How do I remove Apple Intelligence permanently from macOS?

It was not entirely clear to me what you want to remove. AI itself cannot be removed, but you can disable it. The storage space it uses will vary and most likely will diminish once you disable AI. How long that takes is anyone's guess. "Safe Mode" may accelerate that process but Apple is silent on that specific subject. It won't hurt to try it.

Start up your Mac in safe mode - Apple Support

chdsl wrote:

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I have rebooted the Mac, but that information about Apple Intelligence is using 7.06GB is there.

That's not really waiting for the system to clear is it...(?)

I will say patience is a virtue; I would give it 2 or 3 days before you take another peek.

If it is mission critical to your peace of mind— you can try SafeBoot Start up your Mac in safe mode - Apple Support & Reboot  as normal and test.  Caches get rebuilt automatically.

I have no idea if the Ai cache is included here, but you can report back.

chdsl wrote:

John Galt wrote:

System Settings > Apple Intelligence & Siri > Apple Intelligence > off

It's off, but it is there taking space in the disk. Storage > macOS > click on the info button

https://discussions.apple.com/content/attachment/81814a07-d047-4b8b-960d-ac8175b5f232

, it says Apple Intelligence 7.06GB.

Wait for the system to clear the cache...

When you disable Apple Intelligence, the operating system recognizes that the associated files are no longer in use.

It is designed to automatically clear these files when additional storage is needed. 

You're lucky. On my M4mini it takes up 10+GB. Yes, I have it turned off.

I'll let other provide the definitive answer but I suspect those may be just the program files and might be regarded as part of the "system" the way the formerly optional iTunes/Music became part of the system and could not be deleted.

System Settings > Apple Intelligence & Siri > Apple Intelligence > off

Like many macOS components AI will require storage space whether you use it or not.