{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:BGT73LMPV2BIP6EP5EQ247JERO","short_pith_number":"pith:BGT73LMP","canonical_record":{"source":{"id":"2607.19771","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-22T05:39:00Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"7bcab4ebecab087c4661bfa83fb4408b75cf44645e1c274e35d22380463767c2","abstract_canon_sha256":"ca94a256633e1ed5820ad5bc6ffc44ec30df96dfe4ce3284026b1ed1be367d4e"},"schema_version":"1.0"},"canonical_sha256":"09a7fdad8fae8287f88fe921ae7d248baf738d86943286fb4c449847e7778903","source":{"kind":"arxiv","id":"2607.19771","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19771","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19771v1","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19771","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_12","alias_value":"BGT73LMPV2BI","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_16","alias_value":"BGT73LMPV2BIP6EP","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_8","alias_value":"BGT73LMP","created_at":"2026-07-23T01:23:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:BGT73LMPV2BIP6EP5EQ247JERO","target":"record","payload":{"canonical_record":{"source":{"id":"2607.19771","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-22T05:39:00Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"7bcab4ebecab087c4661bfa83fb4408b75cf44645e1c274e35d22380463767c2","abstract_canon_sha256":"ca94a256633e1ed5820ad5bc6ffc44ec30df96dfe4ce3284026b1ed1be367d4e"},"schema_version":"1.0"},"canonical_sha256":"09a7fdad8fae8287f88fe921ae7d248baf738d86943286fb4c449847e7778903","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:23:58.318450Z","signature_b64":"61L3uZCUM4q2cCr7do8ZnLTpNUbRuac8CR9S2J0gh/Rj6TNqQnlL3qY5Fzi3hwYB9wzuPBoUU/hVUW0WnOcPBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09a7fdad8fae8287f88fe921ae7d248baf738d86943286fb4c449847e7778903","last_reissued_at":"2026-07-23T01:23:58.317540Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:23:58.317540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.19771","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":"2026-07-23T01:23:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E+TpsWKrlNWTKiKGguESAcflElHw/zUvWEVZ0jn49mu8uy6m/ADUGZMl3g20kjuJqhAQ6lY0guxn0mqRwd8gAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T04:17:14.710675Z"},"content_sha256":"47caf580145981373891cd7e7a4b997fe2949834fab6e55353c4e136b694a018","schema_version":"1.0","event_id":"sha256:47caf580145981373891cd7e7a4b997fe2949834fab6e55353c4e136b694a018"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:BGT73LMPV2BIP6EP5EQ247JERO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Isotropy-Preserving Spectral Cap for Muon: Theory and Three Case Studies","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Jiachun Li","submitted_at":"2026-07-22T05:39:00Z","abstract_excerpt":"Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19771","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/2607.19771/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":"2026-07-23T01:23:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3bbXc3E5RjH/+KOxFqO8/ggC9Gcl9kLaiyJe/n62+bSlU41yZkg3z4J97zhAgpR7ZRE3DCPH1o0jtTrighN3DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T04:17:14.711190Z"},"content_sha256":"a07a81ae4050000ee2fb65a98e4f323034d4819c2f87b8018c55443a5f9c41c1","schema_version":"1.0","event_id":"sha256:a07a81ae4050000ee2fb65a98e4f323034d4819c2f87b8018c55443a5f9c41c1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BGT73LMPV2BIP6EP5EQ247JERO/bundle.json","state_url":"https://pith.science/pith/BGT73LMPV2BIP6EP5EQ247JERO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BGT73LMPV2BIP6EP5EQ247JERO/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":"2026-08-03T04:17:14Z","links":{"resolver":"https://pith.science/pith/BGT73LMPV2BIP6EP5EQ247JERO","bundle":"https://pith.science/pith/BGT73LMPV2BIP6EP5EQ247JERO/bundle.json","state":"https://pith.science/pith/BGT73LMPV2BIP6EP5EQ247JERO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BGT73LMPV2BIP6EP5EQ247JERO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:BGT73LMPV2BIP6EP5EQ247JERO","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":"ca94a256633e1ed5820ad5bc6ffc44ec30df96dfe4ce3284026b1ed1be367d4e","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-22T05:39:00Z","title_canon_sha256":"7bcab4ebecab087c4661bfa83fb4408b75cf44645e1c274e35d22380463767c2"},"schema_version":"1.0","source":{"id":"2607.19771","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.19771","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"arxiv_version","alias_value":"2607.19771v1","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19771","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_12","alias_value":"BGT73LMPV2BI","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_16","alias_value":"BGT73LMPV2BIP6EP","created_at":"2026-07-23T01:23:58Z"},{"alias_kind":"pith_short_8","alias_value":"BGT73LMP","created_at":"2026-07-23T01:23:58Z"}],"graph_snapshots":[{"event_id":"sha256:a07a81ae4050000ee2fb65a98e4f323034d4819c2f87b8018c55443a5f9c41c1","target":"graph","created_at":"2026-07-23T01:23:58Z","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/2607.19771/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward","authors_text":"Jiachun Li","cross_cats":["cs.AI"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-22T05:39:00Z","title":"An Isotropy-Preserving Spectral Cap for Muon: Theory and Three Case Studies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19771","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":"sha256:47caf580145981373891cd7e7a4b997fe2949834fab6e55353c4e136b694a018","target":"record","created_at":"2026-07-23T01:23:58Z","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":"ca94a256633e1ed5820ad5bc6ffc44ec30df96dfe4ce3284026b1ed1be367d4e","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-07-22T05:39:00Z","title_canon_sha256":"7bcab4ebecab087c4661bfa83fb4408b75cf44645e1c274e35d22380463767c2"},"schema_version":"1.0","source":{"id":"2607.19771","kind":"arxiv","version":1}},"canonical_sha256":"09a7fdad8fae8287f88fe921ae7d248baf738d86943286fb4c449847e7778903","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09a7fdad8fae8287f88fe921ae7d248baf738d86943286fb4c449847e7778903","first_computed_at":"2026-07-23T01:23:58.317540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-23T01:23:58.317540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"61L3uZCUM4q2cCr7do8ZnLTpNUbRuac8CR9S2J0gh/Rj6TNqQnlL3qY5Fzi3hwYB9wzuPBoUU/hVUW0WnOcPBA==","signature_status":"signed_v1","signed_at":"2026-07-23T01:23:58.318450Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.19771","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47caf580145981373891cd7e7a4b997fe2949834fab6e55353c4e136b694a018","sha256:a07a81ae4050000ee2fb65a98e4f323034d4819c2f87b8018c55443a5f9c41c1"],"state_sha256":"dec8739de8deb489278469831cbab6cc946a7ae698903f379d80fe2f675e9165"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FnUL2hI/GuCc+qMefvaCbOI3k7AfPKzz2f3vyx2pjWfKfiLK+jrXHMzG98zYC20pG1e+oOSZRm4yGjJ0jSknCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T04:17:14.716094Z","bundle_sha256":"cd1f43fc8a3390343df2c5b5c7a2e08d6e1e63f883ca16d1f104ca3002a9bd63"}}