{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:JK4Q5BDI7257BSKJ5UUQEEV7PH","short_pith_number":"pith:JK4Q5BDI","canonical_record":{"source":{"id":"1304.5645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-20T16:21:13Z","cross_cats_sorted":[],"title_canon_sha256":"316c1669fda01bdf1b78fe67ae900b63050bdbc3a938ececa93f738f32fb5419","abstract_canon_sha256":"ee3e266e070afc13d40a32bc959e0ba4d4544558f26424870d9ce66d211c224f"},"schema_version":"1.0"},"canonical_sha256":"4ab90e8468febbf0c949ed290212bf79ce5970447fbe3f37f12bd5264880eede","source":{"kind":"arxiv","id":"1304.5645","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.5645","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"arxiv_version","alias_value":"1304.5645v1","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.5645","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"pith_short_12","alias_value":"JK4Q5BDI7257","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JK4Q5BDI7257BSKJ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JK4Q5BDI","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:JK4Q5BDI7257BSKJ5UUQEEV7PH","target":"record","payload":{"canonical_record":{"source":{"id":"1304.5645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-20T16:21:13Z","cross_cats_sorted":[],"title_canon_sha256":"316c1669fda01bdf1b78fe67ae900b63050bdbc3a938ececa93f738f32fb5419","abstract_canon_sha256":"ee3e266e070afc13d40a32bc959e0ba4d4544558f26424870d9ce66d211c224f"},"schema_version":"1.0"},"canonical_sha256":"4ab90e8468febbf0c949ed290212bf79ce5970447fbe3f37f12bd5264880eede","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:39.125300Z","signature_b64":"K3rpUxr2Nd6obKLj/tBdsvVKahw651Umlk7ZsYXoft+4brr7gLQbq6GOwdDI+LO055cuEvdAbaAkDw947TsdAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ab90e8468febbf0c949ed290212bf79ce5970447fbe3f37f12bd5264880eede","last_reissued_at":"2026-05-18T01:19:39.124882Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:39.124882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.5645","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":"2026-05-18T01:19:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MVoEroN43nQVOLm8udWca2zHfmrUXy9jP2OWhA/u3+F7WrwgAhgMxuh1wHLq0tAEiiuln4BayuxbeCKEhjxgCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T04:36:28.869695Z"},"content_sha256":"1966e4c51d6de1df23abec6a3e7d403466b94ed278b5b255e3a188076a1f9710","schema_version":"1.0","event_id":"sha256:1966e4c51d6de1df23abec6a3e7d403466b94ed278b5b255e3a188076a1f9710"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:JK4Q5BDI7257BSKJ5UUQEEV7PH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral expansions of cosmological fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anatoliy Malyarenko","submitted_at":"2013-04-20T16:21:13Z","abstract_excerpt":"We give a review of the theory of random fields defined on the observable part of the Universe that satisfy the cosmological principle, i.e., invariant with respect to the 6-dimensional group $\\mathcal{G}$ of the isometries of the time slice of the Friedmann--Lema\\^{\\i}tre--Robertson--Walker standard chart. Our new results include proof of spectral expansions of scalar and spin weighted $\\mathcal{G}$-invariant cosmological fields in open, flat, and closed cosmological models."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5645","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":""},"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":"2026-05-18T01:19:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lgOiE5RwiqU4S6Dz/wPeiWNRY6qJ/x+r6P8VC29X4fPFKARKiXrWk2dxITsX7a/JgL3vCzqI6jsGJj5ZneOxDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T04:36:28.870034Z"},"content_sha256":"c1c5e3e232e151eafc8ee29ff12cabd1858a272800a300604cb5fe21051501b2","schema_version":"1.0","event_id":"sha256:c1c5e3e232e151eafc8ee29ff12cabd1858a272800a300604cb5fe21051501b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/bundle.json","state_url":"https://pith.science/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/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":"2026-06-26T04:36:28Z","links":{"resolver":"https://pith.science/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH","bundle":"https://pith.science/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/bundle.json","state":"https://pith.science/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JK4Q5BDI7257BSKJ5UUQEEV7PH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:JK4Q5BDI7257BSKJ5UUQEEV7PH","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":"ee3e266e070afc13d40a32bc959e0ba4d4544558f26424870d9ce66d211c224f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-20T16:21:13Z","title_canon_sha256":"316c1669fda01bdf1b78fe67ae900b63050bdbc3a938ececa93f738f32fb5419"},"schema_version":"1.0","source":{"id":"1304.5645","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.5645","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"arxiv_version","alias_value":"1304.5645v1","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.5645","created_at":"2026-05-18T01:19:39Z"},{"alias_kind":"pith_short_12","alias_value":"JK4Q5BDI7257","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"JK4Q5BDI7257BSKJ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"JK4Q5BDI","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:c1c5e3e232e151eafc8ee29ff12cabd1858a272800a300604cb5fe21051501b2","target":"graph","created_at":"2026-05-18T01:19:39Z","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"},"paper":{"abstract_excerpt":"We give a review of the theory of random fields defined on the observable part of the Universe that satisfy the cosmological principle, i.e., invariant with respect to the 6-dimensional group $\\mathcal{G}$ of the isometries of the time slice of the Friedmann--Lema\\^{\\i}tre--Robertson--Walker standard chart. Our new results include proof of spectral expansions of scalar and spin weighted $\\mathcal{G}$-invariant cosmological fields in open, flat, and closed cosmological models.","authors_text":"Anatoliy Malyarenko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-20T16:21:13Z","title":"Spectral expansions of cosmological fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5645","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":"sha256:1966e4c51d6de1df23abec6a3e7d403466b94ed278b5b255e3a188076a1f9710","target":"record","created_at":"2026-05-18T01:19:39Z","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":"ee3e266e070afc13d40a32bc959e0ba4d4544558f26424870d9ce66d211c224f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-20T16:21:13Z","title_canon_sha256":"316c1669fda01bdf1b78fe67ae900b63050bdbc3a938ececa93f738f32fb5419"},"schema_version":"1.0","source":{"id":"1304.5645","kind":"arxiv","version":1}},"canonical_sha256":"4ab90e8468febbf0c949ed290212bf79ce5970447fbe3f37f12bd5264880eede","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ab90e8468febbf0c949ed290212bf79ce5970447fbe3f37f12bd5264880eede","first_computed_at":"2026-05-18T01:19:39.124882Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:39.124882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K3rpUxr2Nd6obKLj/tBdsvVKahw651Umlk7ZsYXoft+4brr7gLQbq6GOwdDI+LO055cuEvdAbaAkDw947TsdAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:39.125300Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.5645","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1966e4c51d6de1df23abec6a3e7d403466b94ed278b5b255e3a188076a1f9710","sha256:c1c5e3e232e151eafc8ee29ff12cabd1858a272800a300604cb5fe21051501b2"],"state_sha256":"b9c4ca5dbc0d07ab6447c86adbbd08234a5d54dc6b987d0a358c7aa682835ab1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AvFWHMOnn6/mehRNf1uQnEm1WSxhWlOloom5zGPP1AczQrzdREbQjlyDTBeYjbfQFehDLUE/5IrqDK0wlrk2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T04:36:28.872043Z","bundle_sha256":"bab27ca7139de739ea060f41b2ef0b890ca9dc044f5c68453ccebbcc5b7f9a91"}}