What do you mean by "flat run"? Can you give an example sentence?
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Fandorin"You have about two minutes until Dr.Bishop dies. At a flat run I estimate about forty seconds until you get back to him. Now hand me your phone and....".This is not an expression I'm very familiar with (per GG, I'd favour "flat-out run"). However, the correct preposition definitely seems to be "at", not "on".