{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:CDWFOC2KNQVQYYWG5XOZIWDCRL","short_pith_number":"pith:CDWFOC2K","canonical_record":{"source":{"id":"1411.6868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"fa810ce90960968c65664084b07db9d5c6c2020ef3f402e50a1e5112eeafc0cb","abstract_canon_sha256":"4e4e2372ec432f36a23dd0ca0c7fb28284d58aa049ad59cade350ff220f952f8"},"schema_version":"1.0"},"canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","source":{"kind":"arxiv","id":"1411.6868","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.6868","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"arxiv_version","alias_value":"1411.6868v1","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.6868","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"pith_short_12","alias_value":"CDWFOC2KNQVQ","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CDWFOC2KNQVQYYWG","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CDWFOC2K","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:CDWFOC2KNQVQYYWG5XOZIWDCRL","target":"record","payload":{"canonical_record":{"source":{"id":"1411.6868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"fa810ce90960968c65664084b07db9d5c6c2020ef3f402e50a1e5112eeafc0cb","abstract_canon_sha256":"4e4e2372ec432f36a23dd0ca0c7fb28284d58aa049ad59cade350ff220f952f8"},"schema_version":"1.0"},"canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:47.108553Z","signature_b64":"jfbwDbo9wtLBXZL2/Zqr5/1d0/IJhPfXKlnYYZbY1IfqWW/IuS1l8B4vHaxukhJtfIjSv1g6Jx0AEeCpfKVnBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","last_reissued_at":"2026-05-18T02:32:47.108029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:47.108029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.6868","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a0fmKktKSeYfDvYNXaW8LRzLGFEBqwVWrf8ZPM4mVQxa1zOtXVmn2DlT1nL8uci85PAtJY8ksgVYt9j4Btw+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:53:09.554959Z"},"content_sha256":"f508841351ab6efa754c7dfc398b3b8d578793607b282417e19aec6f0d10d1c9","schema_version":"1.0","event_id":"sha256:f508841351ab6efa754c7dfc398b3b8d578793607b282417e19aec6f0d10d1c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:CDWFOC2KNQVQYYWG5XOZIWDCRL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bisector energy and few distinct distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Ben Lund, Frank de Zeeuw","submitted_at":"2014-11-25T13:51:36Z","abstract_excerpt":"We introduce the bisector energy of an $n$-point set $P$ in $\\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\\frac{2}{5}}n^{\\frac{12}{5}+\\epsilon} + M(n)n^2).$. We also prove the lower bound $\\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results:\n  (i) If $P$ determines $O(n/\\sqrt{\\log n})$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6868","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:32:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"988oW0xUlw7l9KsS6V+NyABbair25YowN6TzjQJNHuIXuwAKO9jJW9CGGR1Z0y9c9aQ2p3ldCj2iIo+CLJYxDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:53:09.555304Z"},"content_sha256":"85ac2b8ed2cd15e8157567d3c25f841d39e6541a8df947e78c4fec7531be2074","schema_version":"1.0","event_id":"sha256:85ac2b8ed2cd15e8157567d3c25f841d39e6541a8df947e78c4fec7531be2074"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/bundle.json","state_url":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T13:53:09Z","links":{"resolver":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL","bundle":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/bundle.json","state":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CDWFOC2KNQVQYYWG5XOZIWDCRL","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":"4e4e2372ec432f36a23dd0ca0c7fb28284d58aa049ad59cade350ff220f952f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","title_canon_sha256":"fa810ce90960968c65664084b07db9d5c6c2020ef3f402e50a1e5112eeafc0cb"},"schema_version":"1.0","source":{"id":"1411.6868","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.6868","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"arxiv_version","alias_value":"1411.6868v1","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.6868","created_at":"2026-05-18T02:32:47Z"},{"alias_kind":"pith_short_12","alias_value":"CDWFOC2KNQVQ","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CDWFOC2KNQVQYYWG","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CDWFOC2K","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:85ac2b8ed2cd15e8157567d3c25f841d39e6541a8df947e78c4fec7531be2074","target":"graph","created_at":"2026-05-18T02:32:47Z","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"},"paper":{"abstract_excerpt":"We introduce the bisector energy of an $n$-point set $P$ in $\\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\\frac{2}{5}}n^{\\frac{12}{5}+\\epsilon} + M(n)n^2).$. We also prove the lower bound $\\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results:\n  (i) If $P$ determines $O(n/\\sqrt{\\log n})$ ","authors_text":"Adam Sheffer, Ben Lund, Frank de Zeeuw","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","title":"Bisector energy and few distinct distances"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6868","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:f508841351ab6efa754c7dfc398b3b8d578793607b282417e19aec6f0d10d1c9","target":"record","created_at":"2026-05-18T02:32:47Z","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":"4e4e2372ec432f36a23dd0ca0c7fb28284d58aa049ad59cade350ff220f952f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","title_canon_sha256":"fa810ce90960968c65664084b07db9d5c6c2020ef3f402e50a1e5112eeafc0cb"},"schema_version":"1.0","source":{"id":"1411.6868","kind":"arxiv","version":1}},"canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","first_computed_at":"2026-05-18T02:32:47.108029Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:47.108029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jfbwDbo9wtLBXZL2/Zqr5/1d0/IJhPfXKlnYYZbY1IfqWW/IuS1l8B4vHaxukhJtfIjSv1g6Jx0AEeCpfKVnBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:47.108553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.6868","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f508841351ab6efa754c7dfc398b3b8d578793607b282417e19aec6f0d10d1c9","sha256:85ac2b8ed2cd15e8157567d3c25f841d39e6541a8df947e78c4fec7531be2074"],"state_sha256":"fc601f2c9d23d9944462f36f82764297d95b752e34ae53090c6cfa4830ff12dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Hw0FyxzXn3eWVd8ECPfPYedcnWACOyyOGt4jny9XjtnbOlxV890CzMVvzpCzeq9jskfQjTsKB0XKF65mYCWAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T13:53:09.557687Z","bundle_sha256":"2e99760a9db0228ee82cdddc92bea6467f1bc88baa06de7aa2a9addcd8e376a5"}}