{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:OF32YA7VB5KUHH4SU7RM6EVAM7","short_pith_number":"pith:OF32YA7V","canonical_record":{"source":{"id":"2211.08662","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2022-11-16T04:34:58Z","cross_cats_sorted":[],"title_canon_sha256":"4ac2605f1b0f4ebea775a2df03cb0b5b3ae93e39466aeba1ef91ded961743cf2","abstract_canon_sha256":"fb2f74fa9b4c123e927f3a1ba55d13bb9fe00593f91c04bdb0b7201233a33870"},"schema_version":"1.0"},"canonical_sha256":"7177ac03f50f55439f92a7e2cf12a067e3403123880d50b6c20389a69f3400ae","source":{"kind":"arxiv","id":"2211.08662","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.08662","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"arxiv_version","alias_value":"2211.08662v1","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.08662","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_12","alias_value":"OF32YA7VB5KU","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_16","alias_value":"OF32YA7VB5KUHH4S","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_8","alias_value":"OF32YA7V","created_at":"2026-07-05T05:16:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:OF32YA7VB5KUHH4SU7RM6EVAM7","target":"record","payload":{"canonical_record":{"source":{"id":"2211.08662","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2022-11-16T04:34:58Z","cross_cats_sorted":[],"title_canon_sha256":"4ac2605f1b0f4ebea775a2df03cb0b5b3ae93e39466aeba1ef91ded961743cf2","abstract_canon_sha256":"fb2f74fa9b4c123e927f3a1ba55d13bb9fe00593f91c04bdb0b7201233a33870"},"schema_version":"1.0"},"canonical_sha256":"7177ac03f50f55439f92a7e2cf12a067e3403123880d50b6c20389a69f3400ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:16:42.529945Z","signature_b64":"I94bi2e+3qMyU2tVe1e/HSLSK4c1XKlw6YVbzrNbCLrIz+Bq4r3MtJt7rnBXt4TFT9VB1erJv+2DdLKD3x80BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7177ac03f50f55439f92a7e2cf12a067e3403123880d50b6c20389a69f3400ae","last_reissued_at":"2026-07-05T05:16:42.529499Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:16:42.529499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2211.08662","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":"2026-07-05T05:16:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yC00uScMr9pY9+KavW1q/OQ+9olV1OoQcXuZlkG+duNg7d26F0dyibDAZatXkIzhdZn8WbAKmaywoBevf7nUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T01:39:54.594179Z"},"content_sha256":"7f6a92870b210b7b42418edb281ac1d0e0114a56b29a69e3c342036880295b84","schema_version":"1.0","event_id":"sha256:7f6a92870b210b7b42418edb281ac1d0e0114a56b29a69e3c342036880295b84"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:OF32YA7VB5KUHH4SU7RM6EVAM7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Many-to-few for non-local branching Markov process","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Andreas E. Kyprianou, Ellen Powell, Emma Horton, Simon C. Harris","submitted_at":"2022-11-16T04:34:58Z","abstract_excerpt":"We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [14] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.08662","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/2211.08662/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":"2026-07-05T05:16:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SJ685AS3JuLIIQqJBoCVS1+TyKAK5Z2g0QUHRXWN3V/S1yHmcb1InWZGfMqt30qAb0P8Xl4+zh/kUTfvtJKiCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T01:39:54.594575Z"},"content_sha256":"6a45b07a6f05fb33ba0fbf595948052e7f332babef2e0505c3ade160c357be45","schema_version":"1.0","event_id":"sha256:6a45b07a6f05fb33ba0fbf595948052e7f332babef2e0505c3ade160c357be45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/bundle.json","state_url":"https://pith.science/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/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":"2026-07-10T01:39:54Z","links":{"resolver":"https://pith.science/pith/OF32YA7VB5KUHH4SU7RM6EVAM7","bundle":"https://pith.science/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/bundle.json","state":"https://pith.science/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OF32YA7VB5KUHH4SU7RM6EVAM7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:OF32YA7VB5KUHH4SU7RM6EVAM7","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":"fb2f74fa9b4c123e927f3a1ba55d13bb9fe00593f91c04bdb0b7201233a33870","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2022-11-16T04:34:58Z","title_canon_sha256":"4ac2605f1b0f4ebea775a2df03cb0b5b3ae93e39466aeba1ef91ded961743cf2"},"schema_version":"1.0","source":{"id":"2211.08662","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.08662","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"arxiv_version","alias_value":"2211.08662v1","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.08662","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_12","alias_value":"OF32YA7VB5KU","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_16","alias_value":"OF32YA7VB5KUHH4S","created_at":"2026-07-05T05:16:42Z"},{"alias_kind":"pith_short_8","alias_value":"OF32YA7V","created_at":"2026-07-05T05:16:42Z"}],"graph_snapshots":[{"event_id":"sha256:6a45b07a6f05fb33ba0fbf595948052e7f332babef2e0505c3ade160c357be45","target":"graph","created_at":"2026-07-05T05:16:42Z","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/2211.08662/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [14] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population a","authors_text":"Andreas E. Kyprianou, Ellen Powell, Emma Horton, Simon C. Harris","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2022-11-16T04:34:58Z","title":"Many-to-few for non-local branching Markov process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.08662","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":"sha256:7f6a92870b210b7b42418edb281ac1d0e0114a56b29a69e3c342036880295b84","target":"record","created_at":"2026-07-05T05:16:42Z","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":"fb2f74fa9b4c123e927f3a1ba55d13bb9fe00593f91c04bdb0b7201233a33870","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2022-11-16T04:34:58Z","title_canon_sha256":"4ac2605f1b0f4ebea775a2df03cb0b5b3ae93e39466aeba1ef91ded961743cf2"},"schema_version":"1.0","source":{"id":"2211.08662","kind":"arxiv","version":1}},"canonical_sha256":"7177ac03f50f55439f92a7e2cf12a067e3403123880d50b6c20389a69f3400ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7177ac03f50f55439f92a7e2cf12a067e3403123880d50b6c20389a69f3400ae","first_computed_at":"2026-07-05T05:16:42.529499Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:16:42.529499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I94bi2e+3qMyU2tVe1e/HSLSK4c1XKlw6YVbzrNbCLrIz+Bq4r3MtJt7rnBXt4TFT9VB1erJv+2DdLKD3x80BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:16:42.529945Z","signed_message":"canonical_sha256_bytes"},"source_id":"2211.08662","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f6a92870b210b7b42418edb281ac1d0e0114a56b29a69e3c342036880295b84","sha256:6a45b07a6f05fb33ba0fbf595948052e7f332babef2e0505c3ade160c357be45"],"state_sha256":"50076db6e030eab5cf115ea67564e81a7b1a571e6fdab952356298ee5de88dde"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hNObt4oXI4gyvnKqdzdETOUdT1SSwL1eNb7sAL/LsrlbCBJvMzP4Wbgm9nhXSfd32hRmoftJBCiGqspq41wTCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-10T01:39:54.596456Z","bundle_sha256":"a6b6ca5564422c936feed88158ea87edce2554f0412e01532ca575e54ac03c52"}}