Elegantly adapted from David Auburn's Pulitzer Prize-winning play, "Proof" works on so many levels that it shines like a perfected equation. Gwyneth Paltrow previously played her role onstage, and returns here as Catherine, the troubled 27-year-old daughter of Robert, a once-brilliant mathematician (Anthony Hopkins, appearing in flashbacks and imagined visions) who has recently died. What Robert has left behind is an emotionally challenging legacy of genius, mental illness, and unfinished business in the Chicago home where Catherine had cared for him during his erratic final years. Catherine fears she may have inherited her father's unstable condition, and her sister Claire (Hope Davis) arrives from New York with smothering concern and a selfish but well-meaning agenda, while Robert's student and assistant Hal (Jake Gyllenhaal) hopes to find lasting proof of Robert's enduring genius in the piles of notebooks he left behind. Steeped in the authentic atmosphere of advanced academia, revelations of love, fear, regret, and potential recovery unfold with such graceful complexity that "Proof" plays like a thriller, with all the action taking place in the admirable hearts and minds of its characters. The film also has a lot to say about the potential tragedy of assuming mental illness where none exists, while leaving just enough doubt to keep you wondering -- a tribute to the exceptional performances of a first-rate cast, and particularly to Paltrow, whose reunion with "Shakespeare in Love" director John Madden proves equally rewarding for entirely different reasons. "--Jeff Shannon"