That's an insightful perspective and I think I largely agree. But just for fun, I wonder if that isn't an argument in favor of making the function implementation impure. Perhaps "enhancing" all queries with some sort of search result (or query of a giant db) instead of charging for an explicit tool call. Not only is it sorely needed to prevent stale data but (on the process level) it breaks the purity assumption on which the approximation theorem depends (alternatively on the function level it introduces hidden inputs).

This is why every AI company does crawl today.

Do they just reshape the function on the fly or save the process steps? Maybe it doesn't matter anymore. Even Google indexes are more and more spoiled to become representation of the function, because of the AI slop.

Genuine live data is king.