Batman is transported to the real world. He will act in character for the duration of his time here. While he won't have the Batcave transported here, he does have a Rick and Morty style portal gun which he can use to travel to the Batcave and grab anything he might need.

Round 1: Can Batman prove that Jeffrey Epstein was murdered, who murdered him, unravel any larger conspiracy involved, and if so get these people arrested? He was one month.

Round 2: Can Batman prove that Jimmy Hoffa was murdered, where the body was buried, who was involved, and get anyone still alive who was involved in the crime to go to prison? He has one month.

Round 3: Assuming that Lee Harvey Oswald did not act alone, can Batman definitively prove who was involved in the JFK assassination and why, and get anyone who is still alive arrested? He has one month.

Round 4: Can Batman discover the identity of the Zodiac Killer and get him arrested if he is still alive? Batman has one month.

Round 5: Can Batman decipher the Voynich Manuscript? He has one month.

Round 6: Can Batman discover who was behind Cicada 3301? He has one month.

Round 7: Can Batman definitively prove the existence or nonexistence of alien life in our universe(Not the DC Universe). For this one, he has a year.