Weak Quadruple Comparison and Structure Theory Beyond Alexandrov Geometry

We introduce a new four-point comparison principle, called the $(\varepsilon,\delta)$-weak quadruple condition, for non-Riemannian spaces with synthetic non-negative curvature. This condition holds not only for classical Alexandrov spaces with non-negative curvature, but also for many genuinely non-Riemannian spaces. In particular, we show that this condition is intrinsic to spaces satisfying Ohta's $S$-concavity in the full parameter range.
Using this comparison principle, we develop a non-symmetric strainer framework and establish a Burago--Gromov--Perelman-type structure theory for finite-dimensional $S$-concave Busemann concave spaces. We prove that these spaces have constant integer Hausdorff dimension, satisfy the measure contraction property, are rectifiable, and admit unique Banach tangent cones almost everywhere. We further show that each such space contains an open dense topological manifold part of top dimension and full measure. Finally, we establish Hausdorff dimension estimates for singular strata and construct natural measure-theoretic stratifications of these spaces. Our framework includes Alexandrov spaces with non-negative curvature as a special case, and provides tools for studying Finslerian metric spaces whose tangent cones need not be metric cones and angles need not be symmetric.