{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:YLB7CWSF5RFET26SPQYXEVESOQ","short_pith_number":"pith:YLB7CWSF","canonical_record":{"source":{"id":"1502.07657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-26T18:09:28Z","cross_cats_sorted":[],"title_canon_sha256":"5e303f87ae532884f7c9b2b932babfcd356e953fcd63b61d6da029752a4f7519","abstract_canon_sha256":"4dca4234802a3eb93193b40e57734c68db0c17cf2296ca3a8ac1787483f53341"},"schema_version":"1.0"},"canonical_sha256":"c2c3f15a45ec4a49ebd27c3172549274290df799adcfa09d20114b512f8e81eb","source":{"kind":"arxiv","id":"1502.07657","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.07657","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1502.07657v1","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07657","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"YLB7CWSF5RFE","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YLB7CWSF5RFET26S","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YLB7CWSF","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:YLB7CWSF5RFET26SPQYXEVESOQ","target":"record","payload":{"canonical_record":{"source":{"id":"1502.07657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-26T18:09:28Z","cross_cats_sorted":[],"title_canon_sha256":"5e303f87ae532884f7c9b2b932babfcd356e953fcd63b61d6da029752a4f7519","abstract_canon_sha256":"4dca4234802a3eb93193b40e57734c68db0c17cf2296ca3a8ac1787483f53341"},"schema_version":"1.0"},"canonical_sha256":"c2c3f15a45ec4a49ebd27c3172549274290df799adcfa09d20114b512f8e81eb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:03.196865Z","signature_b64":"nSSxvfKWJkODYks3K9yIMXPpg0rsxdbEoLiGOwQFiEw2tJO3+5O3T+nOA7CEtbQoH8q7M6C94+2koof7Mx+MCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2c3f15a45ec4a49ebd27c3172549274290df799adcfa09d20114b512f8e81eb","last_reissued_at":"2026-05-18T02:26:03.196458Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:03.196458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.07657","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":"2026-05-18T02:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sNLmqp15oP6zX80GBcuI55Wm6/rDVSqiXdCmdbBYBtjqoIed5HK75yzmSJJO1ZSUtYaM3RQZfblrllselto0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T09:24:06.442599Z"},"content_sha256":"0791af6f11e082c37a9cd83e31bfbea19de31b9418973acb3580544fac7ca21b","schema_version":"1.0","event_id":"sha256:0791af6f11e082c37a9cd83e31bfbea19de31b9418973acb3580544fac7ca21b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:YLB7CWSF5RFET26SPQYXEVESOQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stochastic Ordering of Infinite Geometric Galton-Watson Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Erik I. Broman","submitted_at":"2015-02-26T18:09:28Z","abstract_excerpt":"We consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\\leq p_1 <p_2 \\leq 1$, there exists a coupling between $T_{\\infty}(p_1)$ and $T_{\\infty}(p_2)$ such that ${\\mathbb P}(T_{\\infty}(p_1) \\subseteq T_{\\infty}(p_2))=1.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07657","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":"2026-05-18T02:26:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BR6HcpGabo6AXlkivoO7EGucr0OBS79LwF99zZIpcYjadnbTIZ58CsQG7BdqZl/jAyDs4eOFZh4FPQY+ccuLCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T09:24:06.442945Z"},"content_sha256":"d44afaa5fefa441b9c9e6803f0cefb312012511e0a7cca244eed94946b3876ed","schema_version":"1.0","event_id":"sha256:d44afaa5fefa441b9c9e6803f0cefb312012511e0a7cca244eed94946b3876ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YLB7CWSF5RFET26SPQYXEVESOQ/bundle.json","state_url":"https://pith.science/pith/YLB7CWSF5RFET26SPQYXEVESOQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YLB7CWSF5RFET26SPQYXEVESOQ/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":"2026-06-25T09:24:06Z","links":{"resolver":"https://pith.science/pith/YLB7CWSF5RFET26SPQYXEVESOQ","bundle":"https://pith.science/pith/YLB7CWSF5RFET26SPQYXEVESOQ/bundle.json","state":"https://pith.science/pith/YLB7CWSF5RFET26SPQYXEVESOQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YLB7CWSF5RFET26SPQYXEVESOQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:YLB7CWSF5RFET26SPQYXEVESOQ","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":"4dca4234802a3eb93193b40e57734c68db0c17cf2296ca3a8ac1787483f53341","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-26T18:09:28Z","title_canon_sha256":"5e303f87ae532884f7c9b2b932babfcd356e953fcd63b61d6da029752a4f7519"},"schema_version":"1.0","source":{"id":"1502.07657","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.07657","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"arxiv_version","alias_value":"1502.07657v1","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07657","created_at":"2026-05-18T02:26:03Z"},{"alias_kind":"pith_short_12","alias_value":"YLB7CWSF5RFE","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YLB7CWSF5RFET26S","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YLB7CWSF","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:d44afaa5fefa441b9c9e6803f0cefb312012511e0a7cca244eed94946b3876ed","target":"graph","created_at":"2026-05-18T02:26:03Z","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 consider Galton-Watson trees with Geom$(p)$ offspring distribution. We let $T_{\\infty}(p)$ denote such a tree conditioned on being infinite. We prove that for any $1/2\\leq p_1 <p_2 \\leq 1$, there exists a coupling between $T_{\\infty}(p_1)$ and $T_{\\infty}(p_2)$ such that ${\\mathbb P}(T_{\\infty}(p_1) \\subseteq T_{\\infty}(p_2))=1.$","authors_text":"Erik I. Broman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-26T18:09:28Z","title":"Stochastic Ordering of Infinite Geometric Galton-Watson Trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07657","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":"sha256:0791af6f11e082c37a9cd83e31bfbea19de31b9418973acb3580544fac7ca21b","target":"record","created_at":"2026-05-18T02:26:03Z","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":"4dca4234802a3eb93193b40e57734c68db0c17cf2296ca3a8ac1787483f53341","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-02-26T18:09:28Z","title_canon_sha256":"5e303f87ae532884f7c9b2b932babfcd356e953fcd63b61d6da029752a4f7519"},"schema_version":"1.0","source":{"id":"1502.07657","kind":"arxiv","version":1}},"canonical_sha256":"c2c3f15a45ec4a49ebd27c3172549274290df799adcfa09d20114b512f8e81eb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2c3f15a45ec4a49ebd27c3172549274290df799adcfa09d20114b512f8e81eb","first_computed_at":"2026-05-18T02:26:03.196458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:03.196458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nSSxvfKWJkODYks3K9yIMXPpg0rsxdbEoLiGOwQFiEw2tJO3+5O3T+nOA7CEtbQoH8q7M6C94+2koof7Mx+MCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:03.196865Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.07657","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0791af6f11e082c37a9cd83e31bfbea19de31b9418973acb3580544fac7ca21b","sha256:d44afaa5fefa441b9c9e6803f0cefb312012511e0a7cca244eed94946b3876ed"],"state_sha256":"2eff6fdb0fb5a2227dd5977864dec3a5387a506d61db67ea39af140f1c002e86"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EAs4rR/zADX0idGzbDQLMNQ8r5P2ZeumTKWXHQFJyfTpVJyPDPlk1GxJSgLk9u6OPmK/9oPRGkmMtwrAmC5hCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T09:24:06.444866Z","bundle_sha256":"a930a097d5adbb8e77b02ebeff78e5a9cf2f4622afb1b4ce42db19780c0f9bbb"}}