How to resolve 'No Permission' error in Numbers on MacBook Pro?

> (When I clicked on Applications/Numbers + opt-comnd-i, I have systems permission to 'read and write'. )

That's irrelevant. That's the Numbers application itself (which you're unlikely to be writing to anyway). It has nothing to do with the permissions of the document(s) you're saving.

What's more relevant is where that document is.. is it on your iCloud Drive? or your local machine?

How are you opening the file? double-click? File -> Open from within Numbers?

Does it happen with all/new documents or only specific ones?

Even if the app permissions look fine, it’s worth double‑checking the file’s own permissions too.

If those seem correct, try saving a copy outside iCloud Drive and see if it saves normally there.