Why exactly is John Proctor sentenced to be hung?
John Proctor initially says that he isn't a witch or bewitched. This is the truth. He was not, we never saw any evidence of him being possessed throughout the entire play. However he does say in the end of Act III that "God is dead!" Although he means that he can see no evidence of God at work among the court because what they are doing is logically and morally wrong, his words are taken as if he is a possessed man full of the Devil's will, not God's. Throughout the play, Proctor uses sarcasm. He could be doing so in the end of Act III. He is sentenced to be hung because of the belief that he is indeed in the Devil's book and at work for the Devil.
Eventually (in the end of Act IV) he is willing to sign his name to the lie that he is a witch. The court officials had agreed that if he would at least give his name and sign a statement he could have his life. After signing they want him to allow the statement to be available for all church members to see. He doesn't want the document to go public and that is what ultimately gets him hung. He holds the document captive from the officials because his reputation is more important. The truth is that he is an upstanding and God-fearing man. Now that his relationship with God is restored and as Elizabeth says, "He [has] his goodness now," he is ready to die.
john proctor is held captive because in Act 3, with the part when the yellow bird comes. Abigail points the attention to Mercy Lewis but makes it as Proctor is performing witchcraft only to save her own well being.
"What did John Proctor lie about", uhh, kinda hard to respond, maybe you could change the question.