The next morning, Catherine is sitting on the porch when Hal, who is partially dressed, steps out to join her. Claire is still sleeping, having drunk heavily with the mathematicians yesterday. There’s a bit of awkwardness between Catherine and Hal, and Hal isn’t sure whether he should stay or go. At last, he tells Catherine that he wants to spend the whole day with her, although he doesn’t want to be too intense. She laughs at his awkwardness—both are relieved, and they kiss.
After sleeping together, both Hal and Catherine are unsure how to act with each other. Each one appears to need encouragement from the other person that they are still interested in building a relationship, particularly one that’s more than just sexual. The implication is that, in building interpersonal connections, one has to continually prove one’s trustworthiness. Hal does this by assuring Catherine that he wants to spend the whole day with her. In return, Catherine kisses him, expressing her desire that he stay.
As they break apart, Hal tells Catherine that last night was incredible. Catherine pauses and, after thinking it through, pulls out a key that’s attached to a necklace that she is wearing. She gives the chain to Hal; he can use it to open a drawer in Robert’s office. After Hal leaves, she smiles, privately excited.
At this point in the play, Catherine’s budding relationship with Hal appears to be the only thing that makes her happy—she at last has someone whom she can trust. To demonstrate this trust, Catherine gives Hal a key to something (the audience doesn’t know what) that she keeps locked away. The implication is that, now that Hal has given her evidence that she can trust him—he is actively working on building a relationship with her—she feels comfortable enough to trust him in return, which she expresses by giving him access to something that she doesn’t trust others with.
Claire comes onto the porch, extremely hungover and cursing the physicists that she had tried to “keep up with” the night before. She tells Catherine that her dress looked good on her; to Claire’s surprise, Catherine thanks her for the dress and says she loves it.
Catherine’s budding relationship with Hal is giving her a lot of joy—so much so that she is even nice to her sister, thanking her for the dress. Catherine’s kindness to Claire can be interpreted as Catherine giving Claire the opportunity to rebuild their relationship. Now that Catherine feels that she can trust one person—Hal—she appears to be more inclined to try building a relationship with Claire. In a way, when someone proves that they can be trusted, they are evidence that people in general can be trustworthy.
Claire takes a deep breath and tells Catherine that she wants her to come to New York City with her. At first, Catherine thinks Claire is talking about the wedding, but she realizes that Claire wants her to move permanently. Catherine politely declines, telling her sister that, after the past few difficult years, she wants some time to stay in Chicago to think things through before making any big decisions. But Claire tells Catherine that she is already in the process of selling the house—Catherine needs to leave soon.
Claire ruins her chance at rebuilding her relationship with her sister by giving Catherine evidence that Claire cannot be trusted to respect Catherine’s wishes. While Catherine believes that the best thing for her would be to stay in Chicago and take some time to reflect on what her next step in life should be, Claire insists that she knows what’s best for Catherine, which is that Catherine should move to New York.
Catherine is furious. Claire claims that she is trying to help and wants to make up for having left Catherine alone with Robert for so many years, but Catherine bitterly asks why she’s trying to “help” now instead of years ago. Claire says she was too busy to live with Robert, then she suggests that their father would have been better off in an institution. But Catherine insists that being by the things that made him happy—the university, his students, his house—allowed him to get better, even if it was only for a short while.
Claire’s version of helping—ignoring others’ wishes when making plans that prioritize herself—hasn’t just affected Catherine. Claire took this same approach when Robert was ill. Whereas Catherine sacrificed her life to give Robert what he wanted—a chance to live at home, surrounded by the things that made him happy—Claire insists that an institution would have been better. Claire may be saying this simply because she feels guilty that she wasn’t as selfless as Catherine. It’s possible that she even deluded herself into thinking that institutionalizing Robert was what was best for him, which demonstrates how easy it is for someone to convince themselves that what they want is what is best for other people. Either way, Claire continues to prove that she doesn’t consider other people’s wishes or opinions when she makes plans that affect them. Through Robert’s journal entry (the one in which he says that Catherine’s decision to give up much of her life to care for him saved him), the play makes it clear that Catherine’s selfless approach to caretaking is the right one.
Nonetheless, Claire believes an institution would have helped Robert more, and also that Catherine may have “been better” in that scenario. Catherine demands to know what Claire means. Uncomfortably, Claire says that Catherine inherited some of Robert’s genius and “instability.” After a moment, Catherine sharply asks if Claire has been looking for institutions for her in New York.
Claire appears to feel guilty for abandoning Catherine to care for Robert on her own, which may be why she is so intent on “helping” Catherine now. Claire is certain that Catherine’s extensive caretaking for Robert had a negative effect on Catherine’s mental health, which suggests that familial influence isn’t only limited to genetics. In other words, Catherine may be experiencing symptoms of Robert’s mental illness not because she inherited the illness genetically, but because she spent a lot of time with Robert and the stress of caring for her ailing father harmed her mental health. In this way, a person’s family influences them not just through genes, but through their environment as well. At the same time, Claire also makes it clear that she thinks genes still play a role: she believes that Catherine has inherited both Robert’s brilliance and his instability. Her statement suggests that, in their family, genius doesn’t come without instability—it’s because Catherine is brilliant that she is showing signs of madness, and vice versa.
At first, Claire tries to deny it, but she does admit that the excellent doctors of New York City are indeed one of her motivations for encouraging Catherine to move. Enraged, Catherine begins to tell her sister that she hates her, but then Hal enters carrying a notebook.
Claire completely breaks any remaining trust that Catherine has in her when she admits that she has an ulterior motive for moving Catherine to New York: she wants to make Catherine get help for her mental illness. Not only does this show how Claire’s doubting Catherine hurts their relationship (Catherine knows she cannot trust Claire to support her), but it also demonstrates another way in which Claire doesn’t respect Catherine’s wishes for her own future. The audience is sympathetic towards Catherine; at this point, Claire’s method of caretaking—not listening to others’ wishes and prioritizing what is most convenient for her—appears to be the wrong thing to do.
Hal asks Catherine how long she has known about the notebook. She says “a while,” and he asks why she didn’t tell him about it sooner. Catherine says that she hadn’t known if she wanted to tell him. After a moment, Hal thanks her effusively. Confused, Claire asks what’s going on.
The notebook is presumably the object that Catherine had locked away with the key that she gave Hal. She hadn’t shown the notebook to him earlier because she wasn’t sure if she could trust him—she waited until he had given her evidence that he was trustworthy. The implication is that one cannot simply claim to be trustworthy; one must prove it.
Hal announces that the notebook contains a very important proof, although he admits that he hasn’t checked it. In fact, the proof is so complex that he isn’t sure that he could verify it. It appears to be a theorem about prime numbers, which, if proven accurate, would be a ground-breaking discovery. Hal tells Claire that Catherine found it. But Catherine says she didn’t find it—she wrote it.
The play never explains what exactly the proof proves, but it does imply that it may prove a pattern for prime numbers. Prime numbers are notoriously difficult numbers in math—while they are a sequence of numbers, there is no proven pattern to them. For Catherine (if she is indeed the author of the proof) to find a pattern to them, she would be seeing something that no one else has been able to. In this way, a unique ability to perceive what others can’t is key to genius.