Proof is a 2005 American drama film directed by John Madden and starring Gwyneth Paltrow, Anthony Hopkins, Jake Gyllenhaal, and Hope Davis; it was written by Rebecca Miller, based on David Auburn's Pulitzer Prize-winning play of the same title.
The film opens with the 27-year-old protagonist, young mathematician Catherine (Gwyneth Paltrow) talking to her father Robert (Anthony Hopkins) after he startles her watching TV in the middle of the night. He gives her a bottle of champagne for her birthday, and they chat for a while about the nature of insanity - ending with the revelation that Robert died last week and his funeral is tomorrow. Awoken from this dream, Catherine realizes that Hal (Jake Gyllenhaal), a former student of Robert's, is still upstairs, reading through Robert's books. Hal believes that somewhere in the gibberish there might be additional products of his mathematical genius. On his way out, he comments on the vast amount of work Robert did. Catherine, suspicious, searches his backpack but finds nothing, but when she picks up Hal's coat to give to him, a notebook falls out. He explains that he wanted to give the notebook as a birthday present because it "had something written in it about her, not math, her". Hal is forced to leave, gifting the notebook as intended, when Catherine calls the police. The next day, for the funeral, Catherine's sister Claire (Hope Davis) arrives in town. A huge contrast to the unkempt Catherine, Claire is an overly put together, neurotic New Yorker. Relations between the sisters are tense, and Catherine can't stand her sister's constant harping on appearance-related issues. Catherine is also upset that Claire didn't care for her father as much as Catherine did in his final years. A series of flashbacks documents Catherine's ultimately futile struggle to keep up with her own studies as she takes care of her father. At the funeral, a lot of frustration on Catherine's part comes out. She is upset at the high number of people there, and she interrupts the string quartet with an impromptu speech, yelling at everyone for not being there for her father while he was alive, saying she's glad her father died, and storming out of the building. Claire is set on selling Robert's house back to the University, and wants Catherine to come with her to New York; Catherine is upset that she will be kicked out of the house. It comes out that Catherine may be struggling with the same mental illnesses that her father struggled with. They have a wake at the house the night after the funeral, and Catherine and Hal end up sleeping together afterwards. After they consummate their affections, Catherine is seemingly comfortable with Hal's intentions toward her family, and gives him a key to her father's desk. He finds a notebook which contains a lengthy but apparently very important proof. He is very excited and shows the discovery to Catherine and Claire. He asks Catherine how long she knew about this and why she didn't tell him about it. She tells him that she wrote it. Catherine claims the work is hers and not her father's despite evidence to the contrary, and neither Hal nor Claire believe her. Hal decides to take it to the math department tomorrow to verify the proof's accuracy. He returns as Claire and Catherine are leaving, with news that the math department believes the proof to be valid. Hal tells Claire that he doesn't think that her father wrote the proof because it employs newer mathematics, and wants Catherine to explain it to him sometime. Catherine remains stung by his earlier lack of trust, and the sisters leave for the airport, but Hal sprints after the car and throws the book through the window and onto Catherine's lap. At the airport, Catherine has another flashback. It is revealed that while living together, her father challenges her to work on math, which she does, and she did in fact write the ground-breaking proof. She goes to tell her father about the breakthrough, but he insists she read aloud his own, nonsensical "proof". Realizing that Robert has slipped back into illness and wouldn't understand her work, she locks her notebook into his desk drawer and keeps the key (which she presents to Hal earlier in the film.) Catherine has begun to come to terms with herself, aided by Hal's confidence in her. She decides that she does not need to go with her sister to New York and runs out of the airport. She returns to University of Chicago, and the film ends with she and Hal meeting up on campus and discussing the proof.