{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:25WQ3K4B7U5VFLOAS5OZ4MB6LP","short_pith_number":"pith:25WQ3K4B","canonical_record":{"source":{"id":"2110.03648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2021-10-07T17:38:43Z","cross_cats_sorted":[],"title_canon_sha256":"18b36d1a57ffc0fc208ad6e4c82bb551fecc3efa0e10a5b9bff69f75c4654617","abstract_canon_sha256":"b6486bc86201a985adbaf35bed3a6961c37b0fa66eacb53cd79a2415b03e4b91"},"schema_version":"1.0"},"canonical_sha256":"d76d0dab81fd3b52adc0975d9e303e5bea59b073f63e356c494ebf24a3d823e3","source":{"kind":"arxiv","id":"2110.03648","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.03648","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"arxiv_version","alias_value":"2110.03648v1","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.03648","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_12","alias_value":"25WQ3K4B7U5V","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_16","alias_value":"25WQ3K4B7U5VFLOA","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_8","alias_value":"25WQ3K4B","created_at":"2026-07-05T03:20:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:25WQ3K4B7U5VFLOAS5OZ4MB6LP","target":"record","payload":{"canonical_record":{"source":{"id":"2110.03648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2021-10-07T17:38:43Z","cross_cats_sorted":[],"title_canon_sha256":"18b36d1a57ffc0fc208ad6e4c82bb551fecc3efa0e10a5b9bff69f75c4654617","abstract_canon_sha256":"b6486bc86201a985adbaf35bed3a6961c37b0fa66eacb53cd79a2415b03e4b91"},"schema_version":"1.0"},"canonical_sha256":"d76d0dab81fd3b52adc0975d9e303e5bea59b073f63e356c494ebf24a3d823e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:20:44.505842Z","signature_b64":"/TZVKlLiIATeCX068KcdIuUD9fIwzHq1gHtbbO01nZHMuQhoLsf6xs6YZLPmkD52G56y8jTqk/lWh0JxzqHvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d76d0dab81fd3b52adc0975d9e303e5bea59b073f63e356c494ebf24a3d823e3","last_reissued_at":"2026-07-05T03:20:44.505250Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:20:44.505250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2110.03648","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":"2026-07-05T03:20:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c7YEHhCWDELZBFlqxNN+WSzO6VG/dsdSYeoyfYpX15zW5kIY9VMjLM+bT/UtNoxtWsAqXp0uMFiUdWETUPG8Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T01:46:02.509326Z"},"content_sha256":"ea02d0f5b323f153079c4d76bfeee179469ddee5ad29dc85f55e69c6c3653b8c","schema_version":"1.0","event_id":"sha256:ea02d0f5b323f153079c4d76bfeee179469ddee5ad29dc85f55e69c6c3653b8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:25WQ3K4B7U5VFLOAS5OZ4MB6LP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Heike Mildenberger, Maxwell Levine","submitted_at":"2021-10-07T17:38:43Z","abstract_excerpt":"Dobrinen, Hathaway and Prikry studied a forcing $\\mathbb{P}_\\kappa$ consisting of perfect trees of height $\\lambda$ and width $\\kappa$ where $\\kappa$ is a singular $\\omega$-strong limit of cofinality $\\lambda$. They showed that if $\\kappa$ is singular of countable cofinality, then $\\mathbb{P}_\\kappa$ is minimal for $\\omega$-sequences assuming that $\\kappa$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption.\n  Prikry proved that $\\mathbb{P}_\\kappa$ is $(\\omega,\\nu)$-distributive for all $\\nu<\\kappa$ given a singular $\\omega$-strong li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.03648","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/2110.03648/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":"2026-07-05T03:20:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zt1Gu8p+UzB1lHo+75GU8Hoiup+WLmZDi2FKYHeDYlTiYTiUzYoIT0WOovIgeM4dpg8RgPKRCofu63zqHjcqDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T01:46:02.509703Z"},"content_sha256":"b48adb2bde32e625fe9e7b73d735a02890461f75180e01e5065cf3a537e2d4f8","schema_version":"1.0","event_id":"sha256:b48adb2bde32e625fe9e7b73d735a02890461f75180e01e5065cf3a537e2d4f8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/bundle.json","state_url":"https://pith.science/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/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":"2026-08-04T01:46:02Z","links":{"resolver":"https://pith.science/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP","bundle":"https://pith.science/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/bundle.json","state":"https://pith.science/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/25WQ3K4B7U5VFLOAS5OZ4MB6LP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:25WQ3K4B7U5VFLOAS5OZ4MB6LP","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":"b6486bc86201a985adbaf35bed3a6961c37b0fa66eacb53cd79a2415b03e4b91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2021-10-07T17:38:43Z","title_canon_sha256":"18b36d1a57ffc0fc208ad6e4c82bb551fecc3efa0e10a5b9bff69f75c4654617"},"schema_version":"1.0","source":{"id":"2110.03648","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.03648","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"arxiv_version","alias_value":"2110.03648v1","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.03648","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_12","alias_value":"25WQ3K4B7U5V","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_16","alias_value":"25WQ3K4B7U5VFLOA","created_at":"2026-07-05T03:20:44Z"},{"alias_kind":"pith_short_8","alias_value":"25WQ3K4B","created_at":"2026-07-05T03:20:44Z"}],"graph_snapshots":[{"event_id":"sha256:b48adb2bde32e625fe9e7b73d735a02890461f75180e01e5065cf3a537e2d4f8","target":"graph","created_at":"2026-07-05T03:20:44Z","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/2110.03648/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Dobrinen, Hathaway and Prikry studied a forcing $\\mathbb{P}_\\kappa$ consisting of perfect trees of height $\\lambda$ and width $\\kappa$ where $\\kappa$ is a singular $\\omega$-strong limit of cofinality $\\lambda$. They showed that if $\\kappa$ is singular of countable cofinality, then $\\mathbb{P}_\\kappa$ is minimal for $\\omega$-sequences assuming that $\\kappa$ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption.\n  Prikry proved that $\\mathbb{P}_\\kappa$ is $(\\omega,\\nu)$-distributive for all $\\nu<\\kappa$ given a singular $\\omega$-strong li","authors_text":"Heike Mildenberger, Maxwell Levine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2021-10-07T17:38:43Z","title":"Distributivity and Minimality in Perfect Tree Forcings for Singular Cardinals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.03648","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:ea02d0f5b323f153079c4d76bfeee179469ddee5ad29dc85f55e69c6c3653b8c","target":"record","created_at":"2026-07-05T03:20:44Z","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":"b6486bc86201a985adbaf35bed3a6961c37b0fa66eacb53cd79a2415b03e4b91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2021-10-07T17:38:43Z","title_canon_sha256":"18b36d1a57ffc0fc208ad6e4c82bb551fecc3efa0e10a5b9bff69f75c4654617"},"schema_version":"1.0","source":{"id":"2110.03648","kind":"arxiv","version":1}},"canonical_sha256":"d76d0dab81fd3b52adc0975d9e303e5bea59b073f63e356c494ebf24a3d823e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d76d0dab81fd3b52adc0975d9e303e5bea59b073f63e356c494ebf24a3d823e3","first_computed_at":"2026-07-05T03:20:44.505250Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:20:44.505250Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/TZVKlLiIATeCX068KcdIuUD9fIwzHq1gHtbbO01nZHMuQhoLsf6xs6YZLPmkD52G56y8jTqk/lWh0JxzqHvCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:20:44.505842Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.03648","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea02d0f5b323f153079c4d76bfeee179469ddee5ad29dc85f55e69c6c3653b8c","sha256:b48adb2bde32e625fe9e7b73d735a02890461f75180e01e5065cf3a537e2d4f8"],"state_sha256":"5ad3f427f79dd3a47d2568feeaeca1b4ba7993c64f65e7d3de47214ffdc1354f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mhpcEjgPQ1HBFLQkjDnUul26SkOjxzE06hdW36X362bfic2xIVEwt8Xh5GNl5s7KR2sJuhlnXBcoB9t2K8bTAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T01:46:02.512355Z","bundle_sha256":"4cd3259c6488ea5f6cfae5153888360da7736a0ae9c6098a2a0093f1e57ccdc2"}}