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>

apps/web/src/app/[locale]/(auth)/sign-in/page.tsx

@@ -1,3 +1,4 @@
+import { enabledSocialProviders } from '@ketopath/auth';
 import { useTranslations } from 'next-intl';
 import { setRequestLocale } from 'next-intl/server';
 
@@ -13,7 +14,7 @@ export default function SignInPage({ params: { locale } }: { params: { locale: s
         <h1 className="text-3xl font-bold tracking-tight">{t('title')}</h1>
         <p className="text-sm text-slate-600">{t('subtitle')}</p>
       </header>
-      <SignInForm />
+      <SignInForm googleEnabled={enabledSocialProviders.includes('google')} />
     </main>
   );
 }

apps/web/src/app/[locale]/(auth)/sign-in/sign-in-form.tsx

@@ -7,7 +7,7 @@ import { useState, type FormEvent } from 'react';
 
 import { authClient } from '@/lib/auth-client';
 
-export function SignInForm() {
+export function SignInForm({ googleEnabled }: { googleEnabled: boolean }) {
   const t = useTranslations('Auth.SignIn');
   const router = useRouter();
   const [email, setEmail] = useState('');
@@ -82,23 +82,27 @@ export function SignInForm() {
           {pending ? t('submitting') : t('submit')}
         </button>
       </form>
-      <div className="relative">
-        <div className="absolute inset-0 flex items-center">
-          <div className="w-full border-t border-slate-200" />
-        </div>
-        <div className="relative flex justify-center">
-          <span className="bg-white px-2 text-xs uppercase tracking-wider text-slate-500">
-            {t('or')}
-          </span>
-        </div>
-      </div>
-      <button
-        type="button"
-        onClick={handleGoogle}
-        className="w-full rounded-md border border-slate-300 bg-white px-5 py-2.5 text-sm font-medium text-slate-700 shadow-sm hover:bg-slate-50"
-      >
-        {t('google')}
-      </button>
+      {googleEnabled ? (
+        <>
+          <div className="relative">
+            <div className="absolute inset-0 flex items-center">
+              <div className="w-full border-t border-slate-200" />
+            </div>
+            <div className="relative flex justify-center">
+              <span className="bg-white px-2 text-xs uppercase tracking-wider text-slate-500">
+                {t('or')}
+              </span>
+            </div>
+          </div>
+          <button
+            type="button"
+            onClick={handleGoogle}
+            className="w-full rounded-md border border-slate-300 bg-white px-5 py-2.5 text-sm font-medium text-slate-700 shadow-sm hover:bg-slate-50"
+          >
+            {t('google')}
+          </button>
+        </>
+      ) : null}
       <p className="text-center text-sm text-slate-600">
         {t.rich('noAccount', {
           link: (chunks) => (

apps/web/src/app/[locale]/(auth)/sign-up/page.tsx

@@ -1,3 +1,4 @@
+import { enabledSocialProviders } from '@ketopath/auth';
 import { useTranslations } from 'next-intl';
 import { setRequestLocale } from 'next-intl/server';
 
@@ -13,7 +14,7 @@ export default function SignUpPage({ params: { locale } }: { params: { locale: s
         <h1 className="text-3xl font-bold tracking-tight">{t('title')}</h1>
         <p className="text-sm text-slate-600">{t('subtitle')}</p>
       </header>
-      <SignUpForm />
+      <SignUpForm googleEnabled={enabledSocialProviders.includes('google')} />
       <p className="text-xs text-slate-500">{t('disclaimer')}</p>
     </main>
   );

apps/web/src/app/[locale]/(auth)/sign-up/sign-up-form.tsx

@@ -7,7 +7,7 @@ import { useState, type FormEvent } from 'react';
 
 import { authClient } from '@/lib/auth-client';
 
-export function SignUpForm() {
+export function SignUpForm({ googleEnabled }: { googleEnabled: boolean }) {
   const t = useTranslations('Auth.SignUp');
   const router = useRouter();
   const [name, setName] = useState('');
@@ -95,23 +95,27 @@ export function SignUpForm() {
           {pending ? t('submitting') : t('submit')}
         </button>
       </form>
-      <div className="relative">
-        <div className="absolute inset-0 flex items-center">
-          <div className="w-full border-t border-slate-200" />
-        </div>
-        <div className="relative flex justify-center">
-          <span className="bg-white px-2 text-xs uppercase tracking-wider text-slate-500">
-            {t('or')}
-          </span>
-        </div>
-      </div>
-      <button
-        type="button"
-        onClick={handleGoogle}
-        className="w-full rounded-md border border-slate-300 bg-white px-5 py-2.5 text-sm font-medium text-slate-700 shadow-sm hover:bg-slate-50"
-      >
-        {t('google')}
-      </button>
+      {googleEnabled ? (
+        <>
+          <div className="relative">
+            <div className="absolute inset-0 flex items-center">
+              <div className="w-full border-t border-slate-200" />
+            </div>
+            <div className="relative flex justify-center">
+              <span className="bg-white px-2 text-xs uppercase tracking-wider text-slate-500">
+                {t('or')}
+              </span>
+            </div>
+          </div>
+          <button
+            type="button"
+            onClick={handleGoogle}
+            className="w-full rounded-md border border-slate-300 bg-white px-5 py-2.5 text-sm font-medium text-slate-700 shadow-sm hover:bg-slate-50"
+          >
+            {t('google')}
+          </button>
+        </>
+      ) : null}
       <p className="text-center text-sm text-slate-600">
         {t.rich('haveAccount', {
           link: (chunks) => (