packages/auth/src/index.ts

@@ -1,2 +1,2 @@
-export { auth, type Auth } from './server.js';
+export { auth, enabledSocialProviders, type Auth, type SocialProvider } from './server.js';
 export { readAuthEnv, type AuthEnv } from './env.js';

packages/auth/src/server.ts

@@ -6,15 +6,20 @@ import { readAuthEnv } from './env.js';
 
 const env = readAuthEnv();
 
-const googleProvider =
-  env.GOOGLE_CLIENT_ID && env.GOOGLE_CLIENT_SECRET
-    ? {
-        google: {
-          clientId: env.GOOGLE_CLIENT_ID,
-          clientSecret: env.GOOGLE_CLIENT_SECRET,
-        },
-      }
-    : undefined;
+const isGoogleConfigured = Boolean(env.GOOGLE_CLIENT_ID && env.GOOGLE_CLIENT_SECRET);
+
+export const enabledSocialProviders = isGoogleConfigured ? (['google'] as const) : ([] as const);
+
+export type SocialProvider = (typeof enabledSocialProviders)[number];
+
+const googleProvider = isGoogleConfigured
+  ? {
+      google: {
+        clientId: env.GOOGLE_CLIENT_ID!,
+        clientSecret: env.GOOGLE_CLIENT_SECRET!,
+      },
+    }
+  : undefined;
 
 export const auth = betterAuth({
   database: prismaAdapter(prisma, { provider: 'postgresql' }),