{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ONLS66R5Y5LBJTPAOIRO5JIWUA","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":"3494f1be637a1fa197e0e89a5ad1cde07d1d43d327cfd4d5b73ac1724fe84fa5","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-08T15:31:00Z","title_canon_sha256":"ec41b76db281bc038ee6d6406dae7f5b5ace6b7322b5d02704105c02c69a8315"},"schema_version":"1.0","source":{"id":"1102.1628","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.1628","created_at":"2026-05-18T02:22:39Z"},{"alias_kind":"arxiv_version","alias_value":"1102.1628v2","created_at":"2026-05-18T02:22:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1628","created_at":"2026-05-18T02:22:39Z"},{"alias_kind":"pith_short_12","alias_value":"ONLS66R5Y5LB","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"ONLS66R5Y5LBJTPA","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"ONLS66R5","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:54bc6838643276c275470bb27c161bfb7192c5c2ec8440188e6c9edf515d19a5","target":"graph","created_at":"2026-05-18T02:22:39Z","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 consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to co","authors_text":"John R. Doyle, Michael Ching","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-08T15:31:00Z","title":"Apollonian circle packings of the half-plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1628","kind":"arxiv","version":2},"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":"sha256:48068fc0d759b6c8f6024269435bb015d9b5fefdc3410b01e76f6e81e363a778","target":"record","created_at":"2026-05-18T02:22:39Z","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":"3494f1be637a1fa197e0e89a5ad1cde07d1d43d327cfd4d5b73ac1724fe84fa5","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-02-08T15:31:00Z","title_canon_sha256":"ec41b76db281bc038ee6d6406dae7f5b5ace6b7322b5d02704105c02c69a8315"},"schema_version":"1.0","source":{"id":"1102.1628","kind":"arxiv","version":2}},"canonical_sha256":"73572f7a3dc75614cde07222eea516a02bcf2a72604d12a5dc59763db85ea533","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73572f7a3dc75614cde07222eea516a02bcf2a72604d12a5dc59763db85ea533","first_computed_at":"2026-05-18T02:22:39.688956Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:22:39.688956Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lOP95X7CIkRY0MO9db2XytqgHcpJ9lPGYGzvOlQoF3vAoCKsATXgYSJ5uz4zKBHU5+O839719u1zUfbu0VsVDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:22:39.689639Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.1628","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:48068fc0d759b6c8f6024269435bb015d9b5fefdc3410b01e76f6e81e363a778","sha256:54bc6838643276c275470bb27c161bfb7192c5c2ec8440188e6c9edf515d19a5"],"state_sha256":"f62cdf8e109d4c24b05e5976f7de4544cff094aca83dacbf5f2271a1bcc3da03"}