{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:H36P5JHVZX6RDJIXZ7ZG7EC37B","short_pith_number":"pith:H36P5JHV","canonical_record":{"source":{"id":"2606.08047","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2026-06-06T08:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"ec8cbafb222bbd0b4fb8b4e49ede3288eb025e7dc785c61c9ab2664e6220c724","abstract_canon_sha256":"1f197cd94ea0c13fd82c4619568fdc034dd490dab4a09696cbe7d18c764edc26"},"schema_version":"1.0"},"canonical_sha256":"3efcfea4f5cdfd11a517cff26f905bf86bdb7bbe5bd369051850730936af9e6c","source":{"kind":"arxiv","id":"2606.08047","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08047","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08047v1","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08047","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_12","alias_value":"H36P5JHVZX6R","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_16","alias_value":"H36P5JHVZX6RDJIX","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_8","alias_value":"H36P5JHV","created_at":"2026-06-09T01:05:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:H36P5JHVZX6RDJIXZ7ZG7EC37B","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08047","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2026-06-06T08:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"ec8cbafb222bbd0b4fb8b4e49ede3288eb025e7dc785c61c9ab2664e6220c724","abstract_canon_sha256":"1f197cd94ea0c13fd82c4619568fdc034dd490dab4a09696cbe7d18c764edc26"},"schema_version":"1.0"},"canonical_sha256":"3efcfea4f5cdfd11a517cff26f905bf86bdb7bbe5bd369051850730936af9e6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:24.414609Z","signature_b64":"5JHruVKQJkEUXmev+YZZpl8iN33TWAOphDj7CrFgx20DSNyhWz5HFF4/Ikqd88Kj1Hn8sQFcxM8xhsgF3vtpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3efcfea4f5cdfd11a517cff26f905bf86bdb7bbe5bd369051850730936af9e6c","last_reissued_at":"2026-06-09T01:05:24.413893Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:24.413893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08047","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":"2026-06-09T01:05:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MA8AcQXmSWTfqN3zc/ofgKKDJtsdx1RoN7+vbmKr37qx3X5/I/uglIVlNDKrhkRxFmLQczaYk4z8MTVEWFF/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T16:32:50.889045Z"},"content_sha256":"d0458ee599e43de77765d2ce9aca73ebf28a1160d583edb983a7d15ceab38383","schema_version":"1.0","event_id":"sha256:d0458ee599e43de77765d2ce9aca73ebf28a1160d583edb983a7d15ceab38383"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:H36P5JHVZX6RDJIXZ7ZG7EC37B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Warped Product Einstein Manifolds in Four Dimensions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"gr-qc","authors_text":"Fedor V. Kusmartsev, Jack C. M. Hughes, Joudy F. Jamal Beek","submitted_at":"2026-06-06T08:20:56Z","abstract_excerpt":"On four-dimensional (pseudo-)Riemannian manifolds $\\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \\times 6$ matrix representation which decomposes into four $3 \\times 3$ blocks. $\\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\\mathcal{M} = F \\times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08047","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/2606.08047/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":"2026-06-09T01:05:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JNMhQq2+aH+lFfW170WT5EP+5vD4CKUpdyZ0WCveB2cJtMm9lxopxnxz3UPRGbfPMw2e8JgjMmC6w/6H5TrwDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T16:32:50.889439Z"},"content_sha256":"f7e2ae7893e17a8b626d4f38df7e98cf22f416fec27fd6595934389a31b66be8","schema_version":"1.0","event_id":"sha256:f7e2ae7893e17a8b626d4f38df7e98cf22f416fec27fd6595934389a31b66be8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/bundle.json","state_url":"https://pith.science/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/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":"2026-06-29T16:32:50Z","links":{"resolver":"https://pith.science/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B","bundle":"https://pith.science/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/bundle.json","state":"https://pith.science/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H36P5JHVZX6RDJIXZ7ZG7EC37B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:H36P5JHVZX6RDJIXZ7ZG7EC37B","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":"1f197cd94ea0c13fd82c4619568fdc034dd490dab4a09696cbe7d18c764edc26","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2026-06-06T08:20:56Z","title_canon_sha256":"ec8cbafb222bbd0b4fb8b4e49ede3288eb025e7dc785c61c9ab2664e6220c724"},"schema_version":"1.0","source":{"id":"2606.08047","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08047","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08047v1","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08047","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_12","alias_value":"H36P5JHVZX6R","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_16","alias_value":"H36P5JHVZX6RDJIX","created_at":"2026-06-09T01:05:24Z"},{"alias_kind":"pith_short_8","alias_value":"H36P5JHV","created_at":"2026-06-09T01:05:24Z"}],"graph_snapshots":[{"event_id":"sha256:f7e2ae7893e17a8b626d4f38df7e98cf22f416fec27fd6595934389a31b66be8","target":"graph","created_at":"2026-06-09T01:05:24Z","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/2606.08047/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"On four-dimensional (pseudo-)Riemannian manifolds $\\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \\times 6$ matrix representation which decomposes into four $3 \\times 3$ blocks. $\\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\\mathcal{M} = F \\times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity","authors_text":"Fedor V. Kusmartsev, Jack C. M. Hughes, Joudy F. Jamal Beek","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2026-06-06T08:20:56Z","title":"Warped Product Einstein Manifolds in Four Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08047","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:d0458ee599e43de77765d2ce9aca73ebf28a1160d583edb983a7d15ceab38383","target":"record","created_at":"2026-06-09T01:05:24Z","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":"1f197cd94ea0c13fd82c4619568fdc034dd490dab4a09696cbe7d18c764edc26","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"gr-qc","submitted_at":"2026-06-06T08:20:56Z","title_canon_sha256":"ec8cbafb222bbd0b4fb8b4e49ede3288eb025e7dc785c61c9ab2664e6220c724"},"schema_version":"1.0","source":{"id":"2606.08047","kind":"arxiv","version":1}},"canonical_sha256":"3efcfea4f5cdfd11a517cff26f905bf86bdb7bbe5bd369051850730936af9e6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3efcfea4f5cdfd11a517cff26f905bf86bdb7bbe5bd369051850730936af9e6c","first_computed_at":"2026-06-09T01:05:24.413893Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:24.413893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5JHruVKQJkEUXmev+YZZpl8iN33TWAOphDj7CrFgx20DSNyhWz5HFF4/Ikqd88Kj1Hn8sQFcxM8xhsgF3vtpCQ==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:24.414609Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08047","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0458ee599e43de77765d2ce9aca73ebf28a1160d583edb983a7d15ceab38383","sha256:f7e2ae7893e17a8b626d4f38df7e98cf22f416fec27fd6595934389a31b66be8"],"state_sha256":"4011ae21a9a5c10fc35c0d46b1e5a03fdae0e6ccf25491c13606aa666a703500"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F+C0M/q2EWdCHYWQSLfGDR12c3z+uOfRGp/PbyFW872W1wnzxpkMhFlxSLurddcG4Rml/sYERjnCLttlAEyvBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T16:32:50.891600Z","bundle_sha256":"d49236bc3dd4f64334fc8d59222a86efc96aa5791b232fe924748e859bcf50f5"}}