The next day, Hal knocks on the door and calls for Catherine. Claire steps onto the porch and explains that she had to delay her flight because Catherine is refusing to eat or leave her room. Hal wants to see Catherine, but Claire doesn’t let him—she thinks it would be a bad idea.
Catherine appears to be undergoing a mental breakdown: she isn’t eating, and she is isolating herself. The cost of Hal and Claire’s skepticism, then, is Catherine’s mental health.
Claire asks why Hal slept with Catherine, suggesting he took advantage of her. But Hal insists that it was consensual and asks again to see Catherine. When Claire doesn’t budge, he angrily tells her it’s not right to take Catherine to New York against her will. Hal suggests that seeing Catherine would help her, but Claire snaps that he—like all mathematicians—isn’t thinking. She promises to give him her New York phone number once she gets Catherine settled there.
The audience gets a glimpse of what Claire is like as a caretaker. She refuses to let Catherine have contact with anyone else, deciding for her that seeing Hal would be a bad idea. She doesn’t try to check in with Catherine to see whether she would like to see Hal, but instead dictates the terms on which Hal can contact Catherine.
Hal agrees, but he doesn’t move, since he has another reason for coming. To his surprise, Claire hands him the notebook, saying he can have it. Hal is confused—he thought that he’d have to argue for it. Claire snidely suggests that the notebook is the main reason he came, which Hal denies.
Claire doesn’t believe that Hal respects her sister. And, indeed, that’s what appears to be the case. Claire knows that her sister is in a fragile state, so it’s reasonable for Claire to worry that a man is exploiting her sister’s vulnerability for his own sexual gain. Additionally, even though Hal denies that the notebook is the main reason for his visit, it does seem that way—although he is willing to leave when he cannot see Catherine, he is ready to argue extensively for the notebook. His claim—that seeing Catherine, and not the notebook, is why he stopped by the house—is meaningless without proof.
Claire says she trusts him with the notebook, adding that he should call her when he has more information. As Hal begins to leave, Claire asks him to explain the proof to her. Hal asks how much math she knows, and she responds that she, a currency analyst, is pretty good with numbers—but she has only a fraction of Robert’s genius. Catherine is more gifted, although Claire isn’t sure how gifted.
While Catherine doesn’t trust Hal with her sister, she does trust him with the proof. It’s not clear why Claire believes that Hal can be trusted with the notebook (and not claim it as his own). Perhaps it is because she noticed Hal’s dedication to giving Robert credit for his genius—his hours of reading the notebooks are evidence of his desire to publish Robert’s work. Or perhaps the play is suggesting that, in times of uncertainty, it is sometimes necessary to make a leap of faith in order to reach one’s goals. Claire, like Hal, wants to know who wrote the proof. In order to know, she has to trust someone else to analyze it—as she says, she isn’t mathematically talented enough to understand something as complex as the proof. Claire’s comment about her intelligence has another significance—it supports the play’s message that genius and mental instability are inextricable from each other. In Catherine and Claire’s family, one either has both or one has neither. While Catherine appears to have inherited Robert’s mental illness along with his brilliance, Claire got neither (or, at any rate, only a fraction of Robert’s talent).