Next.js App Router Edge Middleware Security Guards

Act as a Next.js Security Lead. Architect a highly optimized, Vercel Edge-compatible `middleware.ts` script explicitly engineered to protect the [PROTECTED_ROUTES_ARRAY] within an App Router application. The middleware must autonomously intercept all incoming HTTP requests, extract the authentication cookie, and cryptographically verify the signature utilizing the lightweight `jose` library (since native Node APIs like `jsonwebtoken` are banned at the Edge). If the token is expired or invalid, the script must execute a hyper-fast 302 redirect to the [UNAUTHORIZED_REDIRECT]. Prove mathematically why intercepting malicious traffic at the Edge drastically protects the underlying Server Components from unnecessary compute.