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This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains $n/2$ points of $\\mathcal{P}$, or the number of distinct perpendicular bisectors is $\\Omega(n^2)$.\n  "},"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":"1604.02059","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-04-07T16:14:40Z","cross_cats_sorted":[],"title_canon_sha256":"97daa9aee97d626a2bcba6285a25ac6b2bc2f5d3dd58957d22185cf4880421fd","abstract_canon_sha256":"ca9dc65f7033a50507a1701bf19f947f598dc0f0ca4e6726040675eef0d21d93"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:08.689561Z","signature_b64":"kJPaGdxyvtFm2eUBFRYsVTEH/xJPCQKQq7dUX064WcQlflT1Xa0MN+oNp6j9M75vFXDpmFO4zbT9JrvQrjohCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2bdbed8574b7973462e8b74bf3a475d8720d1897bac645835660fdc4a188ef8","last_reissued_at":"2026-05-17T23:52:08.689151Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:08.689151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A refined energy bound for perpendicular bisectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ben Lund","submitted_at":"2016-04-07T16:14:40Z","abstract_excerpt":"Let $\\mathcal{P}$ be a set of $n$ points in the Euclidean plane. 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