{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1993:HBPQFIMLT6WRIQNKUBAQ4K7GZF","short_pith_number":"pith:HBPQFIML","canonical_record":{"source":{"id":"math/9307214","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1993-07-09T00:00:00Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"80efcb7c83724b657a1f341eaa8e73d8e8e8962feb8c91a1211f201e14d55007","abstract_canon_sha256":"fb95dfa549ae464cd5694f28368b8040b3839d485011ed5f56aaf56767b7563f"},"schema_version":"1.0"},"canonical_sha256":"385f02a18b9fad1441aaa0410e2be6c94592d147ce55887cbb9e415c72f70867","source":{"kind":"arxiv","id":"math/9307214","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9307214","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9307214v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9307214","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"HBPQFIMLT6WR","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"HBPQFIMLT6WRIQNK","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"HBPQFIML","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1993:HBPQFIMLT6WRIQNKUBAQ4K7GZF","target":"record","payload":{"canonical_record":{"source":{"id":"math/9307214","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CA","submitted_at":"1993-07-09T00:00:00Z","cross_cats_sorted":["astro-ph"],"title_canon_sha256":"80efcb7c83724b657a1f341eaa8e73d8e8e8962feb8c91a1211f201e14d55007","abstract_canon_sha256":"fb95dfa549ae464cd5694f28368b8040b3839d485011ed5f56aaf56767b7563f"},"schema_version":"1.0"},"canonical_sha256":"385f02a18b9fad1441aaa0410e2be6c94592d147ce55887cbb9e415c72f70867","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:52.070491Z","signature_b64":"bq3EZ30kmPeuYBdmAw1cQzbGkLqfhXtKxhFzU/SxB7PacCGEUVnKmKChyjwha3jXDKbONe1+xEiDxFUSTZTwBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"385f02a18b9fad1441aaa0410e2be6c94592d147ce55887cbb9e415c72f70867","last_reissued_at":"2026-05-18T01:05:52.069793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:52.069793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9307214","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vgWd5CmfeI+cXSsX/cANMNA8pIgE7UDxHBYtwblJcXbcSmnHTWn0flYgdBVa8xMWL4JbqmO0vtNw4CXd7mFXDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:45:13.601559Z"},"content_sha256":"8201ab753894a81124d2d007d8dac69e96b748a5b2bf85ce0eaadbb8c0b6b6a0","schema_version":"1.0","event_id":"sha256:8201ab753894a81124d2d007d8dac69e96b748a5b2bf85ce0eaadbb8c0b6b6a0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1993:HBPQFIMLT6WRIQNKUBAQ4K7GZF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computational aspects of the gravitational instability problem for a multicomponent cosmological medium","license":"","headline":"","cross_cats":["astro-ph"],"primary_cat":"math.CA","authors_text":"Arak Mathai Mathai, Hans J. Haubold","submitted_at":"1993-07-09T00:00:00Z","abstract_excerpt":"The paper presents results for deriving closed-form analytic solutions of the evolution of inhomogeneities in a homogeneous spatially flat multicomponent cosmological model. Mathematical methods to derive computable forms of the perturbations are outlined."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9307214","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZhckjPGN73J0eMEaOqtui3aHVUL3+YZIOw/o4DDjoQ6ncTbf5ZxPczDWxYUyQK+FEOdBvxI6UAxviJS0xmwjDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T19:45:13.602184Z"},"content_sha256":"5ef62d9ec1bcfac7cab443efaef5545b45bf536b3fc03efb75ffbc025c9a0562","schema_version":"1.0","event_id":"sha256:5ef62d9ec1bcfac7cab443efaef5545b45bf536b3fc03efb75ffbc025c9a0562"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/bundle.json","state_url":"https://pith.science/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/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":"2026-08-16T19:45:13Z","links":{"resolver":"https://pith.science/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF","bundle":"https://pith.science/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/bundle.json","state":"https://pith.science/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HBPQFIMLT6WRIQNKUBAQ4K7GZF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1993:HBPQFIMLT6WRIQNKUBAQ4K7GZF","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":"fb95dfa549ae464cd5694f28368b8040b3839d485011ed5f56aaf56767b7563f","cross_cats_sorted":["astro-ph"],"license":"","primary_cat":"math.CA","submitted_at":"1993-07-09T00:00:00Z","title_canon_sha256":"80efcb7c83724b657a1f341eaa8e73d8e8e8962feb8c91a1211f201e14d55007"},"schema_version":"1.0","source":{"id":"math/9307214","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9307214","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9307214v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9307214","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"HBPQFIMLT6WR","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"HBPQFIMLT6WRIQNK","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"HBPQFIML","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:5ef62d9ec1bcfac7cab443efaef5545b45bf536b3fc03efb75ffbc025c9a0562","target":"graph","created_at":"2026-05-18T01:05:52Z","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":"The paper presents results for deriving closed-form analytic solutions of the evolution of inhomogeneities in a homogeneous spatially flat multicomponent cosmological model. Mathematical methods to derive computable forms of the perturbations are outlined.","authors_text":"Arak Mathai Mathai, Hans J. Haubold","cross_cats":["astro-ph"],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"1993-07-09T00:00:00Z","title":"Computational aspects of the gravitational instability problem for a multicomponent cosmological medium"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9307214","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":"sha256:8201ab753894a81124d2d007d8dac69e96b748a5b2bf85ce0eaadbb8c0b6b6a0","target":"record","created_at":"2026-05-18T01:05:52Z","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":"fb95dfa549ae464cd5694f28368b8040b3839d485011ed5f56aaf56767b7563f","cross_cats_sorted":["astro-ph"],"license":"","primary_cat":"math.CA","submitted_at":"1993-07-09T00:00:00Z","title_canon_sha256":"80efcb7c83724b657a1f341eaa8e73d8e8e8962feb8c91a1211f201e14d55007"},"schema_version":"1.0","source":{"id":"math/9307214","kind":"arxiv","version":1}},"canonical_sha256":"385f02a18b9fad1441aaa0410e2be6c94592d147ce55887cbb9e415c72f70867","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"385f02a18b9fad1441aaa0410e2be6c94592d147ce55887cbb9e415c72f70867","first_computed_at":"2026-05-18T01:05:52.069793Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:52.069793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bq3EZ30kmPeuYBdmAw1cQzbGkLqfhXtKxhFzU/SxB7PacCGEUVnKmKChyjwha3jXDKbONe1+xEiDxFUSTZTwBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:52.070491Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9307214","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8201ab753894a81124d2d007d8dac69e96b748a5b2bf85ce0eaadbb8c0b6b6a0","sha256:5ef62d9ec1bcfac7cab443efaef5545b45bf536b3fc03efb75ffbc025c9a0562"],"state_sha256":"b18c850769494c1e2c2a46bc2f08362364e27b4e227db925990a57014814d94d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tOMLLBc3115i+Qbf/Z8JiDRVnVOWrPRQrB4VJRObMB98QCMkMnzthDyfSyiuXkjTB2Pb9GBjBNZTYC/FOJ9FCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T19:45:13.606854Z","bundle_sha256":"afcbb81a8f0ee98449cd895ced78c81547ca14cb20e81f9b00271d083c4acb34"}}