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visitors who end up at the 404 page are going to be confused enough; adding a search box tellsthem they are not really wanted, doesn't it?
Much better to have a useful and functional page that refers visitors where they want to go *within* your site. That's what a 404 is for!
As for the TOS - I don't know if it breaks the 'letter of the law' - but I'm also sure that's not the kind of thing that Google intends you to do.
Google AdSenseTM Online Standard Terms and Conditions
5. Prohibited Uses. You shall not, and shall not authorize or encourage any third party to:
(v) display any Ad(s), Link(s), or Referral Button(s) on any error page, on any registration or "thank you" page (e.g., a page that thanks a user after he/she has registered with the applicable Web site), on any chat page, in any email, or on any Web page or any Web site that contains any pornographic, hate-related, violent, or illegal content;
Have you e-mailed AdSense support to ask them? Getting a direct answer from the source would be more reliable than any speculation here about exactly what the TOS means.