{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:IPPKFWT7OXIN3RNZ7DJ2POAB2G","short_pith_number":"pith:IPPKFWT7","canonical_record":{"source":{"id":"2604.20308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-04-22T08:09:04Z","cross_cats_sorted":[],"title_canon_sha256":"b5963ee4779632999bcc18c88b0af1e9e80484c837b5940b929e3a2fdae86fe5","abstract_canon_sha256":"f5549ba833101d593dd239458d6397f2e4585a000a195db877d61efbc159d2af"},"schema_version":"1.0"},"canonical_sha256":"43dea2da7f75d0ddc5b9f8d3a7b801d1b619d19a2eb612d61b23e5b439bca9b6","source":{"kind":"arxiv","id":"2604.20308","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.20308","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"arxiv_version","alias_value":"2604.20308v2","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.20308","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_12","alias_value":"IPPKFWT7OXIN","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_16","alias_value":"IPPKFWT7OXIN3RNZ","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_8","alias_value":"IPPKFWT7","created_at":"2026-06-02T01:04:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:IPPKFWT7OXIN3RNZ7DJ2POAB2G","target":"record","payload":{"canonical_record":{"source":{"id":"2604.20308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-04-22T08:09:04Z","cross_cats_sorted":[],"title_canon_sha256":"b5963ee4779632999bcc18c88b0af1e9e80484c837b5940b929e3a2fdae86fe5","abstract_canon_sha256":"f5549ba833101d593dd239458d6397f2e4585a000a195db877d61efbc159d2af"},"schema_version":"1.0"},"canonical_sha256":"43dea2da7f75d0ddc5b9f8d3a7b801d1b619d19a2eb612d61b23e5b439bca9b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:04:16.374647Z","signature_b64":"n//5EiV8mBm9c87woX5x4KxKfuKkb/4ODgPrl870ulCUKgs7YaCT76/OfwX49v76flVPPALGkXr7INpRK//aBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43dea2da7f75d0ddc5b9f8d3a7b801d1b619d19a2eb612d61b23e5b439bca9b6","last_reissued_at":"2026-06-02T01:04:16.374169Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:04:16.374169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.20308","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T01:04:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YxphrccNG8W92HqHDrE51/QxmJ6LYGwrUPS4bk985Uxpt47PCNP6PkhahbeDRddpzcO1bTQL5Dsf5SYlEJA9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:23:42.412687Z"},"content_sha256":"ec7b774532ae4b9d6be3c3b5a5b8bc59d2ee3a9d59de45f6b2af4dbc77cc5282","schema_version":"1.0","event_id":"sha256:ec7b774532ae4b9d6be3c3b5a5b8bc59d2ee3a9d59de45f6b2af4dbc77cc5282"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:IPPKFWT7OXIN3RNZ7DJ2POAB2G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Anna Wienhard, Ce Ju, Diaaeldin Taha, Hao Li, Huitao Feng, Junwen Dong, Kelin Xia, Yuhan Peng, Yuzhi Zeng","submitted_at":"2026-04-22T08:09:04Z","abstract_excerpt":"Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-sp"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sheaf neural networks on the SPD manifold enable strictly more expressive second-order geometric representations than Euclidean versions and achieve SOTA results on most MoleculeNet benchmarks.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5dabebeeec228c7affbf9d0cf4c3a722a864eec41291f3d1af6014d3b209ddbf"},"source":{"id":"2604.20308","kind":"arxiv","version":2},"verdict":{"id":"bce2194c-75a9-4d08-8ebc-3d4555642811","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T00:35:29.506272Z","strongest_claim":"SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations.","one_line_summary":"Sheaf neural networks on the SPD manifold enable strictly more expressive second-order geometric representations than Euclidean versions and achieve SOTA results on most MoleculeNet benchmarks.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space.","pith_extraction_headline":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20308/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T14:43:33.126206Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T02:02:28.303246Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"08710cbd797c96aa848ef86e29f75cd7f391dec743c2f381b128d454162f2873"},"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":"bce2194c-75a9-4d08-8ebc-3d4555642811"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T01:04:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+qi/mCG7qp260pIxzM8NRGLuiE77GgktakesBzk7QCMctZmC4lnuYXf/XXBZwJSWPHg4H8+Wq9rbYCygsgLaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:23:42.413188Z"},"content_sha256":"f6e5fb871a6c98b1e6d51942fe008e279159c585b0e211edf368ce29d2c799ba","schema_version":"1.0","event_id":"sha256:f6e5fb871a6c98b1e6d51942fe008e279159c585b0e211edf368ce29d2c799ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/bundle.json","state_url":"https://pith.science/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/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":"2026-06-22T09:23:42Z","links":{"resolver":"https://pith.science/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G","bundle":"https://pith.science/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/bundle.json","state":"https://pith.science/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IPPKFWT7OXIN3RNZ7DJ2POAB2G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IPPKFWT7OXIN3RNZ7DJ2POAB2G","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":"f5549ba833101d593dd239458d6397f2e4585a000a195db877d61efbc159d2af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-04-22T08:09:04Z","title_canon_sha256":"b5963ee4779632999bcc18c88b0af1e9e80484c837b5940b929e3a2fdae86fe5"},"schema_version":"1.0","source":{"id":"2604.20308","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.20308","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"arxiv_version","alias_value":"2604.20308v2","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.20308","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_12","alias_value":"IPPKFWT7OXIN","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_16","alias_value":"IPPKFWT7OXIN3RNZ","created_at":"2026-06-02T01:04:16Z"},{"alias_kind":"pith_short_8","alias_value":"IPPKFWT7","created_at":"2026-06-02T01:04:16Z"}],"graph_snapshots":[{"event_id":"sha256:f6e5fb871a6c98b1e6d51942fe008e279159c585b0e211edf368ce29d2c799ba","target":"graph","created_at":"2026-06-02T01:04:16Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Sheaf neural networks on the SPD manifold enable strictly more expressive second-order geometric representations than Euclidean versions and achieve SOTA results on most MoleculeNet benchmarks."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot."}],"snapshot_sha256":"5dabebeeec228c7affbf9d0cf4c3a722a864eec41291f3d1af6014d3b209ddbf"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T14:43:33.126206Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T02:02:28.303246Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.20308/integrity.json","findings":[],"snapshot_sha256":"08710cbd797c96aa848ef86e29f75cd7f391dec743c2f381b128d454162f2873","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-sp","authors_text":"Anna Wienhard, Ce Ju, Diaaeldin Taha, Hao Li, Huitao Feng, Junwen Dong, Kelin Xia, Yuhan Peng, Yuzhi Zeng","cross_cats":[],"headline":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-04-22T08:09:04Z","title":"Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.20308","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T00:35:29.506272Z","id":"bce2194c-75a9-4d08-8ebc-3d4555642811","model_set":{"reader":"grok-4.3"},"one_line_summary":"Sheaf neural networks on the SPD manifold enable strictly more expressive second-order geometric representations than Euclidean versions and achieve SOTA results on most MoleculeNet benchmarks.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Sheaf neural networks defined on SPD manifolds represent second-order geometric features that Euclidean sheaves cannot.","strongest_claim":"SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations.","weakest_assumption":"The SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space."}},"verdict_id":"bce2194c-75a9-4d08-8ebc-3d4555642811"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ec7b774532ae4b9d6be3c3b5a5b8bc59d2ee3a9d59de45f6b2af4dbc77cc5282","target":"record","created_at":"2026-06-02T01:04:16Z","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":"f5549ba833101d593dd239458d6397f2e4585a000a195db877d61efbc159d2af","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2026-04-22T08:09:04Z","title_canon_sha256":"b5963ee4779632999bcc18c88b0af1e9e80484c837b5940b929e3a2fdae86fe5"},"schema_version":"1.0","source":{"id":"2604.20308","kind":"arxiv","version":2}},"canonical_sha256":"43dea2da7f75d0ddc5b9f8d3a7b801d1b619d19a2eb612d61b23e5b439bca9b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"43dea2da7f75d0ddc5b9f8d3a7b801d1b619d19a2eb612d61b23e5b439bca9b6","first_computed_at":"2026-06-02T01:04:16.374169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T01:04:16.374169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n//5EiV8mBm9c87woX5x4KxKfuKkb/4ODgPrl870ulCUKgs7YaCT76/OfwX49v76flVPPALGkXr7INpRK//aBQ==","signature_status":"signed_v1","signed_at":"2026-06-02T01:04:16.374647Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.20308","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec7b774532ae4b9d6be3c3b5a5b8bc59d2ee3a9d59de45f6b2af4dbc77cc5282","sha256:f6e5fb871a6c98b1e6d51942fe008e279159c585b0e211edf368ce29d2c799ba"],"state_sha256":"cb9d1f4c0a1fb57f182a597bb9cc5fbebf6a38bd46d3cb80d785997f72ec4b4a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fcaoVRwYyKFOruew+5dHhqn/W+NVzmI0+APFDbYVoIKhVWRyO1HwtEBenvK+rqwPjn7kNjgRK61HQhonZuBqAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:23:42.415607Z","bundle_sha256":"3517453072f47abfab89ba80432ac9b4f13685ad9e7a18914425bd4949538dc2"}}