{"paper":{"title":"Weak Quadruple Comparison and Structure Theory Beyond Alexandrov Geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.MG","authors_text":"Bang-Xian Han, Liming Yin","submitted_at":"2026-06-22T02:35:03Z","abstract_excerpt":"We introduce a new four-point comparison principle, called the weak quadruple condition, for non-Riemannian spaces with synthetic non-negative curvature. This condition is satisfied by classical Alexandrov spaces with non-negative curvature and also by many spaces which may not be infinitesimally Hilbert, including $S$-concave Busemann concave spaces.\n  Using this comparison principle, we develop a non-symmetric strainer theory in the setting of finite-dimensional $S$-concave Busemann concave spaces. We show these spaces have constant integer dimension, satisfy the measure contraction property"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22777","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.22777/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}