Certified upper bounds for two sum-difference exponents

For finite sets A, B of real numbers and G ⊆ A × B, Katz and Tao proved in 1999 that {a − b : (a, b) ∈ G} has at most N11/6 elements whenever A, B and {a + b : (a, b) ∈ G} have at most N elements. We lower the exponent 11/6 to 1.827297. Without non-Shannon inequalities we obtain the exponent 1.829429, but this exponent is valid in all abelian groups. We also prove |X − Y| ≤ |X + Y|1.422300 for all finite sets X, Y of integers. This improves the exponent 1.4543 of Kleinwaks, and through his transfer theorem it lowers the upper bound for the sum-difference constant of Gyarmati, Hennecart and Ruzsa from 1.3124 to 1.296914. Both proofs establish entropy inequalities for a random pair. We construct several copies of the pair on a common probability space, glued by the copy lemma for the first problem and independent for the second, and use cross forms, which are integer combinations of coordinates of different copies. With rational weights found by linear programming, we combine Shannon inequalities among the joint entropies of these forms and, for the first problem, also non-Shannon inequalities that we prove from the copy lemma. For each bound, a checker recomputes the inequalities from their definitions and verifies the combination in exact rational arithmetic, and a second checker confirms the result. The proofs of the exponents 1.827297 and 1.422300 and of the bound 1.296914 have also been checked in the Lean 4 proof assistant. The certificates, the checkers, the Lean proofs and the programs are archived at doi:10.5281/zenodo.23509966 and available at github.com/tejasvi/sum-difference-exponents.