{"paper":{"title":"Global Convergence of the Return Dynamics in the Class $\\mathcal{O}_C$","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"The return dynamics in class O_C domains converge globally to the critical points of the thickness function.","cross_cats":[],"primary_cat":"math.DS","authors_text":"Mohamed El Morsalani, Mohammed Barkatou","submitted_at":"2026-03-30T13:56:24Z","abstract_excerpt":"This research investigates a geometric dynamical mechanism within a specific class of domains that contain a fixed convex core. By using a radial structure that links the boundaries of the core and the outer domain via a thickness function, the authors introduce a \"return map.\" This map is constructed by projecting a point from the core to the outer boundary and then returning to the core by following the inward normals.\n  The main results demonstrate that this motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness. In other words, the syst"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The fixed points of this dynamics coincide with the critical points of the thickness function. The motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The domains belong to the class O_C with a fixed convex core and admit a radial structure defined by a thickness function linking the boundaries.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Return dynamics in the class O_C converge globally to critical points of the thickness function by approximating adaptive gradient descent.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The return dynamics in class O_C domains converge globally to the critical points of the thickness function.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1701ca5d7c08a3bc4c521dfbba5598ff5a354c8fc706b19efb04247a1aff1822"},"source":{"id":"2603.28453","kind":"arxiv","version":3},"verdict":{"id":"7e072702-b353-4ed0-8470-ee9e8ac98749","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T00:30:44.660053Z","strongest_claim":"The fixed points of this dynamics coincide with the critical points of the thickness function. The motion behaves, to a first-order approximation, like an adaptive gradient descent for the domain's thickness.","one_line_summary":"Return dynamics in the class O_C converge globally to critical points of the thickness function by approximating adaptive gradient descent.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The domains belong to the class O_C with a fixed convex core and admit a radial structure defined by a thickness function linking the boundaries.","pith_extraction_headline":"The return dynamics in class O_C domains converge globally to the critical points of the thickness function."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.28453/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"}