We highlight the special case of Valiant's rigidityproblem in which the low-rank matrices are truth-tablesof sparse polynomials. We show that progress on thisspecial case entails that Inner Product is not computableby small \acz circuits with one layer of parity gatesclose to the inputs. We then prove that the sign of any−11 polynomial with s monomials in 2nvariables disagrees with Inner Product in (1s) fraction of inputs, a type of result thatseems unknown in the rigidity setting.