It’s a week after Claire gave Hal the notebook. Claire is on the porch, where she grasps a plane ticket and checks her schedule. Catherine steps outside carrying some bags. Claire gives her a cup of coffee and rambles a bit about a nice coffee shop in New York that that Catherine will enjoy. Catherine seems mildly interested, but there is an awkward pause.
Catherine’s carrying bags onto the porch signifies that she has given in to Claire’s demands: she’s going to move to New York City. No longer resistant to Claire’s arguments, it seems like Catherine’s spirit was crushed once Hal broke her trust. She doesn’t seem to have confidence in herself, either, so she has let Claire dictate her next life step.
Claire is overly attentive to Catherine, asking if she wants to go inside or wear a jacket, or if she wants some time to be alone before they leave. Catherine says she is fine. Claire says that moving is the right choice—while it will be hard to leave, everything will be better once they get to New York.
Now that Claire has got her way—Catherine has agreed to move to New York—she keeps asking Catherine about her preferences for small things, like whether she wants to wear a jacket. At this point in the play, it’s clear that Claire doesn’t actually respect Catherine’s wishes. She just wants affirmation from her sister that her own decisions are the right ones—possibly to assuage her guilt for prioritizing herself over her sister—but she will never actually change her plans to reflect her sister’s desires.
At first, Catherine passively accepts what Claire is saying, but the more Claire tries to empathize with her, the more sarcastic Catherine becomes. Eventually, Catherine exclaims that she can’t wait to blame Claire for all her problems during therapy.
Catherine at last shows a sign of her spunk when she insults Claire, implying that Claire is the reason for all her problems. Catherine may have snapped because Claire was acting as though the two of them are closer than they are. Catherine, however, doesn’t feel close at all to her sister, since Claire broke her trust multiple times, from having left her alone to care for Robert to not believing that she wrote the proof. Catherine’s sarcastic attitude reminds Claire that they do not have a good relationship.
Aggravated, Claire tells Catherine to stay in Chicago if she thinks she can handle it. Catherine insists that while she is going to go to New York, she could stay if she wanted to. But Claire notes that Catherine couldn’t even get out of bed for almost a week. When Catherine insists that she was simply tired and didn’t want to talk to Claire, Claire tearfully tells Catherine to just stay in Chicago. Catherine asks what she would do and Claire tells her to “figure it out” and storms off the porch.
Claire does seem to have good reason for wanting to give her sister medical help—Catherine just suffered a mental breakdown that left her bedridden for a week. But Claire’s problem lies in how she is trying to help her sister. Instead of heeding Catherine’s wishes for how she wants to be cared for, Claire chooses what’s most convenient for her (such as moving Catherine to the city where Claire lives) and then tries to convince Catherine (and perhaps herself) that this is what’s best for her. Claire’s method of “caretaking” has a disastrous outcome: it destroys her and Catherine’s relationship.
Catherine remains on the porch and Hal suddenly appears, sweaty and out of breath from running. He’s relieved that Catherine is still there. Brandishing the notebook, he tells her that the proof is solid. He’s looked it over two times with multiple guys—both old and young. While there are some unconventional moves, the proof checks out. Catherine coolly tells him she already knows.
Hal was clearly rigorous in checking the proof—he’s gone over it multiple times with several different people to make sure that the proof is accurate. Catherine is unimpressed with his “discovery,” since she is presumably the author of the proof and therefore already aware that it works.
When Catherine tells Hal that she’s leaving, he asks her to wait. Apathetic, she tells him to publish the proof, and that she doesn’t care if he publishes it under her father’s name, or even his own. But Hal tells her that he doesn’t think that Robert wrote the proof after all; it uses some recently developed mathematical techniques, ones that Robert likely wouldn’t have known.
Since Hal broke her trust, Catherine is uninterested in what happens to the proof. The proof seems to have been her way of proving her capabilities and talent, so now that she suspects that she won’t get the credit, she doesn’t really care about its fate. She’s clearly aware that people will likely believe that Hal or Robert—both men—wrote the proof. As a man, Hal has the power in this situation; if he tells the math community that he or Robert wrote the proof, he would be readily believed, while Catherine would have no evidence to prove that she is actually the author.
Catherine replies that Robert could have read about the newer techniques. Hal admits that it’s possible, but that the proof—unlike Robert’s other notebooks—is undated. As for the handwriting, “Parents and children sometimes have similar handwriting, especially if they’ve spent a lot of time together.” After a moment, Catherine tells Hal that she already told him all this, but he “blew it.” She sarcastically congratulates him on “[getting] laid and […] [getting] the notebook.”
At last, Hal believes that Catherine wrote the proof because he found evidence to prove that it was hers. This evidence also signifies that Catherine is her own person, and not just a replica of her father. At first glance, Catherine’s work looks a lot like Robert’s, so much so that Hal and Claire mistook it to be his. But, in the end, her work bears signs that someone other than Robert wrote the proof: the books aren’t dated, and the techniques are newer. Hal also concedes that “Parents and children sometimes have similar handwriting, especially if they’ve spent a lot of time together,” which makes the important point that Robert’s influence on Catherine isn’t just genetic—it’s also due to the fact that they lived together for years and had a very close relationship. Hal has been very diligent in gathering information to determine who wrote the proof. Although he had originally dismissed Catherine’s claim, he nevertheless reevaluated it as he kept analyzing the proof. In the end, Hal’s skepticism and relentless analysis of evidence led him to right conclusion (that Catherine is the author), even though his doubt hurts her feelings. While Hal feels victorious in his discovery, Catherine is very cold to him. It doesn’t matter to her that he’s changed his mind to side with her—he broke her trust and destroyed their relationship.
Hal says he would love to at least hear Catherine talk about writing the proof, but she says no and refuses to even take the notebook back. Hal asks her how he can remedy the situation, but Catherine exclaims that he can’t—all of his math and research doesn’t change the fact that he didn’t believe her. Hal concedes that he should have.
Catherine doesn’t care that Hal believes her now that he has the evidence needed to prove that her claim was true. She had trusted him enough to show him her proof, but he broke her trust when he didn’t believe her.
After a moment, Hal asks Catherine if she’s really going to New York City. She says she is, and Hal urges her to stay in Chicago. She says that it might be nice to be taken care of and to get out of the old, drafty house. When Hal quietly assures her that there’s nothing wrong with her, Catherine admits that she’s afraid she is like her father, to which he responds that she is “not him […] Maybe [she]’ll be better.” He holds out the book, and she accepts it.
The only way for Hal to rebuild his relationship with Catherine is to prove to her that she can trust him—he needs to give her evidence that shows that he is trustworthy. He does just this, expressing his desire that she stay in Chicago and assuring her that he has confidence in her talent and her mental health. Catherine, meanwhile, is still anxious that she has inherited Robert’s mental illness. Hal tries to comfort her by telling her that she is her own person and that she may prove herself to be healthier and more talented than Robert. After all, she just demonstrated her genius by writing a groundbreaking proof. What’s more is that she has a unique style and thought process that distinguishes her from her father. At this point, the conversation is less antagonistic, and it seems that Hal has convinced Catherine to give him another chance. While she still seems hesitant, he has nonetheless given her evidence that he is interested in her, believes in her capabilities, and is prepared to continue to prove his worth.
As she traces her fingers over the book, Catherine tells Hal that writing the proof was like “connecting the dots.” She never worked on it with Robert, although sometimes she’d watch television with him in the middle of the night when she felt stuck. She slowly explains that, while her proof works, it isn’t as “elegant” as Robert’s work. Hal encourages her to talk it over with him, that maybe she’ll be able to improve it. Catherine hesitates—but after a moment, she opens the notebook and begins reading.
Once again, genius is associated with perception. Catherine’s proof is a ground-breaking and historic proof; as Hal explained earlier in the play, mathematicians have been trying to prove Catherine’s mathematical theory for centuries, but no one ever managed to do it. Yet, for Catherine, writing the proof was simply a question of perception—as she tells Hal, writing the proof was like “connecting the dots.” The implication is that Catherine can see connections or patterns that others can’t. Significantly, this is similar to both her and her father’s mental illness. For both of them, they often perceived or imagined things that weren’t there. This is what makes it so difficult to distinguish their madness from their genius: in both cases, they are interpreting the world in a unique way. In fact, it is due to this exceptional perception that Catherine is able to write her innovative work. But Catherine doesn’t feel satisfied with her work yet—there are still points where it could be improved. At the end of the play, she makes a leap of faith in deciding to trust Hal again when she agrees to let him help her on her proof. Although he broke her trust before, Catherine is willing to give him another chance. Hal, eager to show her that she can count on him, assures her that he has confidence in her ability to improve it.