Much like the mathematical symbols she jots down, Gwyneth Paltrow’s character Catherine Llewelyn in “Proof” is terribly attractive, but more or less cryptic. And although deciphering Catherine does not require any advanced degrees in math, it can, nevertheless, be as exasperating as any calculus problem.

Directed by John Madden of “Shakespeare in Love” fame and adapted from David Auburn’s Pulitzer Prize-winning play, “Proof” centers on math, or at least on mathematicians. Catherine is the daughter of the late Robert Llewelyn (Anthony Hopkins), a mathematical savant and professor who won great acclaim in his field at a young age and later deteriorated into madness. On her 27th birthday and her father’s funeral day, Catherine says goodbye to the man she looked after for five years.

From the beginning, Catherine’s own sanity is questionable. Once a promising mathematics student at Northwestern University, she is now consumed by crippling melancholy. In the film’s opening scene, she sees and converses with her dead father, prompting her — and us — to wonder if his insanity is somehow genetic. And if the delusion weren’t enough to make us think twice, the doleful stare Paltrow wears like a mask for the majority of the film certainly would.

Catherine appears caught in a private, tortured existence. At the funeral, she mounts the pulpit and speaks accusingly to a large audience. “Where have you been for the last five years?” she demands, and then storms out. On the sidewalk, she is enveloped in dark shadows while the church gleams in the bright sun behind her.

Catherine’s sister Claire (Hope Davis) flies in from New York to attend the funeral, and it’s not difficult to sense the rift between Catherine’s gloom and Claire’s scheduling notebook and specialty conditioner. Claire may be a bit of a caricature with all of her inane mannerisms — including, but not limited to, interminable conversations about roasted coffee — but her contrast at least serves to underscore what an enigma Catherine is.

Harold Dobbs (Jake Gyllenhaal) provides a generous boost towards solving that enigma. Harold, a graduate student and a former pupil of Catherine’s father, has set himself to looking through every one of the 103 notebooks composed by the brilliant but deranged mathematician. In the process, he falls in love with Catherine, who, though perhaps unstable, does manage to look remarkably put-together most of the time.

While Gyllenhaal provides a memorable performance as the math geek with a perpetual five o’clock shadow, he is often forced to breathe life into dead dialogue. Upon viewing the array of math books lining the shelves in Catherine’s room, he can merely muster, “You’re a mysterious person, Catherine, you know that?” As if the preceding 45 minutes hadn’t made that clear.

The burgeoning romance between daughter and student is initially captivating, but when Harold discovers a hidden notebook containing the completed proof for a legendary math problem, an entirely new mystery arises — namely, to whom it belongs. Claire proclaims it to be the work of her father, but Catherine quietly states that it’s her own. Harold pretends to plead ignorance, though he cannot help remarking that the handwriting belongs to his former professor.

Unraveling the truth of the matter occupies the bulk of the rest of the film, and a series of tantalizing flashbacks gradually unfurls the ultimate revelation. Yet Madden’s film hardly encroaches upon M. Night Shyamalan gimmickry. “Proof” is a much quieter film with a refreshing emphasis on character relationships and development. Few things, upon closer inspection, prove to be what they first appear, for so much of the action unfolds in a precarious half-world between madness and normality. No matter how many times Howard harangues Catherine with the words, “There’s nothing wrong with you,” we just can’t bring ourselves to believe it completely.

But however deeply afflicting such burdens may be, Catherine and Harold’s love offers an unexpected method to solve the problem. Although exotic mathematics offer plot framework, it is Catherine’s own inner turmoil that furnishes the greater challenge and, ultimately, the more elegant solutio