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The setting includes dense pairs of o-minimal structures, expansions of $\\mathcal M$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1805.11500","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-05-29T14:30:49Z","cross_cats_sorted":[],"title_canon_sha256":"b052d362ab8a270fec1cc2a28d0e545dd0e83f1107cc51baaeef880ff2396eb4","abstract_canon_sha256":"84b4ab5e19a7d76ed2b5bc0117b661ff5d731c4eaf9ebae49fc639a2a0ad6b64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:21.691455Z","signature_b64":"f44+tg4lLHF/7QbPX8yJUdJzMXnZoXdI+Yirn5Rf8Nf7TORxYa+GLNKRLNMcuk4viuzOi4aPJj5VA1o84chiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c68566d878bc8c6d20f7539a57aee970b81441b221b0d0af1feaa46b1394d40","last_reissued_at":"2026-05-18T00:10:21.690668Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:21.690668Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterizing o-minimal groups in tame expansions of o-minimal structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Pantelis E. 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