Advanced Next.js Edge Middleware JWT Validation

Act as a Next.js Security Lead. Architect a highly optimized, Vercel Edge-compatible `middleware.ts` script explicitly engineered to aggressively guard 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 JWT signature utilizing the lightweight `jose` library (since native Node APIs like `jsonwebtoken` are banned at the Edge). If the token is expired, missing, or maliciously forged, the script must execute a hyper-fast 302 redirect directly to the [UNAUTHORIZED_REDIRECT_PATH]. Prove mathematically why intercepting malicious traffic at the Edge CDN layer drastically protects the underlying Server Components and databases from unnecessary compute load.