{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:BDP5V67MELZEENC5PGY5JWBVYO","short_pith_number":"pith:BDP5V67M","canonical_record":{"source":{"id":"2507.06381","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-08T20:33:15Z","cross_cats_sorted":["cs.AI","math.DS","q-bio.NC"],"title_canon_sha256":"d952646ee7e6ab25c72d7e2f913630c62c22fac6e4c39a48a8334ba33f9f0f88","abstract_canon_sha256":"e11492ad12cd7b1ff5fad79b82bfdd325011f224fa559bf4eb4eb73fbf220509"},"schema_version":"1.0"},"canonical_sha256":"08dfdafbec22f242345d79b1d4d835c380f4c37978bf99e268588e2c728d802c","source":{"kind":"arxiv","id":"2507.06381","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06381","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06381v1","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06381","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"BDP5V67MELZE","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_16","alias_value":"BDP5V67MELZEENC5","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_8","alias_value":"BDP5V67M","created_at":"2026-07-05T11:34:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:BDP5V67MELZEENC5PGY5JWBVYO","target":"record","payload":{"canonical_record":{"source":{"id":"2507.06381","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-08T20:33:15Z","cross_cats_sorted":["cs.AI","math.DS","q-bio.NC"],"title_canon_sha256":"d952646ee7e6ab25c72d7e2f913630c62c22fac6e4c39a48a8334ba33f9f0f88","abstract_canon_sha256":"e11492ad12cd7b1ff5fad79b82bfdd325011f224fa559bf4eb4eb73fbf220509"},"schema_version":"1.0"},"canonical_sha256":"08dfdafbec22f242345d79b1d4d835c380f4c37978bf99e268588e2c728d802c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:34:14.774759Z","signature_b64":"Rb9i03mxpIZjGesT5ETKb0hSn3ZYBVdMuaUn7GGPuubmZyRUOugGWlsafU2vcFwZr58l4uFi1IVuDPptK+qiAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08dfdafbec22f242345d79b1d4d835c380f4c37978bf99e268588e2c728d802c","last_reissued_at":"2026-07-05T11:34:14.774255Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:34:14.774255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.06381","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aLJFk8krMVP6a0FNNud5Kc/6eFwjSsH8yeupQk7+5u/S0+2ZYltEh8iMU9oaA4vk1muSLwT+LyorywKsLDdgAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T02:43:58.041055Z"},"content_sha256":"bbf809a68587f727efe006f44b413854eeea3d3f6a4f173b09f67f3ce310cb42","schema_version":"1.0","event_id":"sha256:bbf809a68587f727efe006f44b413854eeea3d3f6a4f173b09f67f3ce310cb42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:BDP5V67MELZEENC5PGY5JWBVYO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"KPFlow: An Operator Perspective on Dynamic Collapse Under Gradient Descent Training of Recurrent Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","math.DS","q-bio.NC"],"primary_cat":"cs.LG","authors_text":"Eli Shlizerman, Eric Shea-Brown, James Hazelden, Laura Driscoll","submitted_at":"2025-07-08T20:33:15Z","abstract_excerpt":"Gradient Descent (GD) and its variants are the primary tool for enabling efficient training of recurrent dynamical systems such as Recurrent Neural Networks (RNNs), Neural ODEs and Gated Recurrent units (GRUs). The dynamics that are formed in these models exhibit features such as neural collapse and emergence of latent representations that may support the remarkable generalization properties of networks. In neuroscience, qualitative features of these representations are used to compare learning in biological and artificial systems. Despite recent progress, there remains a need for theoretical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06381","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/2507.06381/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":"2026-07-05T11:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0HIIZRpaDOHL0F6lkOA0Y3TZtkoF4MkJFqyYsWh3MO3S09DVBUmqmxfqiS7uVFT8F+ojynjClMEpwXEkCAtaDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T02:43:58.041720Z"},"content_sha256":"d05431d75203a6268e72ef61604bd12b48bf3dd76ae766a9e16efbe405fec772","schema_version":"1.0","event_id":"sha256:d05431d75203a6268e72ef61604bd12b48bf3dd76ae766a9e16efbe405fec772"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BDP5V67MELZEENC5PGY5JWBVYO/bundle.json","state_url":"https://pith.science/pith/BDP5V67MELZEENC5PGY5JWBVYO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BDP5V67MELZEENC5PGY5JWBVYO/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":"2026-08-10T02:43:58Z","links":{"resolver":"https://pith.science/pith/BDP5V67MELZEENC5PGY5JWBVYO","bundle":"https://pith.science/pith/BDP5V67MELZEENC5PGY5JWBVYO/bundle.json","state":"https://pith.science/pith/BDP5V67MELZEENC5PGY5JWBVYO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BDP5V67MELZEENC5PGY5JWBVYO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:BDP5V67MELZEENC5PGY5JWBVYO","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":"e11492ad12cd7b1ff5fad79b82bfdd325011f224fa559bf4eb4eb73fbf220509","cross_cats_sorted":["cs.AI","math.DS","q-bio.NC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-08T20:33:15Z","title_canon_sha256":"d952646ee7e6ab25c72d7e2f913630c62c22fac6e4c39a48a8334ba33f9f0f88"},"schema_version":"1.0","source":{"id":"2507.06381","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06381","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06381v1","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06381","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"BDP5V67MELZE","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_16","alias_value":"BDP5V67MELZEENC5","created_at":"2026-07-05T11:34:14Z"},{"alias_kind":"pith_short_8","alias_value":"BDP5V67M","created_at":"2026-07-05T11:34:14Z"}],"graph_snapshots":[{"event_id":"sha256:d05431d75203a6268e72ef61604bd12b48bf3dd76ae766a9e16efbe405fec772","target":"graph","created_at":"2026-07-05T11:34:14Z","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/2507.06381/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Gradient Descent (GD) and its variants are the primary tool for enabling efficient training of recurrent dynamical systems such as Recurrent Neural Networks (RNNs), Neural ODEs and Gated Recurrent units (GRUs). The dynamics that are formed in these models exhibit features such as neural collapse and emergence of latent representations that may support the remarkable generalization properties of networks. In neuroscience, qualitative features of these representations are used to compare learning in biological and artificial systems. Despite recent progress, there remains a need for theoretical ","authors_text":"Eli Shlizerman, Eric Shea-Brown, James Hazelden, Laura Driscoll","cross_cats":["cs.AI","math.DS","q-bio.NC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-08T20:33:15Z","title":"KPFlow: An Operator Perspective on Dynamic Collapse Under Gradient Descent Training of Recurrent Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06381","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:bbf809a68587f727efe006f44b413854eeea3d3f6a4f173b09f67f3ce310cb42","target":"record","created_at":"2026-07-05T11:34:14Z","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":"e11492ad12cd7b1ff5fad79b82bfdd325011f224fa559bf4eb4eb73fbf220509","cross_cats_sorted":["cs.AI","math.DS","q-bio.NC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-07-08T20:33:15Z","title_canon_sha256":"d952646ee7e6ab25c72d7e2f913630c62c22fac6e4c39a48a8334ba33f9f0f88"},"schema_version":"1.0","source":{"id":"2507.06381","kind":"arxiv","version":1}},"canonical_sha256":"08dfdafbec22f242345d79b1d4d835c380f4c37978bf99e268588e2c728d802c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"08dfdafbec22f242345d79b1d4d835c380f4c37978bf99e268588e2c728d802c","first_computed_at":"2026-07-05T11:34:14.774255Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:34:14.774255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rb9i03mxpIZjGesT5ETKb0hSn3ZYBVdMuaUn7GGPuubmZyRUOugGWlsafU2vcFwZr58l4uFi1IVuDPptK+qiAA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:34:14.774759Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.06381","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bbf809a68587f727efe006f44b413854eeea3d3f6a4f173b09f67f3ce310cb42","sha256:d05431d75203a6268e72ef61604bd12b48bf3dd76ae766a9e16efbe405fec772"],"state_sha256":"c03a98c57be961f7e8dfaad2d6baaec16041997c0ea61a1ef934ff032304dad5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GFKLotQiuxx6faQxARvTue2/P6eN9q4mE0k7orYIB28exfDd/SL8xDDDecWz2rSp2odC3KSOWcXwOgMOdfSbAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T02:43:58.046667Z","bundle_sha256":"45b62f0e83c1253aadc6cd31c955184373abc8682acb549c223a727498d1be1d"}}