Theory of Computing ------------------- Title : Circumventing $d$-to-$1$ for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four Authors : Cenny Wenner Volume : 9 Number : 23 Pages : 703-757 URL : https://theoryofcomputing.org/articles/v009a023 Abstract -------- Hastad established that any predicate $P \subseteq \{0,1\}^m$ containing Parity of width at least three is approximation resistant for almost-satisfiable instances . In comparison to for example the approximation hardness of 3SAT, this general result however left open the hardness of perfectly-satisfiable instances. This limitation was addressed by O'Donnell and Wu, and subsequently generalized by Huang, to show the threshold result that predicates _strictly_ containing Parity of width at least three are approximation resistant also for perfectly-satisfiable instances, assuming the $d$-to-1 Conjecture. We extend modern hardness-of-approximation techniques by Mossel et al., eliminating the dependency on projection degrees for a special case of decoupling/invariance and -- when reducing from _Smooth Label Cover_ -- the dependency on projection degrees for noise introduction. Tools in hand, we prove the same approximation-resistance result for predicates of width at least four, subject only to P \neq NP. A preliminary version of this paper appeared in the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'12).