A lower bound for sets lacking the configuration {x, x+y, x+y³}

We give the first explicit lower bound for sets of integers lacking the configuration {x, x + y, x + y³} that goes beyond what sets without cube differences provide: for every ε > 0 there is A ⊆ {1, …, N} with no such configuration and |A| ≫_ε N^(γ₃−ε), where γ₃ = 2/3 + (3 log 13 + log 28)/(12 log 91) = 0.8703797389…, whereas the best exponent for cube-difference-free sets is 0.8617631…. The construction instantiates Younis's iterated-difference framework with a two-element chain of subsets of ℤ/91ℤ of sizes (13, 28), the first of which is deliberately smaller than the largest cube-difference-free subset of ℤ/91ℤ. We show that the optimisation over all admissible chains modulo a fixed m is a finite dynamic program over 2^(|Pₘ|/2) states, where Pₘ is the set of nonzero cubes, and by solving it exactly we prove that no chain of any length modulo 91 gives a larger exponent. A dependency-free verification script is included.