The title of David Auburn's play "Proof" refers to two things: a specific mathematical proof that comes to dominate the plot as well as the human need to offer proof, to demonstrate to others our value, our worth, the truth of our talents and the sincerity of our emotions. In that sense "Proof" is an especially appropriate title for a film that ended up having a lot to prove.

For despite its Pulitzer Prize-winning pedigree, and in some ways through no fault of its own, the John Madden-directed film raised several potentially troubling questions. Was the delay in releasing a drama originally scheduled for last December due to its quality? Could this quintessentially theatrical vehicle be successfully transferred to the screen? And could star Gwyneth Paltrow handle this kind of powerful, seriously dramatic starring role?

Against considerable odds and despite a shaky start, "Proof" proves itself in every area. Thanks largely to Paltrow's beautifully unadorned performance, an exceptional portrait of psychological fragility that is honest, direct and devastating, this is a film that really has to be seen.

It's a measure of both the tabloidization of our culture and the nature of Paltrow's filmography that her work here will surprise those who only follow her private life and didn't catch her fine work as Sylvia Plath in the little-seen "Sylvia" or don't remember her stark performance in 1993's "Flesh and Bone."

Paltrow's Oscar for "Shakespeare in Love" notwithstanding, it's hard to think of as gifted an actress who's chosen to appear in so much forgettable piffle, and we can only hope that a new, more balanced and fruitful cycle is beginning for her.

Though three of its four main characters are mathematicians, "Proof" is intent not on exploring math itself but on using the subject as a vehicle for investigating issues like the excitement and painful fragility of both creativity and personal relationships and the unnerving proximity of madness and genius.

Screenwriters Auburn and Rebecca Miller have opened up the play by adding new locations and one major new scene. But "Proof's" story is still set primarily in and around a large old house near the University of Chicago, and as such initially feels uneasily stage-bound.

The house has been home to Robert (Anthony Hopkins), a groundbreaking mathematician who made major contributions to three fields and revolutionized one of them twice before he was 22. It's now inhabited by his 27-year-old daughter Catherine (Paltrow), who is about to be visited by her New York-based sister Claire (Hope Davis). Also a visitor is Hal (Jake Gyllenhaal), a young protégé of Robert's who has become a professor in his own right.

"Proof" opens on a rainy night with a bedraggled Catherine absent-mindedly changing the channels on the TV. Robert comes in and engages her in conversation about her forlorn state. From the moment he opens his mouth, something is not right, and in fact it's immediately revealed that Robert has just died, setting "Proof's" plot in motion.

Robert, it turns out, was not just a genius, he also was someone who increasingly went in and out of sanity as he got older. He'd had lucid periods, one lasting a full nine months, but the last three years had been marked by bouts of madness that included filling 103 notebooks with compulsive writing, and a belief that "aliens were sending him messages through the Dewey decimal numbers on library books."

Catherine is in the state she's in because she's been her father's sole caregiver. Once a promising math student who, her father boasts, knew what a prime number was before she could read, she dropped out of school, giving up her career and her life as her father's illness progressed. Very much like him, she has her own demons to fight, including the fear that she has inherited her father's tendency to madness.

The arrival of Claire for their father's funeral only makes things worse. Pulled together and assertive where Catherine is frumpy and distracted, Claire, beautifully played with a fine sisterly edge by Davis, brings out the worst in her sibling.

Catherine comes off to her sister as awkward and humorless, a wary woman with few social graces and multiple hard edges. Both Claire and Hal, her father's former student, for their own reasons desperately want to help Catherine regain her footing. Then, suddenly, something happens that completely changes everyone's equations.

Just as the film's stage-bound qualities seem problematic early on, so does Madden's direction. The man who guided Paltrow in "Shakespeare in Love," he is a filmmaker whose instincts can feel too much on the nose, and in "Proof's" initial stages that tendency makes the characters and the story feel unduly predictable.

But Madden, who also directed Paltrow in a well-reviewed London stage production of "Proof," improves as the film progresses. His familiarity with the material and his history with Paltrow increase the film's intensity to the point it trumps the underlying earnestness. Madden also displays the ability to create a safe place not only for the star but for all his actors to do their best work.

Basically a four-hander, "Proof" is blessed in all its key roles. Hopkins, the heart of the film's extensive flashbacks, has the kind of great-man presence his role demands. Davis takes the character that veers closest to cliché and gives it intensity and believability, while Gyllenhaal in his most adult part to date brings essential sweetness to the contradictory persona of a heartthrob math geek.

Finally, however, "Proof" is incontrovertibly Paltrow's movie. Her performance is so emotionally naked, her on-screen uncertainty and insecurity so piercing, that the raw pain at the core of this story never fades from view. Hers is a performance that redeems everything else about this film, and allows for the hope that there will be many more to come in this gifted actress' career.

'Proof'

MPAA rating: PG-13 for some sexual content, language and drug references

Times guidelines: Mature subject matter

Released by Miramax Films. Director John Madden. Producers Jeffrey Sharp, John N. Hart Jr., Robert Kessel, Alison Owen. Screenplay David Auburn, Rebecca Miller, based on the play by Auburn. Cinematographer Alwin Kuchler. Editor Mick Audsley. Costumes Jill Taylor. Music Stephen Warbeck. Production design Alice Normington. Art director Keith Slote. Set decorator Barbara Herman-Skelding. Running time: 1 hour, 40 minutes. In general release.