{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4WQRP5XORUJEUBPFMQPFSISXGK","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":"15088ea4e625e78c33543cd8745feecaf748148fa25b8e576b60563e5c98ba49","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-04-27T14:24:17Z","title_canon_sha256":"514e18642eaf5072692c9f0eac36707a19e76c40985ab9d91446af2f311ff621"},"schema_version":"1.0","source":{"id":"2604.24522","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.24522","created_at":"2026-05-20T14:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"2604.24522v1","created_at":"2026-05-20T14:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.24522","created_at":"2026-05-20T14:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"4WQRP5XORUJE","created_at":"2026-05-20T14:03:27Z"},{"alias_kind":"pith_short_16","alias_value":"4WQRP5XORUJEUBPF","created_at":"2026-05-20T14:03:27Z"},{"alias_kind":"pith_short_8","alias_value":"4WQRP5XO","created_at":"2026-05-20T14:03:27Z"}],"graph_snapshots":[{"event_id":"sha256:ccfbdff7fbe530699d62ce04adf7db855010519af17c06365e631afdfa3f5de7","target":"graph","created_at":"2026-05-20T14:03:27Z","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 every NTP₂ expansion of (ℝ,<,+) by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP₂ expansions of (ℚ_p,+,·), and all NTP₂ definably complete expansions of ordered groups."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The expansions are NTP2 and the added predicates are constructible sets; the proofs rely on this tameness assumption to conclude that definable sets remain constructible."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"NTP2 expansions of (R,<,+) by constructible sets define only constructible sets with generically piecewise continuous functions, with extensions to Qp and ordered groups."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"NTP2 expansions of the reals by constructible sets define only constructible sets."}],"snapshot_sha256":"a9dbb98a6fef5716d84c74ef098e22268f1ce87cfabdc1a10cbc2b8f8bc4cd49"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T22:02:05.254257Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.24522/integrity.json","findings":[],"snapshot_sha256":"377143b0639eb1b75006e3a6688782e36fc287a96e2c8eb89696edaf0dd2063d","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A subset of a topological space is constructible if it is a finite Boolean combination of closed sets. We prove that every NTP$_2$ expansion of $(\\mathbb{R},<,+)$ by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP$_2$ expansions of $(\\mathbb{Q}_p,+,\\cdot)$, and all NTP$_2$ definably complete expansions of ordered groups. In the latter case, the structure is generically locally o-minimal, has definable choice, and carries a well-behaved notion of naive topological dimension. For NIP uniform ","authors_text":"Pablo and\\'ujar Guerrero","cross_cats":[],"headline":"NTP2 expansions of the reals by constructible sets define only constructible sets.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-04-27T14:24:17Z","title":"NTP2 topological structures"},"references":{"count":3,"internal_anchors":0,"resolved_work":2,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"[FS10] Antongiulio Fornasiero and Tamara Servi","work_id":"a162e8be-311c-4beb-ae2d-ff844fe45b7e","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"A note on topological model theory.Fund","work_id":"","year":1973},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"[Wal22] Erik Walsberg","work_id":"791c1e98-62e7-484d-9d9d-e18612e2dbb5","year":1910}],"snapshot_sha256":"99db164eaff13ddcbc392579b7de6599d79efb87f1602694e58446261a70d949"},"source":{"id":"2604.24522","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-07T17:08:23.267736Z","id":"26719a4b-e0b8-4b8c-8378-1cffadb835f5","model_set":{"reader":"grok-4.3"},"one_line_summary":"NTP2 expansions of (R,<,+) by constructible sets define only constructible sets with generically piecewise continuous functions, with extensions to Qp and ordered groups.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"NTP2 expansions of the reals by constructible sets define only constructible sets.","strongest_claim":"We prove that every NTP₂ expansion of (ℝ,<,+) by constructible sets defines only constructible sets, and that definable functions are generically piecewise continuous. The result also holds for all NTP₂ expansions of (ℚ_p,+,·), and all NTP₂ definably complete expansions of ordered groups.","weakest_assumption":"The expansions are NTP2 and the added predicates are constructible sets; the proofs rely on this tameness assumption to conclude that definable sets remain constructible."}},"verdict_id":"26719a4b-e0b8-4b8c-8378-1cffadb835f5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1f08729167176072c95a12aefb40a4719423bf456ef55c94b05fd27731f6da0e","target":"record","created_at":"2026-05-20T14:03:27Z","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":"15088ea4e625e78c33543cd8745feecaf748148fa25b8e576b60563e5c98ba49","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-04-27T14:24:17Z","title_canon_sha256":"514e18642eaf5072692c9f0eac36707a19e76c40985ab9d91446af2f311ff621"},"schema_version":"1.0","source":{"id":"2604.24522","kind":"arxiv","version":1}},"canonical_sha256":"e5a117f6ee8d124a05e5641e59225732af0144f1f15635ad4a2f48714a5f34c2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5a117f6ee8d124a05e5641e59225732af0144f1f15635ad4a2f48714a5f34c2","first_computed_at":"2026-05-20T14:03:27.074612Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T14:03:27.074612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VqZRHIOhYxmqxMZsIQqQiXsHSLNZYOQTBLi9bY+/rKAoijpA7WFQxHMdUhdNgBpWzhPhKwsQsGNJzIi9euZWBg==","signature_status":"signed_v1","signed_at":"2026-05-20T14:03:27.075152Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.24522","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f08729167176072c95a12aefb40a4719423bf456ef55c94b05fd27731f6da0e","sha256:ccfbdff7fbe530699d62ce04adf7db855010519af17c06365e631afdfa3f5de7"],"state_sha256":"60ddcae304b18535aad0aa09c6bcf8352aa1b6a61e4024dc2a0c6518183db7f3"}