I live in California. The carpet was not dirty or beyond normal wear and tear
Landlord / Tenant Lawyer
Given your fact situation, the landlord may not charge you for cleaning the carpet, but I think he is entitled to one day's rent. Arguably, the parties could quibble about a partial day, but in the real world as I have experienced it, no one ever does. Keep in mind, you are responsible for payment of rent for every day you retain possession, even the final day while you are moving out. Until the landlord has exclusive right of possession to the property, the tenant is responsible for rent.
Real Estate Attorney
If your lease ended May 31 but you did not move out until June1, then you owe one day's rent.