We consider the problem of distribution-free learning for Boolean function\nclasses in the PAC and agnostic models. Generalizing a beautiful work of Malach\nand Shalev-Shwartz (2022) that gave tight correlational SQ (CSQ) lower bounds\nfor learning DNF formulas, we give new proofs that lower bounds on the\nthreshold or approximate degree of any function class directly imply CSQ lower\nbounds for PAC or agnostic learning respectively. While such bounds implicitly\nfollow by combining prior results by Feldman (2008, 2012) and Sherstov (2008,\n2011), to our knowledge the precise statements we give had not appeared in this\nform before. Moreover, our proofs are simple and largely self-contained.\n These lower bounds match corresponding positive results using upper bounds on\nthe threshold or approximate degree in the SQ model for PAC or agnostic\nlearning, and in this sense these results show that the polynomial method is a\nuniversal, best-possible approach for distribution-free CSQ learning.\n