Missing `Cache-Control` header in `/404` rewrite response

Link to the code that reproduces this issue

https://codesandbox.io/p/devbox/nextjs-404-missing-cache-control-xvggzv

To Reproduce

1. Go to terminal
2. Stop Dev server
3. Start Next in production mode: npm run build && npm start
4. Execute curl -I http://localhost:3000/example

Current vs. Expected behavior

Current behavior

The response doesn't include a Cache-Control header:

Expected behavior

The reponse should include an explicit Cache-Control header. I propose private, no-cache, no-store, max-age=0, must-revalidate to avoid caching.

Provide environment information

Operating System:
  Platform: linux
  Arch: x64
  Version: #1 SMP PREEMPT_DYNAMIC Sun Aug  6 20:05:33 UTC 2023
  Available memory (MB): 4242
  Available CPU cores: 2
Binaries:
  Node: 20.12.0
  npm: 10.5.0
  Yarn: 1.22.19
  pnpm: 8.15.6
Relevant Packages:
  next: 15.1.7 // Latest available version is detected (15.1.7).
  eslint-config-next: 15.0.4
  react: 19.0.0
  react-dom: 19.0.0
  typescript: 5.7.3
Next.js Config:
  output: N/A

Which area(s) are affected? (Select all that apply)

Not sure

Which stage(s) are affected? (Select all that apply)

next build (local), next start (local), Other (Deployed)

Additional context

I tested this behavior in Next v14 and v15.

My original motivation for rewriting a path to /404 was rewriting non-existing /_next/static URLs because for those we skip the middleware and they caused 500 errors in my page.tsx.

Start with the linked reproduction, then run the reported npm run build && npm start flow and curl -I http://localhost:3000/example to inspect the /404 rewrite response. Trace the production handling of that rewrite and verify that the response includes an explicit Cache-Control header with the proposed no-cache behavior.