{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GOTXSE5PUKK5BCBQJAR5FMR56W","short_pith_number":"pith:GOTXSE5P","canonical_record":{"source":{"id":"2607.20512","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-06T22:35:57Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"f0c89356cd3c875bb7934638c42d5a216592519c50d3418968515c77ef229c79","abstract_canon_sha256":"2296a0c490be950a41c961eac5cf8b10fbfacc290dd9ef6fb8a1e69387a6014d"},"schema_version":"1.0"},"canonical_sha256":"33a77913afa295d088304823d2b23df593e7e5c82e3d9c14ca3bb37cb753e565","source":{"kind":"arxiv","id":"2607.20512","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20512","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20512v1","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20512","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_12","alias_value":"GOTXSE5PUKK5","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_16","alias_value":"GOTXSE5PUKK5BCBQ","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_8","alias_value":"GOTXSE5P","created_at":"2026-07-24T00:23:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GOTXSE5PUKK5BCBQJAR5FMR56W","target":"record","payload":{"canonical_record":{"source":{"id":"2607.20512","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-06T22:35:57Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"f0c89356cd3c875bb7934638c42d5a216592519c50d3418968515c77ef229c79","abstract_canon_sha256":"2296a0c490be950a41c961eac5cf8b10fbfacc290dd9ef6fb8a1e69387a6014d"},"schema_version":"1.0"},"canonical_sha256":"33a77913afa295d088304823d2b23df593e7e5c82e3d9c14ca3bb37cb753e565","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T00:23:19.730099Z","signature_b64":"9cLtdnZ0b0bwVjeex4YGS6oaIPdofDwc4AXzRmv9KZZZvwkb+WA675obWZitNPLnY3ML73PZmElp2Ijy6UpsDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33a77913afa295d088304823d2b23df593e7e5c82e3d9c14ca3bb37cb753e565","last_reissued_at":"2026-07-24T00:23:19.729243Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T00:23:19.729243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.20512","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":"2026-07-24T00:23:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9eixQft+jQCS/+8+IUBBxLQ1KDJw56PtWoCljm2ZyHBVHkGM7AkH3s50R467F/tvxTtWc4ptRHNOGoDyi0GxCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T02:36:53.761101Z"},"content_sha256":"f4e4e7fa204c6dc768bfe6ec7fc67c87f409bd0eb516b77a88e9c7c630fcc437","schema_version":"1.0","event_id":"sha256:f4e4e7fa204c6dc768bfe6ec7fc67c87f409bd0eb516b77a88e9c7c630fcc437"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GOTXSE5PUKK5BCBQJAR5FMR56W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Active Ingredient in Muon's Grokking","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Yufeng Wang","submitted_at":"2026-07-06T22:35:57Z","abstract_excerpt":"The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW. Prior work attributes this to \"spectral-norm constraints plus orthogonalized momentum\" but does not isolate which mechanism matters. To better understand Moun's behavior, we run multi-seed and multi-learning-rate sweeps to decompose and stress-test the effect. First, an ablation shows the speedup comes from orthogonalization (the Newton-Schulz iteration): orthogonalize-only matches full Muon, whereas spectral-only is no faster than AdamW and is unreliable, and this verdict holds across learning rates. Se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20512","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/2607.20512/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":"2026-07-24T00:23:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oNpKa53VT4kIy2NK/p0v+tAHRUPV4uZBDnpHr9fQpAwdsFgWUM8Z8WQaGSIqyPifAxBbmg1z43pmaHZUviLhDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T02:36:53.761708Z"},"content_sha256":"941fecd4257ac863a070369d124e27ca3a7caf9e4c28c64d579cbf8bf2a3e7b4","schema_version":"1.0","event_id":"sha256:941fecd4257ac863a070369d124e27ca3a7caf9e4c28c64d579cbf8bf2a3e7b4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/bundle.json","state_url":"https://pith.science/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/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":"2026-08-10T02:36:53Z","links":{"resolver":"https://pith.science/pith/GOTXSE5PUKK5BCBQJAR5FMR56W","bundle":"https://pith.science/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/bundle.json","state":"https://pith.science/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GOTXSE5PUKK5BCBQJAR5FMR56W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GOTXSE5PUKK5BCBQJAR5FMR56W","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":"2296a0c490be950a41c961eac5cf8b10fbfacc290dd9ef6fb8a1e69387a6014d","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-06T22:35:57Z","title_canon_sha256":"f0c89356cd3c875bb7934638c42d5a216592519c50d3418968515c77ef229c79"},"schema_version":"1.0","source":{"id":"2607.20512","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20512","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20512v1","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20512","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_12","alias_value":"GOTXSE5PUKK5","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_16","alias_value":"GOTXSE5PUKK5BCBQ","created_at":"2026-07-24T00:23:19Z"},{"alias_kind":"pith_short_8","alias_value":"GOTXSE5P","created_at":"2026-07-24T00:23:19Z"}],"graph_snapshots":[{"event_id":"sha256:941fecd4257ac863a070369d124e27ca3a7caf9e4c28c64d579cbf8bf2a3e7b4","target":"graph","created_at":"2026-07-24T00:23:19Z","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/2607.20512/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW. Prior work attributes this to \"spectral-norm constraints plus orthogonalized momentum\" but does not isolate which mechanism matters. To better understand Moun's behavior, we run multi-seed and multi-learning-rate sweeps to decompose and stress-test the effect. First, an ablation shows the speedup comes from orthogonalization (the Newton-Schulz iteration): orthogonalize-only matches full Muon, whereas spectral-only is no faster than AdamW and is unreliable, and this verdict holds across learning rates. Se","authors_text":"Yufeng Wang","cross_cats":["cs.AI"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-06T22:35:57Z","title":"The Active Ingredient in Muon's Grokking"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20512","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:f4e4e7fa204c6dc768bfe6ec7fc67c87f409bd0eb516b77a88e9c7c630fcc437","target":"record","created_at":"2026-07-24T00:23:19Z","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":"2296a0c490be950a41c961eac5cf8b10fbfacc290dd9ef6fb8a1e69387a6014d","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-06T22:35:57Z","title_canon_sha256":"f0c89356cd3c875bb7934638c42d5a216592519c50d3418968515c77ef229c79"},"schema_version":"1.0","source":{"id":"2607.20512","kind":"arxiv","version":1}},"canonical_sha256":"33a77913afa295d088304823d2b23df593e7e5c82e3d9c14ca3bb37cb753e565","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33a77913afa295d088304823d2b23df593e7e5c82e3d9c14ca3bb37cb753e565","first_computed_at":"2026-07-24T00:23:19.729243Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-24T00:23:19.729243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9cLtdnZ0b0bwVjeex4YGS6oaIPdofDwc4AXzRmv9KZZZvwkb+WA675obWZitNPLnY3ML73PZmElp2Ijy6UpsDA==","signature_status":"signed_v1","signed_at":"2026-07-24T00:23:19.730099Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.20512","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4e4e7fa204c6dc768bfe6ec7fc67c87f409bd0eb516b77a88e9c7c630fcc437","sha256:941fecd4257ac863a070369d124e27ca3a7caf9e4c28c64d579cbf8bf2a3e7b4"],"state_sha256":"4ca81edaadd50da64286aeb9e2bda52ffe6db2cf90bba243ed54d09469221acf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BHZDvoEbCCQlOHrKQuv7PQW28In1/R5DvEGQ/VLOTZYtJJkeWDyY8R5tai/lrsyabs6BUXmaGA69Dw0vxw8sDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T02:36:53.767437Z","bundle_sha256":"7490a50dc0222f9249ed8138f08cb33119dd5ae3c04d1ae46ecadac8bb6519a2"}}