{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:IGC7UAABRI4F3VET5BJLMMWUII","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"626aba2f1af9050741d07bb532c46a05760b39f5394c0c6e7c8b566793e8865c","cross_cats_sorted":["math.AP","math.MG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-25T09:03:53Z","title_canon_sha256":"d141c60bd5685e4c7042a270b91a8cd1d4f6121386769664f8314713fc15af11"},"schema_version":"1.0","source":{"id":"2506.20258","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.20258","created_at":"2026-07-05T11:27:02Z"},{"alias_kind":"arxiv_version","alias_value":"2506.20258v1","created_at":"2026-07-05T11:27:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.20258","created_at":"2026-07-05T11:27:02Z"},{"alias_kind":"pith_short_12","alias_value":"IGC7UAABRI4F","created_at":"2026-07-05T11:27:02Z"},{"alias_kind":"pith_short_16","alias_value":"IGC7UAABRI4F3VET","created_at":"2026-07-05T11:27:02Z"},{"alias_kind":"pith_short_8","alias_value":"IGC7UAAB","created_at":"2026-07-05T11:27:02Z"}],"graph_snapshots":[{"event_id":"sha256:3eef247eb49b3fb803673166199090e903f36b723553d91c1d87a3c939dde7e8","target":"graph","created_at":"2026-07-05T11:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.20258/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via maximal monotone operator theory, their extension to general metric spaces, particularly Wasserstein spaces, has remained largely unexplored. In this paper, we develop a mathematical theory of GDA flows on the product of two complete metric spaces, formulating them as solutions to a system of evolution variational inequalities (EVIs) driven by a proper, close","authors_text":"Noboru Isobe, Sho Shimoyama","cross_cats":["math.AP","math.MG","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-25T09:03:53Z","title":"On gradient descent-ascent flows in metric spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20258","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0e7724fabd2ce7ec483bb5e579dc43849d5f4a077b6ebcb34772f421c10de18f","target":"record","created_at":"2026-07-05T11:27:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"626aba2f1af9050741d07bb532c46a05760b39f5394c0c6e7c8b566793e8865c","cross_cats_sorted":["math.AP","math.MG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-06-25T09:03:53Z","title_canon_sha256":"d141c60bd5685e4c7042a270b91a8cd1d4f6121386769664f8314713fc15af11"},"schema_version":"1.0","source":{"id":"2506.20258","kind":"arxiv","version":1}},"canonical_sha256":"4185fa00018a385dd493e852b632d44236f51b577124411fb2a3beecc667f4ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4185fa00018a385dd493e852b632d44236f51b577124411fb2a3beecc667f4ae","first_computed_at":"2026-07-05T11:27:02.760864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:27:02.760864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nFygr33oEhekMEha7uC9xCkCeodUqpSEDQ4cvYTYIfm4EsNo7MPnZ1qmrLiR6qBOOs4WaY5EjUQqGl27IqcVBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:27:02.761569Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.20258","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e7724fabd2ce7ec483bb5e579dc43849d5f4a077b6ebcb34772f421c10de18f","sha256:3eef247eb49b3fb803673166199090e903f36b723553d91c1d87a3c939dde7e8"],"state_sha256":"e5eaa364258b05b19f49f4814c03c4d0150e31d02743fabf420a280c9ca810d9"}