The 'proof' of the title is an unequivocal mathematical result reached via a hypothesis formulated from a specific series of complex maths applications. Head hurting yet?
It's a tidy title concept for a film that carefully balances out the Oscar equation: intellectual and classy + plenty of redemptive suffering. Given it boasts a sturdy, upscale cast and director, with a script adapted by playwright royalty Rebecca Miller, it would be easy to dismiss Proof as pretentious, stagey and calculated. And it's hard to feel sympathy for maths nerds crying over their calculus.
But at the square root, if you will, of this enterprise is Paltrow's raw, tempestuous portrait of a woman paralysed by anger, fear and grief. Having previously played the role at London's Donmar Warehouse (with Madden directing), Paltrow grew even closer to her character when she began the film version, as she battled real grief over her father's recent death. This gives her already affecting scenes with Hopkins an added poignancy, and her teary mourning a curious, voyeuristic quality. It may explain why the usually glacial Gwyneth manages to inject tangible bile and passion into what could have been an overly dreamy performance, off-setting Hopkins' penchant for theatrical roaring and making Davis a rewardingly humorous foil as her well-meaning but infuriating sister.
Bitter, fierce and vulnerable, Paltrow is magnificent when raging at a memorial service of fairweather friends and intriguingly inscrutable as she makes claims of her own abilities. It's a turn that could certainly net her a second Oscar nod, especially as she's toying with that Academy favourite, mental illness.
A disappointment, then, that the romance between Paltrow and Gyllenhaal is less convincing (though he brings a real warmth to the otherwise frosty backdrop) and, after a compelling whodunnit set-up, the last 20 minutes simply trails off in an oppressive haze of glumness and - sorry, but it's true - fails to really add up.