A Uniform Gap Below One Half for Sum-Free Subsets of Euclidean Balls

A measurable subset of a Euclidean ball is sum-free if it contains no three points x, y, and z satisfying x + y = z, including when the two summands are equal. We prove that every measurable sum-free subset of the unit ball in dimension two or higher occupies at most 49.9% of the ball's volume. In dimensions three and higher, we obtain the stronger upper bound of approximately 49.8197%. The same bounds hold for every nondegenerate ellipsoid centered at the origin. For dimensions three and higher, the proof lifts the problem to a higher-dimensional sphere while preserving normalized measure. We then combine harmonic analysis, rotation parity, a finite inequality for the projective 600-cell, and two Boolean sampling inequalities to obtain a uniform bound strictly below one half. The two-dimensional disk requires a separate argument. We show that occupied points along a ray impose simultaneous restrictions on parallel chords. These restrictions lead to covariogram capacity inequalities, a curvature identity, and a finite collection of exact geometric comparisons that establish the disk bound. All finite computations are certified using integer or rational arithmetic, and the infinite spectral estimates are controlled analytically. The result establishes a dimension-independent gap below one half for sum-free subsets of Euclidean balls, contributing to the study of sum-free sets in centrally symmetric convex regions.