Building a Streaming AI Chat App with Next.js 16 + Claude API — Complete App Router Guide

Search for "Next.js AI chat" in 2026 and Vercel AI SDK still comes up as the de facto standard. Nothing wrong with it, but relying on the SDK means you often don't understand what's happening under the hood — how streaming actually works, what the Route Handler is doing behind the scenes. I built this from scratch in a sandbox using only the Anthropic SDK. The entire flow: create-next-app, adding @anthropic-ai/sdk, implementing the Route Handler. It turned out simpler than I expected. About 50...