{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DCXPIFM3OKID4CKGMFPLYCCOKS","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":"c5e97016fa00c44db0d4831d74ba4ab4ad73a3f384e642c7b7784557280486d4","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T09:40:30Z","title_canon_sha256":"3a091e6fe2f2ea838c36eb88a992eec7bdb3816ed9cb3f01097a2c91984dee6f"},"schema_version":"1.0","source":{"id":"2606.04662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04662","created_at":"2026-06-04T01:09:23Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04662v1","created_at":"2026-06-04T01:09:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04662","created_at":"2026-06-04T01:09:23Z"},{"alias_kind":"pith_short_12","alias_value":"DCXPIFM3OKID","created_at":"2026-06-04T01:09:23Z"},{"alias_kind":"pith_short_16","alias_value":"DCXPIFM3OKID4CKG","created_at":"2026-06-04T01:09:23Z"},{"alias_kind":"pith_short_8","alias_value":"DCXPIFM3","created_at":"2026-06-04T01:09:23Z"}],"graph_snapshots":[{"event_id":"sha256:0f835a76582b315d62a41905d03380dbdd944cb4dd215b08f8d234715a56f522","target":"graph","created_at":"2026-06-04T01:09:23Z","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/2606.04662/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this cur","authors_text":"Dirk Bergemann, Fengzhuo Zhang, Jiaxiang Li, Shuche Wang, Zhuoran Yang","cross_cats":["cs.AI"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T09:40:30Z","title":"Why Muon Outperforms Adam: A Curvature Perspective"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04662","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":"sha256:6544a9f1b9bb4af499dc1474f4104f1b691bd6278b9449faf7f6ec643f5a8d2e","target":"record","created_at":"2026-06-04T01:09:23Z","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":"c5e97016fa00c44db0d4831d74ba4ab4ad73a3f384e642c7b7784557280486d4","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T09:40:30Z","title_canon_sha256":"3a091e6fe2f2ea838c36eb88a992eec7bdb3816ed9cb3f01097a2c91984dee6f"},"schema_version":"1.0","source":{"id":"2606.04662","kind":"arxiv","version":1}},"canonical_sha256":"18aef4159b72903e0946615ebc084e549562527f25e315fbf763707f1f7c5553","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18aef4159b72903e0946615ebc084e549562527f25e315fbf763707f1f7c5553","first_computed_at":"2026-06-04T01:09:23.806967Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:23.806967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YttiNstAKWv/RSlf8He/yrO/c+L5STx+LiZfROgI6R3BgVsiIu4uxdq8psQ8tqhMR+ZIwzxHvE97f9uksLpCDA==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:23.807782Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04662","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6544a9f1b9bb4af499dc1474f4104f1b691bd6278b9449faf7f6ec643f5a8d2e","sha256:0f835a76582b315d62a41905d03380dbdd944cb4dd215b08f8d234715a56f522"],"state_sha256":"6f1337c63dd0e39549fa30ea8b4841a550b38dd17b9c5ba1158ad2880ad773f3"}