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Proof is a dramatic film released in 2005, directed by John Madden and starring Gwyneth Paltrow, Anthony Hopkins, Hope Davis, and Jake Gyllenhaal. The movie is an adaptation of the Tony Award-winning play with the same name, written by David Auburn. The story revolves around Catherine, a young woman played by Gwyneth Paltrow, who struggles to cope with the death of her father, a brilliant mathematician named Robert, played by Anthony Hopkins. After his death, Catherine finds herself in a state of depression, feeling lost and uncertain about her future.

Catherine's sister Claire, played by Hope Davis, arrives to help her with funeral arrangements and advises Catherine to move to New York with her. However, Catherine is torn between staying in her childhood house and honoring her father's work that she discovered after his death in a pile of notebooks filled with complex mathematical proofs. The notebooks reveal the extraordinary mind of Robert, and they become the focal point of the film.

Throughout the film, Catherine struggles to convince the rest of the world that the proofs are her own doing and that they are not the product of a deteriorating mental state brought upon by grief. Her relationship with her father becomes increasingly strained as she realizes that his illness and obsession with mathematics might have been passed down to her. Meanwhile, she forms a deeper connection with Hal, a former student of Robert's, played by Jake Gyllenhaal, who begins to uncover secrets about Catherine's father's life and work.

The film revolves around themes of love, loss, identity, and truth. It explores the complexity of familial relationships and the impact that mental illness can have on individuals and their loved ones. The mathematical language throughout the film is used to symbolize the struggle of trying to grasp the intangible concepts of life and answer questions that don't have easy answers.

Gwyneth Paltrow delivers a powerful performance as Catherine, capturing the character's vulnerability and courage in equal measure. Her portrayal of a young woman dealing with mental illness and intense grief is nuanced and realistic, lending the film an emotional weight that is felt throughout. Anthony Hopkin's Robert is a complex and layered character, haunted by his past and the legacy he leaves behind. Hope Davis's portrayal of Claire provides a counterpoint to Catherine's character with her sharp wit and control over her emotions.

The film is visually stunning, with its cinematography capturing the stark beauty of Chicago in the winter. The use of light and shadow is masterful, creating a sense of intimacy and intensity throughout the film's most pivotal moments. The film's score, composed by Stephen Warbeck, is both haunting and beautiful, adding to the emotional weight of the film.

In conclusion, Proof is a well-crafted film adaptation of the Tony Award-winning play of the same name. With its strong performances, beautiful cinematography, and gripping story, it is a highly emotional and thought-provoking work that explores the complex nature of human relationships and the challenges of dealing with loss and mental illness. The film's themes of love, identity, and truth are universal, making it a powerful and enduring work that resonates with audiences to this day.

Proof is a 2005 drama with a runtime of 1 hour and 39 minutes. It has received moderate reviews from critics and viewers, who have given it an IMDb score of 6.7 and a MetaScore of 64.