{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:K57ODBMBQMSKOWU6CMNU7L63PZ","short_pith_number":"pith:K57ODBMB","canonical_record":{"source":{"id":"2407.20199","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-07-29T17:28:58Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"524c0fdde5fe1f1ad955548396fd79cf9ba8bb1f95de9e5258c73bfe2aa55f9b","abstract_canon_sha256":"0b7ae370a1bacddeea6f1a0eeb59a894548a21e349bd2f4210b70f6bd08c14c4"},"schema_version":"1.0"},"canonical_sha256":"577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac","source":{"kind":"arxiv","id":"2407.20199","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.20199","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"arxiv_version","alias_value":"2407.20199v3","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.20199","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_12","alias_value":"K57ODBMBQMSK","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_16","alias_value":"K57ODBMBQMSKOWU6","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_8","alias_value":"K57ODBMB","created_at":"2026-07-05T11:33:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:K57ODBMBQMSKOWU6CMNU7L63PZ","target":"record","payload":{"canonical_record":{"source":{"id":"2407.20199","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-07-29T17:28:58Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"524c0fdde5fe1f1ad955548396fd79cf9ba8bb1f95de9e5258c73bfe2aa55f9b","abstract_canon_sha256":"0b7ae370a1bacddeea6f1a0eeb59a894548a21e349bd2f4210b70f6bd08c14c4"},"schema_version":"1.0"},"canonical_sha256":"577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:59.163698Z","signature_b64":"3ngVfg7Tj+Yy5fRjOttWyrAHtDUzdEihp0iPWVPqNG6Oyx0s8IgdpkRCmWLgoXhB/CWGlGnR0Cg8XXb0l+LkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac","last_reissued_at":"2026-07-05T11:33:59.163208Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:59.163208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.20199","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:33:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/TJNBxDUx4q/l+9z+9hbjWJRJkVN5lCiv6z/deHypQq/srTH/rtQcQnHazUGED0l8CUKJnvfK6ter0RxGfVwBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T14:55:02.615078Z"},"content_sha256":"0cfa33a5df611bbf2c106f73e2d18bbdb0ba796ae60cec232931f3af88b0e7c0","schema_version":"1.0","event_id":"sha256:0cfa33a5df611bbf2c106f73e2d18bbdb0ba796ae60cec232931f3af88b0e7c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:K57ODBMBQMSKOWU6CMNU7L63PZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Emergence in non-neural models: grokking modular arithmetic via average gradient outer product","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Adityanarayanan Radhakrishnan, Daniel Beaglehole, Libin Zhu, Mikhail Belkin, Neil Mallinar, Parthe Pandit","submitted_at":"2024-07-29T17:28:58Z","abstract_excerpt":"Neural networks trained to solve modular arithmetic tasks exhibit grokking, a phenomenon where the test accuracy starts improving long after the model achieves 100% training accuracy in the training process. It is often taken as an example of \"emergence\", where model ability manifests sharply through a phase transition. In this work, we show that the phenomenon of grokking is not specific to neural networks nor to gradient descent-based optimization. Specifically, we show that this phenomenon occurs when learning modular arithmetic with Recursive Feature Machines (RFM), an iterative algorithm "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.20199","kind":"arxiv","version":3},"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/2407.20199/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":"2026-07-05T11:33:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IFUGr6D8JZNwKEn4qNgFqIYqgJYB20K240HRerRDciaQt4gD1A5eJx5G0kdnRBE6NsrIbKmftikK2IgxYASqDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T14:55:02.616514Z"},"content_sha256":"e743b787b5c134af414e4cf49c55dee026ac9b5926d1568eb560b7b81347085b","schema_version":"1.0","event_id":"sha256:e743b787b5c134af414e4cf49c55dee026ac9b5926d1568eb560b7b81347085b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/bundle.json","state_url":"https://pith.science/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/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":"2026-08-09T14:55:02Z","links":{"resolver":"https://pith.science/pith/K57ODBMBQMSKOWU6CMNU7L63PZ","bundle":"https://pith.science/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/bundle.json","state":"https://pith.science/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K57ODBMBQMSKOWU6CMNU7L63PZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:K57ODBMBQMSKOWU6CMNU7L63PZ","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":"0b7ae370a1bacddeea6f1a0eeb59a894548a21e349bd2f4210b70f6bd08c14c4","cross_cats_sorted":["cs.LG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-07-29T17:28:58Z","title_canon_sha256":"524c0fdde5fe1f1ad955548396fd79cf9ba8bb1f95de9e5258c73bfe2aa55f9b"},"schema_version":"1.0","source":{"id":"2407.20199","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.20199","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"arxiv_version","alias_value":"2407.20199v3","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.20199","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_12","alias_value":"K57ODBMBQMSK","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_16","alias_value":"K57ODBMBQMSKOWU6","created_at":"2026-07-05T11:33:59Z"},{"alias_kind":"pith_short_8","alias_value":"K57ODBMB","created_at":"2026-07-05T11:33:59Z"}],"graph_snapshots":[{"event_id":"sha256:e743b787b5c134af414e4cf49c55dee026ac9b5926d1568eb560b7b81347085b","target":"graph","created_at":"2026-07-05T11:33:59Z","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/2407.20199/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Neural networks trained to solve modular arithmetic tasks exhibit grokking, a phenomenon where the test accuracy starts improving long after the model achieves 100% training accuracy in the training process. It is often taken as an example of \"emergence\", where model ability manifests sharply through a phase transition. In this work, we show that the phenomenon of grokking is not specific to neural networks nor to gradient descent-based optimization. Specifically, we show that this phenomenon occurs when learning modular arithmetic with Recursive Feature Machines (RFM), an iterative algorithm ","authors_text":"Adityanarayanan Radhakrishnan, Daniel Beaglehole, Libin Zhu, Mikhail Belkin, Neil Mallinar, Parthe Pandit","cross_cats":["cs.LG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-07-29T17:28:58Z","title":"Emergence in non-neural models: grokking modular arithmetic via average gradient outer product"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.20199","kind":"arxiv","version":3},"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":"sha256:0cfa33a5df611bbf2c106f73e2d18bbdb0ba796ae60cec232931f3af88b0e7c0","target":"record","created_at":"2026-07-05T11:33:59Z","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":"0b7ae370a1bacddeea6f1a0eeb59a894548a21e349bd2f4210b70f6bd08c14c4","cross_cats_sorted":["cs.LG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-07-29T17:28:58Z","title_canon_sha256":"524c0fdde5fe1f1ad955548396fd79cf9ba8bb1f95de9e5258c73bfe2aa55f9b"},"schema_version":"1.0","source":{"id":"2407.20199","kind":"arxiv","version":3}},"canonical_sha256":"577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"577ee185818324a75a9e131b4fafdb7e5c904ab97c949825bee554cf8f3af4ac","first_computed_at":"2026-07-05T11:33:59.163208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:33:59.163208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3ngVfg7Tj+Yy5fRjOttWyrAHtDUzdEihp0iPWVPqNG6Oyx0s8IgdpkRCmWLgoXhB/CWGlGnR0Cg8XXb0l+LkBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:33:59.163698Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.20199","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0cfa33a5df611bbf2c106f73e2d18bbdb0ba796ae60cec232931f3af88b0e7c0","sha256:e743b787b5c134af414e4cf49c55dee026ac9b5926d1568eb560b7b81347085b"],"state_sha256":"e2080ef2a7f7f53db76e511fdeec6668d92c49c2978cc63c3569a2bb98d05e69"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c1CzJpNEJYT9n5qMYfYilZZPqFRNNTXjbVj/S495A6GTKypBcKoRACOS8sMI6VbuV69av4qKEhveol6o7Q2+Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T14:55:02.624211Z","bundle_sha256":"21c6e8742abb1a2941590df788bc4abb3432f9cc4ccd0de1b4f6b0c398fa138b"}}