An improved lower bound in the nonlinear Roth theorem

We construct, for every ε > 0, a subset of {1, …, N} of size ≫_ε N^(γ−ε) containing no nontrivial configuration {x, x + y, x + y²}, where γ = 1/2 + (30 log 10 + 15 log 28 + 10 log 20 + 4 log 21 + log 22)/(120 log 145) = 0.77509… This improves the exponent 0.7685038… of Younis. The construction instantiates Younis's iterated-difference framework at the modulus 145 with a chain of subsets of ℤ/145ℤ of sizes 10, 28 followed by the period-four pattern 20, 21, 20, 22, and every admissibility condition is verified by exhaustive finite computation. We show that the optimisation over all admissible chains at a fixed modulus is a dynamic program over forbidden-set states, and we solve it exactly, with big-integer certificates, at the moduli 65 and 85: the optimum at Younis's modulus 65 is 0.7732006…, attained only by the periodic chain of sizes (7, 17, 10, 15), and the optimum at 85 is 0.7626068…. At 145 the state space is too large for the exact program and the chain above comes from a restricted search, so the exact optimum there, and the optimisation over the modulus, remain open. A short, dependency-free verification script is included. The authors are listed in alphabetical order and contributed equally.