Proof

The film opens with a touching late-night conversation between Catherine and her father, Robert. He surprises her with a bottle of champagne for her birthday, which she critiques humorously, leading to a deeper discussion about sanity and perception of mental illness.

In a shocking twist, Robert reveals to Catherine that he passed away the previous week and his funeral is scheduled for the following day. This unexpected confession adds a layer of complexity to their discussion about sanity and madness.

Catherine awakens from the dream to find Hal, a former student of Robert's, still in the house. Hal reveals his aspirations as a mathematician and expresses his belief that there are undiscovered notes of Robert’s genius among his belongings.

As Catherine interrogates Hal, the tension escalates between them as she searches through his backpack. A notebook falls out, intended as a gift for her, which contains a compliment that unexpectedly highlights the developing bond between them.

The following day, the family gathers for Robert's funeral, revealing the strained relationship between Catherine and her sister Claire. Claire's lavish arrival contrasts sharply with Catherine's disheveled appearance, signifying their differing approaches to family responsibilities.

During the funeral service, Catherine's frustration reaches a boiling point. She publicly vents her anger at the attendees who neglected Robert in his final years, even expressing relief over his passing, leading to a dramatic exit.

After the funeral, Claire reveals her plans to sell Robert's house back to the University, leaving Catherine distressed about losing her home. This moment amplifies Katherine's fears of inheriting her father's mental health issues.

On the night of the wake, while Claire mingles with academics, Hal and Catherine find comfort in each other's company. This shared experience leads to an intimate encounter that deepens their connection amid the chaos surrounding them.

Following their encounter, Hal gains access to Robert's desk and uncovers a monumental mathematical proof. He excitedly presents his findings to Catherine and Claire, but they respond with disbelief, leading to a pivotal confrontation.

In a heated confrontation, Catherine claims authorship over the mathematical proof, which triggers doubt from Hal and Claire regarding her capabilities. This moment further explores insecurities and the impact of familial expectations on her mental state.

As the tension escalates, Catherine experiences an emotional breakdown, revealing her internal struggles and guilt. She confesses to Claire, 'I stole it from him,' bringing to light the heavy burdens she has been carrying throughout her father’s illness.

Determined to prove the proof's legitimacy, Hal takes it to his colleagues for validation. Their approval serves as a crucial moment for Catherine as she grapples with her identity and the pressures surrounding her father’s legacy.

Claire's pressure leads Catherine to consider a departure to New York, amplifying her feelings of inadequacy. However, Hal intervenes, urging her to recognize and embrace her innate worth, which pushes her towards self-affirmation.

At the airport, Catherine reflects on her journey and begins to acknowledge her own talents. This moment symbolizes her growth and her decision to carve her own path, independent from Claire’s expectations.

Rushing back to the University of Chicago, Catherine achieves a heartfelt reunion with Hal. Their conversation about the proof signifies her acceptance of her genius and the budding relationship they share, amidst the bustling campus life.