{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:CWB2YPF7LW5UZBHOQAJA3NRPNE","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":"944ba432807ae4011c6d90634ad160e75c6b85a92ff683b2aa1b2781b4c98b69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-08-24T11:17:32Z","title_canon_sha256":"f7079e24ce8009021156f05b9bee419753774427f39ff69cbfa60c7534169d35"},"schema_version":"1.0","source":{"id":"0908.3409","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.3409","created_at":"2026-05-18T04:06:46Z"},{"alias_kind":"arxiv_version","alias_value":"0908.3409v3","created_at":"2026-05-18T04:06:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.3409","created_at":"2026-05-18T04:06:46Z"},{"alias_kind":"pith_short_12","alias_value":"CWB2YPF7LW5U","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"CWB2YPF7LW5UZBHO","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"CWB2YPF7","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:25fc5c8df936244ddfecd3ae7e994faf65b62843d9dc5234f1a89d45bf726ecf","target":"graph","created_at":"2026-05-18T04:06:46Z","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":"Given a homogeneous Poisson process on ${\\mathbb{R}}^d$ with intensity $\\lambda$, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to $\\lambda$. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60--69], who proved that in $d=1$, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process o","authors_text":"Alexander E. Holroyd, Russell Lyons, Terry Soo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-08-24T11:17:32Z","title":"Poisson splitting by factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.3409","kind":"arxiv","version":3},"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":"sha256:27b7c80182a267208c1f853b75b57afcc6d2bbb82cf7ee4e1cd8a6da0894b5e1","target":"record","created_at":"2026-05-18T04:06:46Z","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":"944ba432807ae4011c6d90634ad160e75c6b85a92ff683b2aa1b2781b4c98b69","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-08-24T11:17:32Z","title_canon_sha256":"f7079e24ce8009021156f05b9bee419753774427f39ff69cbfa60c7534169d35"},"schema_version":"1.0","source":{"id":"0908.3409","kind":"arxiv","version":3}},"canonical_sha256":"1583ac3cbf5dbb4c84ee80120db62f690a2814ba6c752903fe6785f055fb699c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1583ac3cbf5dbb4c84ee80120db62f690a2814ba6c752903fe6785f055fb699c","first_computed_at":"2026-05-18T04:06:46.370539Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:46.370539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PPwId5+/R3ux+RwC6E/WDw8VGNGnBazV0Bw+gaTwRxwGRQWx/nKaBvgX3nWzXtlYAzL+ZqvpF+uktd7fI5lrDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:46.371009Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.3409","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27b7c80182a267208c1f853b75b57afcc6d2bbb82cf7ee4e1cd8a6da0894b5e1","sha256:25fc5c8df936244ddfecd3ae7e994faf65b62843d9dc5234f1a89d45bf726ecf"],"state_sha256":"f259a6df1342d4bbe3ae0436d4145dd7cea92b627d0a14e7523824d4e7db4f41"}