{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7BBZDEBOWFCUVTHTG6E7TX2AO2","short_pith_number":"pith:7BBZDEBO","canonical_record":{"source":{"id":"2607.03444","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-07-03T15:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"a1e2d9af67454809ea3f43b3d94953b39c80e3bd10d7dafc30a8720214493a3e","abstract_canon_sha256":"0d1dbe378e9445720d5fdf8be823bbad43876e4c59899c40b31563db8e797575"},"schema_version":"1.0"},"canonical_sha256":"f84391902eb1454accf33789f9df4076a296d07d9ac3b913156ed322e9ec7334","source":{"kind":"arxiv","id":"2607.03444","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.03444","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.03444v1","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.03444","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_12","alias_value":"7BBZDEBOWFCU","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_16","alias_value":"7BBZDEBOWFCUVTHT","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_8","alias_value":"7BBZDEBO","created_at":"2026-07-07T02:17:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7BBZDEBOWFCUVTHTG6E7TX2AO2","target":"record","payload":{"canonical_record":{"source":{"id":"2607.03444","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-07-03T15:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"a1e2d9af67454809ea3f43b3d94953b39c80e3bd10d7dafc30a8720214493a3e","abstract_canon_sha256":"0d1dbe378e9445720d5fdf8be823bbad43876e4c59899c40b31563db8e797575"},"schema_version":"1.0"},"canonical_sha256":"f84391902eb1454accf33789f9df4076a296d07d9ac3b913156ed322e9ec7334","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:47.090191Z","signature_b64":"IjRTV57IR4fhnJCs9gCxRyT3bBduHnZa9pKwo/G1kXfeCcuMybVcAy/XrKzEhqHstvEhU5oDF7o6SmWwMGUHDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f84391902eb1454accf33789f9df4076a296d07d9ac3b913156ed322e9ec7334","last_reissued_at":"2026-07-07T02:17:47.089301Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:47.089301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.03444","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":"2026-07-07T02:17:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DSiIi9WZjTANGN0ESDgaPj/glns0IAOW4cmMpWhPeYXenF4QJZwSnvM/GJ3xMChZLRK94H8du3pjJEstHgbbCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T05:27:59.213152Z"},"content_sha256":"faffcd9df90f37aa4465d19055f6abbdc32a93c663e59e60e053203ab536497a","schema_version":"1.0","event_id":"sha256:faffcd9df90f37aa4465d19055f6abbdc32a93c663e59e60e053203ab536497a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7BBZDEBOWFCUVTHTG6E7TX2AO2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Determinacy for Cut and Choose Games of Uncountable Length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Maxwell Levine","submitted_at":"2026-07-03T15:56:26Z","abstract_excerpt":"We obtain results on cut and choose games for complete Boolean algebras. Zapletal proved that there is a Boolean algebra $\\mathbb{B}$ such that $\\mathcal{G}_\\omega^\\textsf{candc}(\\mathbb{B})$, the version of the game which ends on the $\\omega$'th round, is undetermined. We prove that, assuming the consistency of a proper class of supercompact cardinals, the limit version $\\mathcal{G}^{\\textsf{candc}}_{< \\lambda}(\\mathbb{B})$, in which there are $\\lambda$-many rounds but no concluding round, is consistently determined for all complete Boolean algebras $\\mathbb{B}$ and all successor cardinals $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03444","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/2607.03444/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":"2026-07-07T02:17:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ieHKYJidei8VPTcO9lRci1u/ogngHfY7FkokIf+x20K60x+xQtNAwqGqkgBaz1sH8gCUC/tQb7Gg130HoBNFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T05:27:59.213572Z"},"content_sha256":"25c08b1b562bf70ea970ee8eb69e72156ec4f40e65804a3fb9a63303aa753b5c","schema_version":"1.0","event_id":"sha256:25c08b1b562bf70ea970ee8eb69e72156ec4f40e65804a3fb9a63303aa753b5c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/bundle.json","state_url":"https://pith.science/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/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":"2026-08-04T05:27:59Z","links":{"resolver":"https://pith.science/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2","bundle":"https://pith.science/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/bundle.json","state":"https://pith.science/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7BBZDEBOWFCUVTHTG6E7TX2AO2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7BBZDEBOWFCUVTHTG6E7TX2AO2","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":"0d1dbe378e9445720d5fdf8be823bbad43876e4c59899c40b31563db8e797575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-07-03T15:56:26Z","title_canon_sha256":"a1e2d9af67454809ea3f43b3d94953b39c80e3bd10d7dafc30a8720214493a3e"},"schema_version":"1.0","source":{"id":"2607.03444","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.03444","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.03444v1","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.03444","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_12","alias_value":"7BBZDEBOWFCU","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_16","alias_value":"7BBZDEBOWFCUVTHT","created_at":"2026-07-07T02:17:47Z"},{"alias_kind":"pith_short_8","alias_value":"7BBZDEBO","created_at":"2026-07-07T02:17:47Z"}],"graph_snapshots":[{"event_id":"sha256:25c08b1b562bf70ea970ee8eb69e72156ec4f40e65804a3fb9a63303aa753b5c","target":"graph","created_at":"2026-07-07T02:17:47Z","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/2607.03444/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We obtain results on cut and choose games for complete Boolean algebras. Zapletal proved that there is a Boolean algebra $\\mathbb{B}$ such that $\\mathcal{G}_\\omega^\\textsf{candc}(\\mathbb{B})$, the version of the game which ends on the $\\omega$'th round, is undetermined. We prove that, assuming the consistency of a proper class of supercompact cardinals, the limit version $\\mathcal{G}^{\\textsf{candc}}_{< \\lambda}(\\mathbb{B})$, in which there are $\\lambda$-many rounds but no concluding round, is consistently determined for all complete Boolean algebras $\\mathbb{B}$ and all successor cardinals $\\","authors_text":"Maxwell Levine","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-07-03T15:56:26Z","title":"On Determinacy for Cut and Choose Games of Uncountable Length"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03444","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:faffcd9df90f37aa4465d19055f6abbdc32a93c663e59e60e053203ab536497a","target":"record","created_at":"2026-07-07T02:17:47Z","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":"0d1dbe378e9445720d5fdf8be823bbad43876e4c59899c40b31563db8e797575","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-07-03T15:56:26Z","title_canon_sha256":"a1e2d9af67454809ea3f43b3d94953b39c80e3bd10d7dafc30a8720214493a3e"},"schema_version":"1.0","source":{"id":"2607.03444","kind":"arxiv","version":1}},"canonical_sha256":"f84391902eb1454accf33789f9df4076a296d07d9ac3b913156ed322e9ec7334","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f84391902eb1454accf33789f9df4076a296d07d9ac3b913156ed322e9ec7334","first_computed_at":"2026-07-07T02:17:47.089301Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:17:47.089301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IjRTV57IR4fhnJCs9gCxRyT3bBduHnZa9pKwo/G1kXfeCcuMybVcAy/XrKzEhqHstvEhU5oDF7o6SmWwMGUHDg==","signature_status":"signed_v1","signed_at":"2026-07-07T02:17:47.090191Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.03444","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:faffcd9df90f37aa4465d19055f6abbdc32a93c663e59e60e053203ab536497a","sha256:25c08b1b562bf70ea970ee8eb69e72156ec4f40e65804a3fb9a63303aa753b5c"],"state_sha256":"8ce99eb4880871807cfb48a410609f5e0f1aaa7c5992c1408c49d19114774d82"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wsK6TcmNmJ/U7k8IwgBqpvef/HeBgpThUO29/8IarF3LTqjngTbKLyst0x68oAk5LKaKsblnh123QDc5PBpLBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T05:27:59.216648Z","bundle_sha256":"df5cdd77b16c00fe6a1ec6b3a33161aa0adec0f4c20cfb2c43aee39999a11957"}}