{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GXJW2Y4RDBIH4KGFIOYBWM7PEY","short_pith_number":"pith:GXJW2Y4R","canonical_record":{"source":{"id":"2607.01946","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T09:38:01Z","cross_cats_sorted":[],"title_canon_sha256":"4008afe3ce6ae242a3fe62fe2c81c90e80f858a3b7552e4d3393a03a6147b270","abstract_canon_sha256":"d2cc84d38e7e90508347177061c2fbcd6c67064ff53489476ff81d630da62fb9"},"schema_version":"1.0"},"canonical_sha256":"35d36d639118507e28c543b01b33ef262207fc3d97cddb1b968bd18e3e1cf294","source":{"kind":"arxiv","id":"2607.01946","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01946","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01946v1","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01946","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_12","alias_value":"GXJW2Y4RDBIH","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_16","alias_value":"GXJW2Y4RDBIH4KGF","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_8","alias_value":"GXJW2Y4R","created_at":"2026-07-03T01:17:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GXJW2Y4RDBIH4KGFIOYBWM7PEY","target":"record","payload":{"canonical_record":{"source":{"id":"2607.01946","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T09:38:01Z","cross_cats_sorted":[],"title_canon_sha256":"4008afe3ce6ae242a3fe62fe2c81c90e80f858a3b7552e4d3393a03a6147b270","abstract_canon_sha256":"d2cc84d38e7e90508347177061c2fbcd6c67064ff53489476ff81d630da62fb9"},"schema_version":"1.0"},"canonical_sha256":"35d36d639118507e28c543b01b33ef262207fc3d97cddb1b968bd18e3e1cf294","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:34.675334Z","signature_b64":"yk1WfD535HlhpUzuCBN1QZimMSBT0AHWTQDGkEaVJd+WS7+7oRxs7emp14vfWhuBA+vWjCX65fYjvtKzee9TDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"35d36d639118507e28c543b01b33ef262207fc3d97cddb1b968bd18e3e1cf294","last_reissued_at":"2026-07-03T01:17:34.674899Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:34.674899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.01946","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":"2026-07-03T01:17:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NIy7zCBwlUjEt3/eI4RGesj3kKaRo3TxqbmtN7cWye+k73QcdTY98cFpnAZZQMABXGRxu5ynFa3+NTCgrlKHCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:47:23.978640Z"},"content_sha256":"1dd6738d53f593d8da6e74700be0cf1e267f28f54f54a159d6bbf3e84dc2c8d8","schema_version":"1.0","event_id":"sha256:1dd6738d53f593d8da6e74700be0cf1e267f28f54f54a159d6bbf3e84dc2c8d8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GXJW2Y4RDBIH4KGFIOYBWM7PEY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Second-order fluctuations for a phase transition in random partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jaime Garza, Yizao Wang","submitted_at":"2026-07-02T09:38:01Z","abstract_excerpt":"In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameter $\\alpha\\in(0,1)$ and $\\theta>-\\alpha$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\\{1,\\ldots,n\\}$ and consider $j=j_n\\to\\infty$ as $n\\to\\infty$. They revealed a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\\sim rn^{\\alpha/(1+\\alpha)}$ for some $r>0$. A natural next question is to understand the corresponding second-order "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01946","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/2607.01946/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":"2026-07-03T01:17:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lQR75pBBkzdmdBkjMNMf4fe27PgfGbM136DUTMQGLZCYAQtPOWUjJK03Ee/hTyQhR4ucfEV84IH0oa+CDkQMAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:47:23.979018Z"},"content_sha256":"3d1add68c31b3bd955bf14236232f47dd0d303c07a5ed7d178f3b64d9fdf0500","schema_version":"1.0","event_id":"sha256:3d1add68c31b3bd955bf14236232f47dd0d303c07a5ed7d178f3b64d9fdf0500"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/bundle.json","state_url":"https://pith.science/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/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":"2026-07-03T16:47:23Z","links":{"resolver":"https://pith.science/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY","bundle":"https://pith.science/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/bundle.json","state":"https://pith.science/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GXJW2Y4RDBIH4KGFIOYBWM7PEY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GXJW2Y4RDBIH4KGFIOYBWM7PEY","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":"d2cc84d38e7e90508347177061c2fbcd6c67064ff53489476ff81d630da62fb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T09:38:01Z","title_canon_sha256":"4008afe3ce6ae242a3fe62fe2c81c90e80f858a3b7552e4d3393a03a6147b270"},"schema_version":"1.0","source":{"id":"2607.01946","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.01946","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"arxiv_version","alias_value":"2607.01946v1","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01946","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_12","alias_value":"GXJW2Y4RDBIH","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_16","alias_value":"GXJW2Y4RDBIH4KGF","created_at":"2026-07-03T01:17:34Z"},{"alias_kind":"pith_short_8","alias_value":"GXJW2Y4R","created_at":"2026-07-03T01:17:34Z"}],"graph_snapshots":[{"event_id":"sha256:3d1add68c31b3bd955bf14236232f47dd0d303c07a5ed7d178f3b64d9fdf0500","target":"graph","created_at":"2026-07-03T01:17:34Z","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/2607.01946/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameter $\\alpha\\in(0,1)$ and $\\theta>-\\alpha$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\\{1,\\ldots,n\\}$ and consider $j=j_n\\to\\infty$ as $n\\to\\infty$. They revealed a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\\sim rn^{\\alpha/(1+\\alpha)}$ for some $r>0$. A natural next question is to understand the corresponding second-order ","authors_text":"Jaime Garza, Yizao Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T09:38:01Z","title":"Second-order fluctuations for a phase transition in random partitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01946","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":"sha256:1dd6738d53f593d8da6e74700be0cf1e267f28f54f54a159d6bbf3e84dc2c8d8","target":"record","created_at":"2026-07-03T01:17:34Z","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":"d2cc84d38e7e90508347177061c2fbcd6c67064ff53489476ff81d630da62fb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T09:38:01Z","title_canon_sha256":"4008afe3ce6ae242a3fe62fe2c81c90e80f858a3b7552e4d3393a03a6147b270"},"schema_version":"1.0","source":{"id":"2607.01946","kind":"arxiv","version":1}},"canonical_sha256":"35d36d639118507e28c543b01b33ef262207fc3d97cddb1b968bd18e3e1cf294","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35d36d639118507e28c543b01b33ef262207fc3d97cddb1b968bd18e3e1cf294","first_computed_at":"2026-07-03T01:17:34.674899Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:34.674899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yk1WfD535HlhpUzuCBN1QZimMSBT0AHWTQDGkEaVJd+WS7+7oRxs7emp14vfWhuBA+vWjCX65fYjvtKzee9TDA==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:34.675334Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.01946","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1dd6738d53f593d8da6e74700be0cf1e267f28f54f54a159d6bbf3e84dc2c8d8","sha256:3d1add68c31b3bd955bf14236232f47dd0d303c07a5ed7d178f3b64d9fdf0500"],"state_sha256":"676a32f7168ec274829e13fa695e7b71c3d3cdbed8f1cfd749ddf01be5645904"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T9KsyNMuKNfr0UK9KTja4ePH4ZJD/SPkwUGatqa5C312xKDPQP+f0+hgmB+mbNJcg217tgtWMnxZ4hi0MntIBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T16:47:23.981145Z","bundle_sha256":"86b7c255aff6bb7002f136918f9293f133cb0aeeb4e5f86a9e4b792b157b68ee"}}