{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:SX7JHMITZE6H7TJEANG3MJHMRO","short_pith_number":"pith:SX7JHMIT","canonical_record":{"source":{"id":"0910.1710","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-09T11:09:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"fe0de5d270c34c5939b6ab864d6b4136429ac699bf538d281891f5bf94d25479","abstract_canon_sha256":"114f30a0d7dbd9e964a2e93bf0f25374bbcf0e32091695245928298a0d9b5184"},"schema_version":"1.0"},"canonical_sha256":"95fe93b113c93c7fcd24034db624ec8b8e2e12d8eb4e6d075d08e5f24a30f439","source":{"kind":"arxiv","id":"0910.1710","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.1710","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"arxiv_version","alias_value":"0910.1710v2","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1710","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"pith_short_12","alias_value":"SX7JHMITZE6H","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SX7JHMITZE6H7TJE","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SX7JHMIT","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:SX7JHMITZE6H7TJEANG3MJHMRO","target":"record","payload":{"canonical_record":{"source":{"id":"0910.1710","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-09T11:09:52Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"fe0de5d270c34c5939b6ab864d6b4136429ac699bf538d281891f5bf94d25479","abstract_canon_sha256":"114f30a0d7dbd9e964a2e93bf0f25374bbcf0e32091695245928298a0d9b5184"},"schema_version":"1.0"},"canonical_sha256":"95fe93b113c93c7fcd24034db624ec8b8e2e12d8eb4e6d075d08e5f24a30f439","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:04.754409Z","signature_b64":"+FBm2Hzcp99SlLIGWrUwGbKrtUPMqYTo4FczNqyVRpvz2TLVpCFPOEdRq0QbTck3fpsZsxuiFJJxyyxhQAiKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95fe93b113c93c7fcd24034db624ec8b8e2e12d8eb4e6d075d08e5f24a30f439","last_reissued_at":"2026-05-18T04:15:04.753786Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:04.753786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.1710","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":"2026-05-18T04:15:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HQaHEDk+dJqmSQ5ZKli20/PM//3AQGzCSzPAV7o33DEKhvMMgt3nT86/Rabz/8DYg33zb/xbtSJk3pPZ5eMHAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:51:55.695599Z"},"content_sha256":"a193a6e8a31e1d19024a84639fed60662b50a7c1581eb747267e520684d3a951","schema_version":"1.0","event_id":"sha256:a193a6e8a31e1d19024a84639fed60662b50a7c1581eb747267e520684d3a951"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:SX7JHMITZE6H7TJEANG3MJHMRO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Necessary and sufficient conditions for realizability of point processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Eugene R. Speer, Joel L. Lebowitz, Tobias Kuna","submitted_at":"2009-10-09T11:09:52Z","abstract_excerpt":"We give necessary and sufficient conditions for a pair of (generalized) functions $\\rho_1(\\mathbf{r}_1)$ and $\\rho_2(\\mathbf{r}_1,\\mathbf{r}_2)$, $\\mathbf{r}_i\\in X$, to be the density and pair correlations of some point process in a topological space $X$, for example, $\\mathbb {R}^d$, $\\mathbb {Z}^d$ or a subset of these. This is an infinite-dimensional version of the classical \"truncated moment\" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of $X$. Without this restriction we obtain, for compact $X$, strengthened co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1710","kind":"arxiv","version":2},"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":"2026-05-18T04:15:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n+ltmGH6maclqmetqOZChQpTwFy17vuDE6GPbcA2/CNEaPCitV90et0QTcUyg97WO0WMSJXcfHdorTZ9X/d9Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:51:55.695948Z"},"content_sha256":"428e7ca29f38baaa88a356b362642743a1c52341bdea4c0c258f43b6b4c14393","schema_version":"1.0","event_id":"sha256:428e7ca29f38baaa88a356b362642743a1c52341bdea4c0c258f43b6b4c14393"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SX7JHMITZE6H7TJEANG3MJHMRO/bundle.json","state_url":"https://pith.science/pith/SX7JHMITZE6H7TJEANG3MJHMRO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SX7JHMITZE6H7TJEANG3MJHMRO/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":"2026-06-22T10:51:55Z","links":{"resolver":"https://pith.science/pith/SX7JHMITZE6H7TJEANG3MJHMRO","bundle":"https://pith.science/pith/SX7JHMITZE6H7TJEANG3MJHMRO/bundle.json","state":"https://pith.science/pith/SX7JHMITZE6H7TJEANG3MJHMRO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SX7JHMITZE6H7TJEANG3MJHMRO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:SX7JHMITZE6H7TJEANG3MJHMRO","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":"114f30a0d7dbd9e964a2e93bf0f25374bbcf0e32091695245928298a0d9b5184","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-09T11:09:52Z","title_canon_sha256":"fe0de5d270c34c5939b6ab864d6b4136429ac699bf538d281891f5bf94d25479"},"schema_version":"1.0","source":{"id":"0910.1710","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.1710","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"arxiv_version","alias_value":"0910.1710v2","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1710","created_at":"2026-05-18T04:15:04Z"},{"alias_kind":"pith_short_12","alias_value":"SX7JHMITZE6H","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SX7JHMITZE6H7TJE","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SX7JHMIT","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:428e7ca29f38baaa88a356b362642743a1c52341bdea4c0c258f43b6b4c14393","target":"graph","created_at":"2026-05-18T04:15:04Z","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 necessary and sufficient conditions for a pair of (generalized) functions $\\rho_1(\\mathbf{r}_1)$ and $\\rho_2(\\mathbf{r}_1,\\mathbf{r}_2)$, $\\mathbf{r}_i\\in X$, to be the density and pair correlations of some point process in a topological space $X$, for example, $\\mathbb {R}^d$, $\\mathbb {Z}^d$ or a subset of these. This is an infinite-dimensional version of the classical \"truncated moment\" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of $X$. Without this restriction we obtain, for compact $X$, strengthened co","authors_text":"Eugene R. Speer, Joel L. Lebowitz, Tobias Kuna","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-09T11:09:52Z","title":"Necessary and sufficient conditions for realizability of point processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1710","kind":"arxiv","version":2},"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:a193a6e8a31e1d19024a84639fed60662b50a7c1581eb747267e520684d3a951","target":"record","created_at":"2026-05-18T04:15:04Z","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":"114f30a0d7dbd9e964a2e93bf0f25374bbcf0e32091695245928298a0d9b5184","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-10-09T11:09:52Z","title_canon_sha256":"fe0de5d270c34c5939b6ab864d6b4136429ac699bf538d281891f5bf94d25479"},"schema_version":"1.0","source":{"id":"0910.1710","kind":"arxiv","version":2}},"canonical_sha256":"95fe93b113c93c7fcd24034db624ec8b8e2e12d8eb4e6d075d08e5f24a30f439","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95fe93b113c93c7fcd24034db624ec8b8e2e12d8eb4e6d075d08e5f24a30f439","first_computed_at":"2026-05-18T04:15:04.753786Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:04.753786Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+FBm2Hzcp99SlLIGWrUwGbKrtUPMqYTo4FczNqyVRpvz2TLVpCFPOEdRq0QbTck3fpsZsxuiFJJxyyxhQAiKCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:04.754409Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.1710","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a193a6e8a31e1d19024a84639fed60662b50a7c1581eb747267e520684d3a951","sha256:428e7ca29f38baaa88a356b362642743a1c52341bdea4c0c258f43b6b4c14393"],"state_sha256":"cc1a60e0ac54656b8b48c8592acd2104852b1649a2dfa3890a16e3865c3f4737"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AO4I6uD3v4FGsEjp7VPXUTumXWz7DQc43yqrhoSVBTGIdNW1OYf6S80wzEgP8/mG5fljLKHgLjR8GTL0L1pHAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T10:51:55.697921Z","bundle_sha256":"23974f28b00b20d5372e287000deb1af3dfc1ca43221f384eea14672189c6e57"}}