{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:OCDDK7JKLSFUREB4AYAZE6IWKC","short_pith_number":"pith:OCDDK7JK","canonical_record":{"source":{"id":"2412.09655","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-12T02:02:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7752decb51544bcc1bf3efe987c67e05cd807752e620bcfb4005e364c918c2b8","abstract_canon_sha256":"4a7566d1e71ebd3fa7fd04a899421516d2a4abbe92207127f096d79efa74a203"},"schema_version":"1.0"},"canonical_sha256":"7086357d2a5c8b48903c06019279165081436a714da113330634169d7131d53c","source":{"kind":"arxiv","id":"2412.09655","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.09655","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"arxiv_version","alias_value":"2412.09655v1","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.09655","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_12","alias_value":"OCDDK7JKLSFU","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_16","alias_value":"OCDDK7JKLSFUREB4","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_8","alias_value":"OCDDK7JK","created_at":"2026-07-05T09:48:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:OCDDK7JKLSFUREB4AYAZE6IWKC","target":"record","payload":{"canonical_record":{"source":{"id":"2412.09655","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-12T02:02:03Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7752decb51544bcc1bf3efe987c67e05cd807752e620bcfb4005e364c918c2b8","abstract_canon_sha256":"4a7566d1e71ebd3fa7fd04a899421516d2a4abbe92207127f096d79efa74a203"},"schema_version":"1.0"},"canonical_sha256":"7086357d2a5c8b48903c06019279165081436a714da113330634169d7131d53c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:48:39.663469Z","signature_b64":"GLwPTuT7k4kEyBb9gku9hxGppvit4LF4sSGzIAp6DOgCwyzJ+YfIoo3SHUqivayN3LUtBY505ABa62rRjz7pDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7086357d2a5c8b48903c06019279165081436a714da113330634169d7131d53c","last_reissued_at":"2026-07-05T09:48:39.662980Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:48:39.662980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.09655","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":"2026-07-05T09:48:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sCLwYhD9mPdM5eJ94XCANuUgZ9SzT4MKVG06yfJbxb6jjtSfDzXIZjvtMrfYZRDO7FxI4tY5EREIF+doZmmiBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:22:16.535637Z"},"content_sha256":"17f1ad1466e982a4eeca850c897ca076f3e05e4085f7503d1cb1ee81cc62db56","schema_version":"1.0","event_id":"sha256:17f1ad1466e982a4eeca850c897ca076f3e05e4085f7503d1cb1ee81cc62db56"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:OCDDK7JKLSFUREB4AYAZE6IWKC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Critical Beta-splitting Random Tree III: The exchangeable partition representation and the fringe tree","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"David J. Aldous, Svante Janson","submitted_at":"2024-12-12T02:02:03Z","abstract_excerpt":"In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\\propto 1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of the model is an active research topic. It turns out that many results have several different proofs, and detailed studies of analytic proofs are given elsdewhere (via analysis of recursions and via Mellin transforms). This article describes two core probabilistic methods for studying $n \\to \\inf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.09655","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/2412.09655/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":"2026-07-05T09:48:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cgTTuFUT7xwEJV9BkMt9kPJJedOMQWxR06beMEdU59M/hOFXZGsdfUycfleAKkvDRe95uSKE1OOYTRz3nhytBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:22:16.536134Z"},"content_sha256":"28bd68c4f57ecbbf4a47ef4cfcd44b4494bbb5dceaccb6b3fa6d86a33f6215f2","schema_version":"1.0","event_id":"sha256:28bd68c4f57ecbbf4a47ef4cfcd44b4494bbb5dceaccb6b3fa6d86a33f6215f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/bundle.json","state_url":"https://pith.science/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/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":"2026-08-08T10:22:16Z","links":{"resolver":"https://pith.science/pith/OCDDK7JKLSFUREB4AYAZE6IWKC","bundle":"https://pith.science/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/bundle.json","state":"https://pith.science/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OCDDK7JKLSFUREB4AYAZE6IWKC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:OCDDK7JKLSFUREB4AYAZE6IWKC","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":"4a7566d1e71ebd3fa7fd04a899421516d2a4abbe92207127f096d79efa74a203","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-12T02:02:03Z","title_canon_sha256":"7752decb51544bcc1bf3efe987c67e05cd807752e620bcfb4005e364c918c2b8"},"schema_version":"1.0","source":{"id":"2412.09655","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.09655","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"arxiv_version","alias_value":"2412.09655v1","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.09655","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_12","alias_value":"OCDDK7JKLSFU","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_16","alias_value":"OCDDK7JKLSFUREB4","created_at":"2026-07-05T09:48:39Z"},{"alias_kind":"pith_short_8","alias_value":"OCDDK7JK","created_at":"2026-07-05T09:48:39Z"}],"graph_snapshots":[{"event_id":"sha256:28bd68c4f57ecbbf4a47ef4cfcd44b4494bbb5dceaccb6b3fa6d86a33f6215f2","target":"graph","created_at":"2026-07-05T09:48:39Z","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/2412.09655/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\\propto 1/(i(m-i))$. Study of structure theory and explicit quantitative aspects of the model is an active research topic. It turns out that many results have several different proofs, and detailed studies of analytic proofs are given elsdewhere (via analysis of recursions and via Mellin transforms). This article describes two core probabilistic methods for studying $n \\to \\inf","authors_text":"David J. Aldous, Svante Janson","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-12T02:02:03Z","title":"The Critical Beta-splitting Random Tree III: The exchangeable partition representation and the fringe tree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.09655","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":"sha256:17f1ad1466e982a4eeca850c897ca076f3e05e4085f7503d1cb1ee81cc62db56","target":"record","created_at":"2026-07-05T09:48:39Z","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":"4a7566d1e71ebd3fa7fd04a899421516d2a4abbe92207127f096d79efa74a203","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2024-12-12T02:02:03Z","title_canon_sha256":"7752decb51544bcc1bf3efe987c67e05cd807752e620bcfb4005e364c918c2b8"},"schema_version":"1.0","source":{"id":"2412.09655","kind":"arxiv","version":1}},"canonical_sha256":"7086357d2a5c8b48903c06019279165081436a714da113330634169d7131d53c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7086357d2a5c8b48903c06019279165081436a714da113330634169d7131d53c","first_computed_at":"2026-07-05T09:48:39.662980Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:48:39.662980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GLwPTuT7k4kEyBb9gku9hxGppvit4LF4sSGzIAp6DOgCwyzJ+YfIoo3SHUqivayN3LUtBY505ABa62rRjz7pDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:48:39.663469Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.09655","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17f1ad1466e982a4eeca850c897ca076f3e05e4085f7503d1cb1ee81cc62db56","sha256:28bd68c4f57ecbbf4a47ef4cfcd44b4494bbb5dceaccb6b3fa6d86a33f6215f2"],"state_sha256":"8c7655cd131a67f39f622411997d47766e3f84f7fea06cee80b9a32c8f013b89"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7lzZrhQNI+3oK++69enNgzh0z2/XBoYL4Pnwp8NcnFj39O8esD7AazOLcumFpXaVsYTBtiLjsWpf27a14ivtAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T10:22:16.540564Z","bundle_sha256":"f6f8a35cd11118b356547f5691f8feb5ca8e31585256fe2c24972d6b2a071e0d"}}