Add GitHub star count to site header (#673)

* Add GitHub star count to site header

Fetches star count from GitHub API via /api/github-stars with 5-minute
server-side caching (ISR + stale-while-revalidate). Shows formatted
count (e.g. "2.2k") next to the GitHub link in both desktop nav and
mobile drawer.

* Move star count to separate badge left of download button

GitHub icon + formatted count as its own clickable element in the
right header section, separate from the nav links. Desktop only.

* Center GitHub stars badge vertically in header

* Add right padding to GitHub stars badge

web/app/api/github-stars/route.ts

@@ -0,0 +1,33 @@
+import { NextResponse } from "next/server";
+
+export const revalidate = 300; // ISR: regenerate every 5 minutes
+
+export async function GET() {
+  try {
+    const res = await fetch(
+      "https://api.github.com/repos/manaflow-ai/cmux",
+      {
+        headers: { Accept: "application/vnd.github.v3+json" },
+        next: { revalidate: 300 },
+      }
+    );
+
+    if (!res.ok) {
+      return NextResponse.json({ stars: null }, { status: 502 });
+    }
+
+    const data = await res.json();
+    const stars: number = data.stargazers_count;
+
+    return NextResponse.json(
+      { stars },
+      {
+        headers: {
+          "Cache-Control": "public, s-maxage=300, stale-while-revalidate=600",
+        },
+      }
+    );
+  } catch {
+    return NextResponse.json({ stars: null }, { status: 502 });
+  }
+}