{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JR4FZD3VDPNE5OTMANE2CMTXWP","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":"9d019558f754a86bb43112e6bc88e696fabaefbac9968c34dbd0aea6df55cb42","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-28T17:18:11Z","title_canon_sha256":"ecef0cf3d7339074a6915452b7d88983159823e7752a5e44144333624e0304df"},"schema_version":"1.0","source":{"id":"2607.26001","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26001","created_at":"2026-07-29T01:26:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26001v1","created_at":"2026-07-29T01:26:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26001","created_at":"2026-07-29T01:26:18Z"},{"alias_kind":"pith_short_12","alias_value":"JR4FZD3VDPNE","created_at":"2026-07-29T01:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"JR4FZD3VDPNE5OTM","created_at":"2026-07-29T01:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"JR4FZD3V","created_at":"2026-07-29T01:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:3d4abf12c2aead9cdc9a925a5869634f059b7420a4246f35d5f605febb97f0a4","target":"graph","created_at":"2026-07-29T01:26:18Z","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/2607.26001/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Sharpness-Aware Minimization (SAM) aims to improve generalization by encouraging insensitivity to small, worst-case parameter perturbations. However, the notion of a \"small\" perturbation is inherently geometry-dependent: while existing SAM variants have explored a wide range of choices, a clear perspective on which geometries are most effective in practice remains elusive. Recent work on matrix-aware optimization, particularly the Muon optimizer, suggests that respecting the matrix structure of hidden-layer weights can lead to strong empirical performance. Motivated by this, we study matrix-aw","authors_text":"Edward Milsom, Michael Murray, Wenzhi Zhong","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-28T17:18:11Z","title":"Sharpness-Aware Minimization and Muon: Robustness under the Spectral Norm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26001","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:b6e9a685242f70a4752183db3002051a7b756a10531d419067135e959bda88a2","target":"record","created_at":"2026-07-29T01:26:18Z","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":"9d019558f754a86bb43112e6bc88e696fabaefbac9968c34dbd0aea6df55cb42","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-07-28T17:18:11Z","title_canon_sha256":"ecef0cf3d7339074a6915452b7d88983159823e7752a5e44144333624e0304df"},"schema_version":"1.0","source":{"id":"2607.26001","kind":"arxiv","version":1}},"canonical_sha256":"4c785c8f751bda4eba6c0349a13277b3e0517157b33371fb042b952bf40bd01a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4c785c8f751bda4eba6c0349a13277b3e0517157b33371fb042b952bf40bd01a","first_computed_at":"2026-07-29T01:26:18.748912Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:26:18.748912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ybt/LbA35iZQq8+/dOqNAlcSBMLFdnM5bvSGgQ733xxg3NpQMka99eEz4irK7J+UylCybnSICeQpyYtY86M1BQ==","signature_status":"signed_v1","signed_at":"2026-07-29T01:26:18.749821Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.26001","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6e9a685242f70a4752183db3002051a7b756a10531d419067135e959bda88a2","sha256:3d4abf12c2aead9cdc9a925a5869634f059b7420a4246f35d5f605febb97f0a4"],"state_sha256":"0be559c3e5054c5972e940519278a2a856b9fcaf7b8c40defeed56a2e9d6b11c"}