{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:CYJHYOVH5PF7TTVSWXO3KPNLQL","short_pith_number":"pith:CYJHYOVH","canonical_record":{"source":{"id":"2307.04204","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023","abstract_canon_sha256":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f"},"schema_version":"1.0"},"canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","source":{"kind":"arxiv","id":"2307.04204","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"arxiv_version","alias_value":"2307.04204v2","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_12","alias_value":"CYJHYOVH5PF7","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_16","alias_value":"CYJHYOVH5PF7TTVS","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_8","alias_value":"CYJHYOVH","created_at":"2026-07-05T07:05:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:CYJHYOVH5PF7TTVSWXO3KPNLQL","target":"record","payload":{"canonical_record":{"source":{"id":"2307.04204","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023","abstract_canon_sha256":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f"},"schema_version":"1.0"},"canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:05:31.557017Z","signature_b64":"3MgYxCdsYfnJRr5OCyoNhC+9ELJNS0aGMHiNL8VqbaxyF+koxKy5ELdemBOQhEgF+AypkrxfjYbC0rTB2VtbBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","last_reissued_at":"2026-07-05T07:05:31.556550Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:05:31.556550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2307.04204","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:05:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s9kddvNidbBoDHxVGP5CqR6Yy1mWHjg4o2CMd2xm1c9eNdEp/eJvPIt1Kf4CfXg7PnqLeJ7O5dm0GirkkDf7Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:54:28.740312Z"},"content_sha256":"ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48","schema_version":"1.0","event_id":"sha256:ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:CYJHYOVH5PF7TTVSWXO3KPNLQL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Chulhee Yun, Minhak Song","submitted_at":"2023-07-09T15:16:45Z","abstract_excerpt":"Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon. The sharpness increases at the early phase of training (referred to as progressive sharpening), and eventually saturates close to the threshold of $2 / \\text{(step size)}$. In this paper, we start by demonstrating through empirical studies that when the EoS phenomenon occurs, different GD trajectories (after a proper reparameterization) align on a specific bifurcation diagram in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.04204","kind":"arxiv","version":2},"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/2307.04204/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:05:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nH3A5skQqqkM2GcVnvYzfNxgG9G9zBVCZXbA7K7qmEm9ea4hJSlZ7Ertfp5JnKDuUsrrv8glbkboROuTVwfNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:54:28.740639Z"},"content_sha256":"fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e","schema_version":"1.0","event_id":"sha256:fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/bundle.json","state_url":"https://pith.science/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-14T20:54:28Z","links":{"resolver":"https://pith.science/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL","bundle":"https://pith.science/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/bundle.json","state":"https://pith.science/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYJHYOVH5PF7TTVSWXO3KPNLQL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:CYJHYOVH5PF7TTVSWXO3KPNLQL","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":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023"},"schema_version":"1.0","source":{"id":"2307.04204","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"arxiv_version","alias_value":"2307.04204v2","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.04204","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_12","alias_value":"CYJHYOVH5PF7","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_16","alias_value":"CYJHYOVH5PF7TTVS","created_at":"2026-07-05T07:05:31Z"},{"alias_kind":"pith_short_8","alias_value":"CYJHYOVH","created_at":"2026-07-05T07:05:31Z"}],"graph_snapshots":[{"event_id":"sha256:fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e","target":"graph","created_at":"2026-07-05T07:05:31Z","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/2307.04204/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon. The sharpness increases at the early phase of training (referred to as progressive sharpening), and eventually saturates close to the threshold of $2 / \\text{(step size)}$. In this paper, we start by demonstrating through empirical studies that when the EoS phenomenon occurs, different GD trajectories (after a proper reparameterization) align on a specific bifurcation diagram in","authors_text":"Chulhee Yun, Minhak Song","cross_cats":["math.OC","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title":"Trajectory Alignment: Understanding the Edge of Stability Phenomenon via Bifurcation Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.04204","kind":"arxiv","version":2},"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:ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48","target":"record","created_at":"2026-07-05T07:05:31Z","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":"b8ce54b006bb98edbfd0a8deb52d9945d5453fa7beda643287966e9477d8520f","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-07-09T15:16:45Z","title_canon_sha256":"fb4e396a9125c7c1eb46ff83fbe2c0394995360145bd545d6b14590df88f0023"},"schema_version":"1.0","source":{"id":"2307.04204","kind":"arxiv","version":2}},"canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16127c3aa7ebcbf9ceb2b5ddb53dab82cc7d8835de8af652c18b3eae69503322","first_computed_at":"2026-07-05T07:05:31.556550Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:05:31.556550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3MgYxCdsYfnJRr5OCyoNhC+9ELJNS0aGMHiNL8VqbaxyF+koxKy5ELdemBOQhEgF+AypkrxfjYbC0rTB2VtbBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T07:05:31.557017Z","signed_message":"canonical_sha256_bytes"},"source_id":"2307.04204","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef0bb8f4821889d2e8400a104f9332166f27ef475e87135e0f111a6069b4bf48","sha256:fd6988d800c731d5f18b3596b942e273412c3000397bc4f929267c2e6259e24e"],"state_sha256":"bb7fb1a652fccee318288d7445ca436e197f87d8159e6b3aacca3d1e64a0ca41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RludA+ZreRSFScQu+mH428XhDHybzUXlkBI6ZJe971SeSWSnilia/ttJNlj7DIcFf7PeP4XUxC8gQK2hYb59Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T20:54:28.744171Z","bundle_sha256":"3b09d9cd2f0a76681f4a47495a43d1806347b1358dd36fcb5c593a59e1704133"}}