Underneath is just a system prompt, or more likely a prompt layered on top "You are a frontend engineer, competent in react and Next.js, tailwind-css" - the stack details and project layout, key information is already in the CLAUDE.md. For more stuff the model is going to call file-read tools etc.
I think its more theatre then utilty.
What I have taken to doing is having a parent folder and then frontend/ backend/ infra/ etc as children.
parent/CLAUDE.md frontend/CLAUDE.md backend/CLAUDE.md
The parent/CLAUDE.md provides a highlevel view of the stack "FastAPI backend with postgres, Next.js frontend using with tailwind, etc". The parent/CLAUDE.md also points to the childrens CLAUDE.md's which have more granular information.
I then just spawn a claude in the parent folder, set up plan mode, go back and forth on a design and then have it dump out to markdown to RFC/ and after that go to work. I find it does really well then as all changes it makes are made with a context of the other service.