Lossless sixth-moment bounds for Frostman measures on the parabola

Let 2/3 < s < 1, and let μ be a probability measure on [−1, 1] with upper Frostman constant A at exponent s. We prove that the sixth Fourier moment of its pushforward to the parabola on a ball of radius R is at most C_s A^6 R^(1−s). The corresponding six-variable collision bound is C_s A^6 δ^(s+1). The proof uses a point–line estimate governed by the average small-ball pairings of the two parameter measures. Localizing all six variables together preserves the two moment precisions, and a finite iteration reaches a summable frequency regime.