Finish interrupted features: tab prefetch, pet viewport, and home logo.

Add shared quiz/habits tab cache with nav hover prefetch and AppDataPrefetcher,
make the pet screen fill and center the viewport, and point the HaBit logo to /.

app/(app)/avatar/page.tsx

@@ -8,6 +8,8 @@ export default function AvatarPage() {
     <AppShell
       title="Your Habit Pet"
       description="Watch your pet roam, breathe, and react to how you care for them."
+      fillViewport
+      centered
     >
       <Suspense fallback={<p className="text-sm text-muted-foreground">Loading...</p>}>
         <AvatarPageContent />

app/globals.css

@@ -116,6 +116,21 @@
     max-width: calc(32rem - 2.5rem - 2rem);
   }
 
+  .pet-viewport-stage {
+    @apply flex min-h-0 w-full flex-1 flex-col items-center justify-center;
+  }
+
+  .tamagotchi-shell-fill {
+    @apply flex min-h-0 w-full max-w-4xl flex-1 flex-col;
+  }
+
+  .tamagotchi-lcd-pet-fill {
+    @apply relative min-h-0 w-full flex-1;
+    min-height: 16rem;
+    max-width: none;
+    aspect-ratio: auto;
+  }
+
   .customize-body {
     min-height: 26rem;
   }