[BUG] fnm global install will not trigger auto updates

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

If I use npm install -g @anthropic-ai/claude-code in fnm's npm, which will create in ~/.local/state/fnm_multishells/32677_1757783734660/bin/claude, auto-update do not work.

If I use npm install -g @anthropic-ai/claude-code in nvm's npm, which will create in ~/.nvm/versions/node/v22.19.0/bin/claude, auto-update works well.

What Should Happen?

fnm should works well like nvm

Steps to Reproduce

fnm

1. uninstall nvm, install fnm
2. fnm use 22
3. npm install -g @anthropic-ai/claude-code@1.0.112
4. claude
5. /status
6. Check claude version is 1.0.112
7. Wait for 60 second
8. Ctrl+D, Ctrl+D
9. claude
10. /status
11. Check claude version is still 1.0.112

nvm

1. uninstall fnm, install nvm
2. nvm install 22
3. npm install -g @anthropic-ai/claude-code@1.0.112
4. claude
5. /status
6. Check claude version is 1.0.112
7. Wait for 60 second
8. Ctrl+D, Ctrl+D
9. claude
10. /status
11. Check claude version is change to 1.0.113

Claude Model

None

Is this a regression?

No, this never worked

Last Working Version

_No response_

Claude Code Version

1.0.113

Platform

Anthropic API

Operating System

macOS

Terminal/Shell

Terminal.app (macOS)

Additional Information

_No response_