Vector-valued smoothing for finite Sidon sets

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0, 1, \dots, N - 1\}$. We prove $$F(N) \le N^{1/2} + 0.94601 N^{1/4} + O(1).$$ This improves the recently announced coefficient $0.97633$ obtained by Carter, Georgiev, G\'{o}mez-Serrano, Hunter, O'Bryant, Tao and Wagner. It is also very close to, and numerically below, the tentatively reported value of approximately $0.947$. The argument is based on a vector-valued convolution inequality: several smoothing kernels share the task of producing a boundary majorant, while their $L^2$ energies are averaged. The analytic reduction is elementary. The final constant is supplied by a finite rational certificate, verified by a short program using exact arithmetic only.