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).