re: How to handle bill from AC service company if nothing was fixed?

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They have to honor the work and come back out for no charge.

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This. It is called a re-call. Always make sure a service company or plumber details the work they did on their invoice. If they have to come back out and make a repair on the same part, a reputable company will cover it. If it is a different part that breaks, they will charge you to repair that part, but usually will waive the service call fee.