One way to see the bottle of champagne that Robert presents to Catherine is as a symbol of the proof that he gives her later in the play.
At the beginning of the play, it is just after midnight, and Catherine has turned 25. Her father, Robert, gives her what he says is a bottle of champagne. She opens it. She tells him that it isn't really champagne. It's just some kind of sparkling wine. She has to drink from the bottle, because he forgot to bring glasses. He says that he's glad he's not a wine savant and that he's never been able to stand that kind of person. Of course, Robert isn't really there; he died a week ago. Catherine is just seeing him in her grief. The champagne and Robert are a sign that she might be in a mental decline like her father, which she truly fears.
Later in the play, Catherine comes home from school and finds her father doing work. He thinks that he's regained his ability to come up with brilliant mathematical theories. He wants them to work together. When Catherine reads his work, however, it's just rambling nonsense. Like the champagne, it's a sign that Robert isn't mentally well and is falling apart. Though he either isn't aware of it—or isn't willing to admit it—Robert is in decline. The proof, like the champagne, is a sign of it.