Theory: Frank Costello knew Billy Costigan was a cop in The Departed

Hey everyone, just watched The Departed (2006) for the first time, and a detail struck me about Frank Costello that completely changes the movie.

I’m convinced Frank knew Billy Costigan was the mole all along, because he left the recorded tape of Colin Sullivan to Billy.

After Frank dies, his lawyer tracks down Billy and hands over a recorded tape documenting Frank's conversations with his police rat, Colin Sullivan. Billy explicitly tells Colin later: “Costello's lawyer came to me... Costello trusted me more than anyone.” Billy then mails that tape to Colin’s apartment, which is the envelope Madolyn intercepts and plays. Even if Costello told his lawyer to hand it over to a random survivor in his crew, why not choose seasoned veterans like Fitzy or Tino who were still alive? Why choose Billy?

If Frank genuinely believed Billy was just a regular street thug, leaving him that tape makes zero sense. Why would a regular gangster care about exposing a corrupt detective?

But if Frank knew Billy was an undercover cop, the move changes how a base interpretation would paint Costellos character. I believe Frank used Billy's real identity as a law enforcement pipeline to ensure Colin Sullivan would be destroyed.

PS: I have never used Reddit in my 18 years of life, but I looked everywhere and have not seen this theory at all. Please let me know what you guys think!