{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:GCQTTUQXUSTNDVEIQIPGILM6IF","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":"921024d821322d1622853fdf0b172a9ad59192063d0614b4a5580ae135ff8d77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2025-07-28T22:24:56Z","title_canon_sha256":"617939bc0087d994111fb811af2b9f4050ab5ab2a278c9ecbb4769879b0efeae"},"schema_version":"1.0","source":{"id":"2507.21366","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.21366","created_at":"2026-07-05T11:44:58Z"},{"alias_kind":"arxiv_version","alias_value":"2507.21366v1","created_at":"2026-07-05T11:44:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.21366","created_at":"2026-07-05T11:44:58Z"},{"alias_kind":"pith_short_12","alias_value":"GCQTTUQXUSTN","created_at":"2026-07-05T11:44:58Z"},{"alias_kind":"pith_short_16","alias_value":"GCQTTUQXUSTNDVEI","created_at":"2026-07-05T11:44:58Z"},{"alias_kind":"pith_short_8","alias_value":"GCQTTUQX","created_at":"2026-07-05T11:44:58Z"}],"graph_snapshots":[{"event_id":"sha256:b011a8c5cba9755e4a52e6c2a274edd01afb01ff811ce04dc7615e396edfdaf9","target":"graph","created_at":"2026-07-05T11:44:58Z","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/2507.21366/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a combinatorial consistency-inconsistency configuration that is equivalent to the failure of the following form of Kim's lemma for a given $k$: $(\\star)$ For any set of parameters $A$, formula $\\varphi(x,b)$, and $A$-bi-invariant types $p$ and $q$ extending $\\mathrm{tp}(b/A)$, if $\\varphi(x,b)$ $k$-divides along $p$, then it divides along $q$.\n  We then give an equivalent technical variant of $(\\star)$ that is non-trivial over arbitrary invariance bases. We also show that the failure of weaker versions of $(\\star)$ entails the existence of stronger combinatorial configurations, the str","authors_text":"James E. Hanson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2025-07-28T22:24:56Z","title":"A combinatorial characterization of Kim's lemma for pairs of bi-invariant types"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.21366","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:e19dfe8d9f5f16366617ddf696b7708108e1287753b95d1936de5ba60e440dd8","target":"record","created_at":"2026-07-05T11:44:58Z","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":"921024d821322d1622853fdf0b172a9ad59192063d0614b4a5580ae135ff8d77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2025-07-28T22:24:56Z","title_canon_sha256":"617939bc0087d994111fb811af2b9f4050ab5ab2a278c9ecbb4769879b0efeae"},"schema_version":"1.0","source":{"id":"2507.21366","kind":"arxiv","version":1}},"canonical_sha256":"30a139d217a4a6d1d488821e642d9e415117048dda418548858cbcae30acc3c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"30a139d217a4a6d1d488821e642d9e415117048dda418548858cbcae30acc3c8","first_computed_at":"2026-07-05T11:44:58.445355Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:44:58.445355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5F3sKQlHIni3NcAL3aJBijv39eDB9E3VqLQ4y8DOpz635GPxf3ZOkALH3PYuD8E7FbXJ86vcLUq1liER9/pLBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:44:58.445773Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.21366","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e19dfe8d9f5f16366617ddf696b7708108e1287753b95d1936de5ba60e440dd8","sha256:b011a8c5cba9755e4a52e6c2a274edd01afb01ff811ce04dc7615e396edfdaf9"],"state_sha256":"d95ec8ac680739d708b33be20b9f216010276013e948c458a477d2feee7145b0"}