Advanced AI Prompt for Next.js Middleware Edge Auth Routing

Act as a Next.js Security Architect. Develop a highly optimized, Vercel Edge-compatible `middleware.ts` file strictly engineered to guard the [PROTECTED_ROUTES] array. The script must autonomously intercept incoming requests, extract the [AUTH_TOKEN_COOKIE], and mathematically verify its signature using the lightweight `jose` library (since standard `jsonwebtoken` relies on Node APIs unavailable at the Edge). If the token is invalid or missing, execute a hyper-fast 302 redirect directly to the [LOGIN_URL]. Prove mathematically why executing this logic at the Edge drastically improves TTFB (Time to First Byte) compared to doing authorization checks inside the Server Components.