{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UU4NYTSYFOTQ2UCEVEN7GM5VY5","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":"89bef574563cf71890aabfd3b0e6036370373b054079d2fbfa8550167bdb614b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-05-13T15:40:59Z","title_canon_sha256":"2083b3332ed3074063e6d5a12257c76f8cdac59602e3921f4fdeb7f683b1e3d8"},"schema_version":"1.0","source":{"id":"2605.13683","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13683","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13683v1","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13683","created_at":"2026-05-18T02:44:17Z"},{"alias_kind":"pith_short_12","alias_value":"UU4NYTSYFOTQ","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"UU4NYTSYFOTQ2UCE","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"UU4NYTSY","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:796ced6be33be914d7b74a502dc9d67dd702e55dcc31b01eddbc0f91c84323c0","target":"graph","created_at":"2026-05-18T02:44:17Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure (Q,<) that has an o-minimal open core, but some of its elementary superstructures do not."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the specific expansion of (Q,<) can be chosen so its open core is o-minimal while some elementary extension has a non-o-minimal open core; this relies on the construction preserving the necessary first-order properties."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"O-minimality of the open core is not an elementary property, shown via a counterexample expansion of (Q, <) whose elementary superstructures can lack the property."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Having an o-minimal open core is not an elementary property."}],"snapshot_sha256":"02d65a4522b694389957bb75f36d469ea1de43da45469d6b92b2ed13cff66ab6"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Given a structure $\\mathcal{M}$ with a definable topology, its open core is a structure defined on the same universe whose language consists of all open sets of all arities definable in $\\mathcal{M}$. In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure $(\\mathbb{Q},<)$ that has an o-minimal open core, but some of its elementary superstructures do not.","authors_text":"Alexi Block Gorman, Esther Elbaz Saban","cross_cats":[],"headline":"Having an o-minimal open core is not an elementary property.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-05-13T15:40:59Z","title":"O-minimal open core is not an elementary property"},"references":{"count":9,"internal_anchors":0,"resolved_work":9,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Alexi Block Gorman, Philipp Hieronymi, and Elliot Kaplan, Pairs of Theories Satisfying a Mordell-Lang Condition. Fund. Math., (2) 251:131-160, (2020)","work_id":"235c927d-d80f-4ae4-88b0-ce2c19f73f09","year":2020},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Gareth Boxall and Philipp Hieronymi, Expansions which introduce no new open sets.J. Symb. Log., 77(1):111-121, (2012)","work_id":"c3356608-a2e0-4f33-928a-34e4e822b62f","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Alfred Dolich, Chris Miller, and Charles Steinhorn, Structures having o-minimal open core.Trans. Am. Math. Soc., 362(3):1371–1411, (2010)","work_id":"42532b51-913e-4918-9a6d-7d11d98b6dee","year":2010},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Alfred Dolich, Chris Miller, and Charles Steinhorn, Expansions of o-minimal structures by dense independent sets.Ann. Pure Appl. Logic, 167(8):684–706, (2016)","work_id":"86c87324-a358-470f-bce3-60ad41d4e8b3","year":2016},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Philipp Hieronymi, Travis Nell, and Erik Walsberg, Wild theories with o-minimal open core.Ann. Pure Appl. Logic, (2) 169:146–163, (2018)","work_id":"294e3dce-bbe7-4243-84b1-58947123a720","year":2018}],"snapshot_sha256":"e2bb3aa9189ff03f2a03d469e037b6f231b76520a3a8803fa61c63c781c9d521"},"source":{"id":"2605.13683","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:04:46.125770Z","id":"5f6869f5-ddd1-4c85-a085-b75adf9303ff","model_set":{"reader":"grok-4.3"},"one_line_summary":"O-minimality of the open core is not an elementary property, shown via a counterexample expansion of (Q, <) whose elementary superstructures can lack the property.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Having an o-minimal open core is not an elementary property.","strongest_claim":"we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure (Q,<) that has an o-minimal open core, but some of its elementary superstructures do not.","weakest_assumption":"That the specific expansion of (Q,<) can be chosen so its open core is o-minimal while some elementary extension has a non-o-minimal open core; this relies on the construction preserving the necessary first-order properties."}},"verdict_id":"5f6869f5-ddd1-4c85-a085-b75adf9303ff"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:602e0fab8c8b3a682f12581263fdb82fc9395fb83057c866a26fb6e8066a546f","target":"record","created_at":"2026-05-18T02:44:17Z","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":"89bef574563cf71890aabfd3b0e6036370373b054079d2fbfa8550167bdb614b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2026-05-13T15:40:59Z","title_canon_sha256":"2083b3332ed3074063e6d5a12257c76f8cdac59602e3921f4fdeb7f683b1e3d8"},"schema_version":"1.0","source":{"id":"2605.13683","kind":"arxiv","version":1}},"canonical_sha256":"a538dc4e582ba70d5044a91bf333b5c746e10b927de7a7d465cba5f412da95fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a538dc4e582ba70d5044a91bf333b5c746e10b927de7a7d465cba5f412da95fa","first_computed_at":"2026-05-18T02:44:17.016208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:17.016208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LXGXnnydMRW/5ue10Yioln7NNpL6LaPk8X4/rX5HoaqmZQ+DrFgL6hg9SqJc3nFj0Bqux0EEOSWog+FPymjBBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:17.016711Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13683","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:602e0fab8c8b3a682f12581263fdb82fc9395fb83057c866a26fb6e8066a546f","sha256:796ced6be33be914d7b74a502dc9d67dd702e55dcc31b01eddbc0f91c84323c0"],"state_sha256":"7fb81bc7136e988a84bb427564f0c8822f5e459e4b70f181c9dd1c79e46dc3c8"}