{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:FQ2SNRVF3GBZM7UO7YR3FLLXVN","short_pith_number":"pith:FQ2SNRVF","canonical_record":{"source":{"id":"2606.04405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T03:33:45Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"9bbe28fdfe39a3bdf9e9b0bd1b24f8de782a9a785516119db5423b3a7c0b8b73","abstract_canon_sha256":"92dfb718f7e10669d7d1bc98718ee716581fb23fcd1ff3a0fdaddb2fb426890c"},"schema_version":"1.0"},"canonical_sha256":"2c3526c6a5d983967e8efe23b2ad77ab7dbfb4375acc2016cca92218eb12a5c3","source":{"kind":"arxiv","id":"2606.04405","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04405","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04405v1","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04405","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_12","alias_value":"FQ2SNRVF3GBZ","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_16","alias_value":"FQ2SNRVF3GBZM7UO","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_8","alias_value":"FQ2SNRVF","created_at":"2026-06-04T01:09:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:FQ2SNRVF3GBZM7UO7YR3FLLXVN","target":"record","payload":{"canonical_record":{"source":{"id":"2606.04405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T03:33:45Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"9bbe28fdfe39a3bdf9e9b0bd1b24f8de782a9a785516119db5423b3a7c0b8b73","abstract_canon_sha256":"92dfb718f7e10669d7d1bc98718ee716581fb23fcd1ff3a0fdaddb2fb426890c"},"schema_version":"1.0"},"canonical_sha256":"2c3526c6a5d983967e8efe23b2ad77ab7dbfb4375acc2016cca92218eb12a5c3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:06.853581Z","signature_b64":"haZlJHvot10PMX+6jivquu9DSvmO0Tch1g9kpP5XZO+yITNwAxdf5H38lj1I7pVKN+oYEVnp2IyYS99MNwJvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c3526c6a5d983967e8efe23b2ad77ab7dbfb4375acc2016cca92218eb12a5c3","last_reissued_at":"2026-06-04T01:09:06.853090Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:06.853090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.04405","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":"2026-06-04T01:09:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1iGlkcz1GcRD1kU7AvlErUUFS4L6BsMNbA0NhRnI+GYm4m7B74H8vCvNgrP96rSH7vk8UIvXFIVeD5ElDfjCAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:14:24.341549Z"},"content_sha256":"733ea403a94660f168433639d5a4755dc02532dd5dd7a8f9150d0a6568f1fdc7","schema_version":"1.0","event_id":"sha256:733ea403a94660f168433639d5a4755dc02532dd5dd7a8f9150d0a6568f1fdc7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:FQ2SNRVF3GBZM7UO7YR3FLLXVN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Low-Rank Decay for Grokking in Scale-Invariant Transformers: A Spectral-Geometric View","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Mingyu Li","submitted_at":"2026-06-03T03:33:45Z","abstract_excerpt":"Modern Transformer architectures frequently employ normalization mechanisms such as RMSNorm and Query-Key Normalization, making parts of the model approximately scale-invariant with respect to weight magnitudes. In this regime, standard Frobenius-norm weight decay acts purely along the radial direction of the weight space and cannot directly simplify the function represented by the normalized layer. We study grokking in small algorithmic tasks through this lens and propose \\emph{Low-Rank Decay} (LRD), a nuclear-norm-like spectral regularizer whose subgradient -- the polar factor $UV^\\top$ -- r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04405","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/2606.04405/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":"2026-06-04T01:09:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vskl9aQ9UPmbV2IAJYYquMrlGF6L4J9rC8m0GjzZntY2sfkpvPJYGu6Sf8m+VVsFr5b3kNVGkkzRgVfV0QPYCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T13:14:24.341951Z"},"content_sha256":"a321f7be36b1e0a4056e58eda95df6708d55a8aa43b5fdf16a10d53ece2e55d4","schema_version":"1.0","event_id":"sha256:a321f7be36b1e0a4056e58eda95df6708d55a8aa43b5fdf16a10d53ece2e55d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/bundle.json","state_url":"https://pith.science/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/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":"2026-07-03T13:14:24Z","links":{"resolver":"https://pith.science/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN","bundle":"https://pith.science/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/bundle.json","state":"https://pith.science/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FQ2SNRVF3GBZM7UO7YR3FLLXVN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FQ2SNRVF3GBZM7UO7YR3FLLXVN","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":"92dfb718f7e10669d7d1bc98718ee716581fb23fcd1ff3a0fdaddb2fb426890c","cross_cats_sorted":["cs.AI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T03:33:45Z","title_canon_sha256":"9bbe28fdfe39a3bdf9e9b0bd1b24f8de782a9a785516119db5423b3a7c0b8b73"},"schema_version":"1.0","source":{"id":"2606.04405","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04405","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04405v1","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04405","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_12","alias_value":"FQ2SNRVF3GBZ","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_16","alias_value":"FQ2SNRVF3GBZM7UO","created_at":"2026-06-04T01:09:06Z"},{"alias_kind":"pith_short_8","alias_value":"FQ2SNRVF","created_at":"2026-06-04T01:09:06Z"}],"graph_snapshots":[{"event_id":"sha256:a321f7be36b1e0a4056e58eda95df6708d55a8aa43b5fdf16a10d53ece2e55d4","target":"graph","created_at":"2026-06-04T01:09:06Z","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/2606.04405/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Modern Transformer architectures frequently employ normalization mechanisms such as RMSNorm and Query-Key Normalization, making parts of the model approximately scale-invariant with respect to weight magnitudes. In this regime, standard Frobenius-norm weight decay acts purely along the radial direction of the weight space and cannot directly simplify the function represented by the normalized layer. We study grokking in small algorithmic tasks through this lens and propose \\emph{Low-Rank Decay} (LRD), a nuclear-norm-like spectral regularizer whose subgradient -- the polar factor $UV^\\top$ -- r","authors_text":"Mingyu Li","cross_cats":["cs.AI"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T03:33:45Z","title":"Low-Rank Decay for Grokking in Scale-Invariant Transformers: A Spectral-Geometric View"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04405","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":"sha256:733ea403a94660f168433639d5a4755dc02532dd5dd7a8f9150d0a6568f1fdc7","target":"record","created_at":"2026-06-04T01:09:06Z","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":"92dfb718f7e10669d7d1bc98718ee716581fb23fcd1ff3a0fdaddb2fb426890c","cross_cats_sorted":["cs.AI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-06-03T03:33:45Z","title_canon_sha256":"9bbe28fdfe39a3bdf9e9b0bd1b24f8de782a9a785516119db5423b3a7c0b8b73"},"schema_version":"1.0","source":{"id":"2606.04405","kind":"arxiv","version":1}},"canonical_sha256":"2c3526c6a5d983967e8efe23b2ad77ab7dbfb4375acc2016cca92218eb12a5c3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c3526c6a5d983967e8efe23b2ad77ab7dbfb4375acc2016cca92218eb12a5c3","first_computed_at":"2026-06-04T01:09:06.853090Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:06.853090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"haZlJHvot10PMX+6jivquu9DSvmO0Tch1g9kpP5XZO+yITNwAxdf5H38lj1I7pVKN+oYEVnp2IyYS99MNwJvCA==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:06.853581Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:733ea403a94660f168433639d5a4755dc02532dd5dd7a8f9150d0a6568f1fdc7","sha256:a321f7be36b1e0a4056e58eda95df6708d55a8aa43b5fdf16a10d53ece2e55d4"],"state_sha256":"cdc606bc24565696b550ffa69dbe264f4bf871cf441fcfad849bd99ead5799fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FxFL8eoas8lZ6MSorhRYbHumkDjkih7R14JOs3NfiNOaX/k/CGdxQhksWL3YF6KGvtehGYwkeQw6COFXQDrSBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T13:14:24.344289Z","bundle_sha256":"372ba50b43af1d3d9014acfebb69dc59c8f90b08d60a312841607e8025674d5b"}}