{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:HIMDRPSVDM6SWNKPEGXMZUXMTD","short_pith_number":"pith:HIMDRPSV","canonical_record":{"source":{"id":"2210.15435","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2022-10-26T11:07:04Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"d5be81131b242d5b512dfbdab527ce03a5dbee06edd5aca2ef78a18419dc1b15","abstract_canon_sha256":"710c1fc39114b0acd8efaf1e16d6a68cd3def7a8fccede1f16e063c7fa9fdaaf"},"schema_version":"1.0"},"canonical_sha256":"3a1838be551b3d2b354f21aeccd2ec98f39f89a6060c073fe9f8dca5b21649bb","source":{"kind":"arxiv","id":"2210.15435","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.15435","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"arxiv_version","alias_value":"2210.15435v1","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.15435","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_12","alias_value":"HIMDRPSVDM6S","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_16","alias_value":"HIMDRPSVDM6SWNKP","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_8","alias_value":"HIMDRPSV","created_at":"2026-07-05T05:11:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:HIMDRPSVDM6SWNKPEGXMZUXMTD","target":"record","payload":{"canonical_record":{"source":{"id":"2210.15435","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2022-10-26T11:07:04Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"d5be81131b242d5b512dfbdab527ce03a5dbee06edd5aca2ef78a18419dc1b15","abstract_canon_sha256":"710c1fc39114b0acd8efaf1e16d6a68cd3def7a8fccede1f16e063c7fa9fdaaf"},"schema_version":"1.0"},"canonical_sha256":"3a1838be551b3d2b354f21aeccd2ec98f39f89a6060c073fe9f8dca5b21649bb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:11:06.196563Z","signature_b64":"HZMlo23oa+CikL2xe41iuvo9/PrX0RzRbaX8QgcM/zSTaTJO6p+dk9tKImINe90eDPn8bG1LEgBGcIaGLcCLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a1838be551b3d2b354f21aeccd2ec98f39f89a6060c073fe9f8dca5b21649bb","last_reissued_at":"2026-07-05T05:11:06.196080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:11:06.196080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2210.15435","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":"2026-07-05T05:11:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NZY+pvKINmwZGDJXBssrG4D9PTw+Ezdsu828UIWE+fOJEi6UdNHWA7O/rGkrmi0LcKOJWA5kzvotKaCQzyKTDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T20:21:05.430597Z"},"content_sha256":"cfa07ad952d3294774585520add884b76dfb0687a71c23b95c742776585db9d7","schema_version":"1.0","event_id":"sha256:cfa07ad952d3294774585520add884b76dfb0687a71c23b95c742776585db9d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:HIMDRPSVDM6SWNKPEGXMZUXMTD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Grokking phase transitions in learning local rules with gradient descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cond-mat.stat-mech","authors_text":"Bojan \\v{Z}unkovi\\v{c}, Enej Ilievski","submitted_at":"2022-10-26T11:07:04Z","abstract_excerpt":"We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.15435","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/2210.15435/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":"2026-07-05T05:11:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rHQ7pMveCoShAo0FzTcINBnftpBtqKs7Rt6LxTPuClb6QPWxX/oI4VM7xHHnlEXPRKQbBYXE0rhyO2YuByXOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T20:21:05.431124Z"},"content_sha256":"025802dfda4486eda607aa4d4ade8db01fefd4985fa85f9926d0d8f9c3c1a1d7","schema_version":"1.0","event_id":"sha256:025802dfda4486eda607aa4d4ade8db01fefd4985fa85f9926d0d8f9c3c1a1d7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/bundle.json","state_url":"https://pith.science/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/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":"2026-08-07T20:21:05Z","links":{"resolver":"https://pith.science/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD","bundle":"https://pith.science/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/bundle.json","state":"https://pith.science/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HIMDRPSVDM6SWNKPEGXMZUXMTD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:HIMDRPSVDM6SWNKPEGXMZUXMTD","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":"710c1fc39114b0acd8efaf1e16d6a68cd3def7a8fccede1f16e063c7fa9fdaaf","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2022-10-26T11:07:04Z","title_canon_sha256":"d5be81131b242d5b512dfbdab527ce03a5dbee06edd5aca2ef78a18419dc1b15"},"schema_version":"1.0","source":{"id":"2210.15435","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.15435","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"arxiv_version","alias_value":"2210.15435v1","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.15435","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_12","alias_value":"HIMDRPSVDM6S","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_16","alias_value":"HIMDRPSVDM6SWNKP","created_at":"2026-07-05T05:11:06Z"},{"alias_kind":"pith_short_8","alias_value":"HIMDRPSV","created_at":"2026-07-05T05:11:06Z"}],"graph_snapshots":[{"event_id":"sha256:025802dfda4486eda607aa4d4ade8db01fefd4985fa85f9926d0d8f9c3c1a1d7","target":"graph","created_at":"2026-07-05T05:11:06Z","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/2210.15435/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and ","authors_text":"Bojan \\v{Z}unkovi\\v{c}, Enej Ilievski","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2022-10-26T11:07:04Z","title":"Grokking phase transitions in learning local rules with gradient descent"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.15435","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:cfa07ad952d3294774585520add884b76dfb0687a71c23b95c742776585db9d7","target":"record","created_at":"2026-07-05T05:11:06Z","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":"710c1fc39114b0acd8efaf1e16d6a68cd3def7a8fccede1f16e063c7fa9fdaaf","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2022-10-26T11:07:04Z","title_canon_sha256":"d5be81131b242d5b512dfbdab527ce03a5dbee06edd5aca2ef78a18419dc1b15"},"schema_version":"1.0","source":{"id":"2210.15435","kind":"arxiv","version":1}},"canonical_sha256":"3a1838be551b3d2b354f21aeccd2ec98f39f89a6060c073fe9f8dca5b21649bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a1838be551b3d2b354f21aeccd2ec98f39f89a6060c073fe9f8dca5b21649bb","first_computed_at":"2026-07-05T05:11:06.196080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:11:06.196080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HZMlo23oa+CikL2xe41iuvo9/PrX0RzRbaX8QgcM/zSTaTJO6p+dk9tKImINe90eDPn8bG1LEgBGcIaGLcCLAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:11:06.196563Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.15435","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfa07ad952d3294774585520add884b76dfb0687a71c23b95c742776585db9d7","sha256:025802dfda4486eda607aa4d4ade8db01fefd4985fa85f9926d0d8f9c3c1a1d7"],"state_sha256":"2a9e927b3f4994716d70fc4550180866beda7d4f04d6bc7d63dd94a123fbd2eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c7DDfgmz9LZSYN+HjsyzdCt+3w4F6IidmHRldPkYwo/JDGq8ae0OQFY2UrXmwF8/A7PlX7zA9i8RjE1o+MZSDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T20:21:05.437087Z","bundle_sha256":"b02c37d0cd90a9281b5013975023dc8486cd0f3e24a58d4e3d8d0150da572dff"}}