{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XA34OSLWVP23VP4UMDBLU4YJZS","short_pith_number":"pith:XA34OSLW","canonical_record":{"source":{"id":"1311.0417","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-02T23:24:25Z","cross_cats_sorted":[],"title_canon_sha256":"99bd4d214ef76250d3e29e1a5611db325642e0b98f23daca7c04e35d8a584dd1","abstract_canon_sha256":"f86cfc0a9298b7540c4046fbc59710ec26980ef81dced9dcb163880aa8e9d096"},"schema_version":"1.0"},"canonical_sha256":"b837c74976abf5babf9460c2ba7309cc8f6b4a24b24432760f63f7fe1aeba7d0","source":{"kind":"arxiv","id":"1311.0417","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0417","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0417v3","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0417","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"XA34OSLWVP23","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XA34OSLWVP23VP4U","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XA34OSLW","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XA34OSLWVP23VP4UMDBLU4YJZS","target":"record","payload":{"canonical_record":{"source":{"id":"1311.0417","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-02T23:24:25Z","cross_cats_sorted":[],"title_canon_sha256":"99bd4d214ef76250d3e29e1a5611db325642e0b98f23daca7c04e35d8a584dd1","abstract_canon_sha256":"f86cfc0a9298b7540c4046fbc59710ec26980ef81dced9dcb163880aa8e9d096"},"schema_version":"1.0"},"canonical_sha256":"b837c74976abf5babf9460c2ba7309cc8f6b4a24b24432760f63f7fe1aeba7d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:00:59.188759Z","signature_b64":"5X1UJB9qvCF0zAyLwgnL45hH+stFaaJ9LO6lNxJCx0Td5JtrxIe7Zj3J4jj44LfS+7Z18tB6vJx7VTH6TiGtAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b837c74976abf5babf9460c2ba7309cc8f6b4a24b24432760f63f7fe1aeba7d0","last_reissued_at":"2026-05-18T02:00:59.188179Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:00:59.188179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.0417","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p/VYaVgQyeAVd1X25W6AnOojhN6Vv0potm67xgNoJ/PNFWWT+KRVKP6kvomDjb/0EzR4pja2zIS0zc7kjMiPDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:39:57.069332Z"},"content_sha256":"f43fdfdd256888a3e9f1439777c5acd60f8f42069a3f29e8084944d6dfa067b6","schema_version":"1.0","event_id":"sha256:f43fdfdd256888a3e9f1439777c5acd60f8f42069a3f29e8084944d6dfa067b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XA34OSLWVP23VP4UMDBLU4YJZS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A particle system with cooperative branching and coalescence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anja Sturm, Jan M. Swart","submitted_at":"2013-11-02T23:24:25Z","abstract_excerpt":"In this paper, we introduce a one-dimensional model of particles performing independent random walks, where only pairs of particles can produce offspring (\"cooperative branching\"), and particles that land on an occupied site merge with the particle present on that site (\"coalescence\"). We show that the system undergoes a phase transition as the branching rate is increased. For small branching rates, the upper invariant law is trivial, and the process started with finitely many particles a.s. ends up with a single particle. Both statements are not true for high branching rates. An interesting f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0417","kind":"arxiv","version":3},"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":"2026-05-18T02:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TjC7IJ2K/DfUot2daoOJiu7WKnzmSo0N9jMCiy+5vVaEh/7oURIXabrGESDfFZHhoOXNZmRPfDEt1UaHIh9aBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:39:57.069974Z"},"content_sha256":"a8f1e48bff57ca6c4bf5ce4cb0e66b93e3f3f2726a9d9e708c37a12d2a0834cb","schema_version":"1.0","event_id":"sha256:a8f1e48bff57ca6c4bf5ce4cb0e66b93e3f3f2726a9d9e708c37a12d2a0834cb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XA34OSLWVP23VP4UMDBLU4YJZS/bundle.json","state_url":"https://pith.science/pith/XA34OSLWVP23VP4UMDBLU4YJZS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XA34OSLWVP23VP4UMDBLU4YJZS/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":"2026-05-28T08:39:57Z","links":{"resolver":"https://pith.science/pith/XA34OSLWVP23VP4UMDBLU4YJZS","bundle":"https://pith.science/pith/XA34OSLWVP23VP4UMDBLU4YJZS/bundle.json","state":"https://pith.science/pith/XA34OSLWVP23VP4UMDBLU4YJZS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XA34OSLWVP23VP4UMDBLU4YJZS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XA34OSLWVP23VP4UMDBLU4YJZS","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":"f86cfc0a9298b7540c4046fbc59710ec26980ef81dced9dcb163880aa8e9d096","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-02T23:24:25Z","title_canon_sha256":"99bd4d214ef76250d3e29e1a5611db325642e0b98f23daca7c04e35d8a584dd1"},"schema_version":"1.0","source":{"id":"1311.0417","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.0417","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"1311.0417v3","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.0417","created_at":"2026-05-18T02:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"XA34OSLWVP23","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XA34OSLWVP23VP4U","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XA34OSLW","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:a8f1e48bff57ca6c4bf5ce4cb0e66b93e3f3f2726a9d9e708c37a12d2a0834cb","target":"graph","created_at":"2026-05-18T02:00:59Z","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":"In this paper, we introduce a one-dimensional model of particles performing independent random walks, where only pairs of particles can produce offspring (\"cooperative branching\"), and particles that land on an occupied site merge with the particle present on that site (\"coalescence\"). We show that the system undergoes a phase transition as the branching rate is increased. For small branching rates, the upper invariant law is trivial, and the process started with finitely many particles a.s. ends up with a single particle. Both statements are not true for high branching rates. An interesting f","authors_text":"Anja Sturm, Jan M. Swart","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-02T23:24:25Z","title":"A particle system with cooperative branching and coalescence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0417","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:f43fdfdd256888a3e9f1439777c5acd60f8f42069a3f29e8084944d6dfa067b6","target":"record","created_at":"2026-05-18T02:00:59Z","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":"f86cfc0a9298b7540c4046fbc59710ec26980ef81dced9dcb163880aa8e9d096","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-02T23:24:25Z","title_canon_sha256":"99bd4d214ef76250d3e29e1a5611db325642e0b98f23daca7c04e35d8a584dd1"},"schema_version":"1.0","source":{"id":"1311.0417","kind":"arxiv","version":3}},"canonical_sha256":"b837c74976abf5babf9460c2ba7309cc8f6b4a24b24432760f63f7fe1aeba7d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b837c74976abf5babf9460c2ba7309cc8f6b4a24b24432760f63f7fe1aeba7d0","first_computed_at":"2026-05-18T02:00:59.188179Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:00:59.188179Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5X1UJB9qvCF0zAyLwgnL45hH+stFaaJ9LO6lNxJCx0Td5JtrxIe7Zj3J4jj44LfS+7Z18tB6vJx7VTH6TiGtAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:00:59.188759Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.0417","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f43fdfdd256888a3e9f1439777c5acd60f8f42069a3f29e8084944d6dfa067b6","sha256:a8f1e48bff57ca6c4bf5ce4cb0e66b93e3f3f2726a9d9e708c37a12d2a0834cb"],"state_sha256":"38885883c1bf9662f1e9ff5f04121c36a4fb01bd1b00a89cb93a91c1712a0409"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ox9XNGILILTPPhQK4C+nacpBAqKk0DyK0c5jHzR6x/RbCIiA0RZM+D5AA6j/6oYn2WsP7hQBf/OaxIvX2k4mBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:39:57.073110Z","bundle_sha256":"7cc616713c03d71ef7667bb6946a5c1850a7bdd4daea635101e31cb3783b86fc"}}