Complex, engaging, surprisingly witty and for the most part emotionally true, "Proof" is proof indeed that stage plays can be brought successfully to the screen, that they can take on cinematic strength, that they don't have to have that air of "it's a movie" desperation that too commonly destroys works in transition.
Credit director John Madden and playwright David Auburn for that. And credit Madden especially for guiding Gwyneth Paltrow through her finest role since Madden directed her Oscar-winning turn in "Shakespeare in Love." The two movies could hardly be more different, but each perfectly matches the Paltrow of their time. It's been a long drought between knockouts, but Paltrow certainly shows she's still got the stuff in "Proof." In fact, she has more stuff.
Paltrow stars as Catherine, the daughter of onetime mathematical genius Robert (Anthony Hopkins). Robert's mind disintegrated over the years and Catherine had to give up her own academic aspirations to take care of him. Now he has just died and she has been left not only wondering about her future, but also worrying whether her own mind might begin disintegrating the way her father's did.
It's the day before Robert's funeral, and one of his last student's, Hal (a fine, conflicted Jake Gyllenhaal), is going through papers in his study, trying to salvage some real math out of the notebooks filled with ramblings. Catherine's prim sister Claire (Hope Davis) will arrive in the morning to commandeer the funeral arrangements.
Eventually, Hal will stumble on a mathematical proof in Robert's office, a work of genius. And Catherine will declare that she wrote it, not her father, even though the handwriting looks like his. Hal and Claire will doubt her word, which will get her doubting her own sanity. And proof of any kind will be scarce.
Comparisons to Ron Howard's "A Beautiful Mind" will be inevitable, since there isn't exactly an abundance of movies exploring math and insanity. And the two films do share a concern with the fine line between genius and madness. But while "Mind" is epic and stretched over decades, "Proof" is contained for the most part, taking place in a matter of days aside from the flashbacks.
Moreover, "Proof" is essentially about trust, between father and daughter, sister and sister, lover and lover. And trusting in one's self, even when reality seems shaky and mean.
As a result, the entire film is dependent on Catherine's state of mind, the balance of fragility and inner resolve, the pain she feels after her father's death as well as the relief, the apparent betrayal of both family and lover. This is where Paltrow is startling, showing years absorbed and demons considered.
She is still young and beautiful, but the dewy innocence of "Shakespeare" has been replaced by natural wear and depth. Her Catherine is whip-smart and wounded, anti-social and craving connection, and Paltrow brings her to full life.
Hopkins appears in visions and flashbacks, and it's the best role he's had in a while as well, mad yet considered, joyously confident while unaware. And Gyllenhaal -- who'll be having some major months here with this, Ang Lee's "Brokeback Mountain" and "Jarhead," all Oscar contenders -- is something of a surprise, handling the back-and-forth role of Hal without ever tripping over the contradictions. It's smooth stuff.
The film's only glaring weakness, which does glare quite a bit, is the somewhat one-dimensional Claire. It has nothing to do with Davis, it's just that the character as written comes off shrill, especially when set beside the nuances of the others.
Claire starts out as someone you want to slap and ends that way as well; in a film this good she should have gone somewhere, learned something, confronted something. She doesn't.
Still, "Proof" is crackling good work, emotionally and intellectually interesting and grandly entertaining, saved from pomposity a good deal of the time by its sense of humor. It's been seven years since "Shakespeare," Gwyneth. You and Mr. Madden might not want to wait that long again.
You can reach Tom Long at tlong@ detnews.com.