Must the Runner Clearly beat the ball to the bag???

[Skarecrow]

I have researched the rule book (can't find my case book), and can't find the wording or rule that defines "The runner (in order to be safe), must clearly beat the ball to the bag." Hence, ties do NOT go to the runner.....Am I correct in remembering the "tie to the runner rule?" Is this an old-wives' tale I am perpetuating? Bang-bang plays are out or safe....One bang (TIE) is an out...Correct??? And if so, where is the rule supporting that? Thanks for your help.....