This paper is the second in a series of three papers devoted to sums squares and hypoellipticity infinitely degenerate operators. In first we established sharp ω-monotonicity criterion for writing smooth nonnegative function f that flat at, positive away from, origin, as finite sum C2,δ functions some δ>0, namely ω-monotone Hölder modulus continuity ω. Counterexamples were provided any larger c...