{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:MMJC7CWU6QLJMMTHVB7VVQBVQQ","short_pith_number":"pith:MMJC7CWU","canonical_record":{"source":{"id":"1302.1813","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","cross_cats_sorted":[],"title_canon_sha256":"dc3feed24c7dba6acfbe83b058078e91687e8ad9df76ba32dbf77633aa4a7b1f","abstract_canon_sha256":"470b410c86a4b5eaf1e69f711fdf903f17cc134ddc186edaad91a1b362a3fd54"},"schema_version":"1.0"},"canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","source":{"kind":"arxiv","id":"1302.1813","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.1813","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1302.1813v1","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.1813","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"MMJC7CWU6QLJ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MMJC7CWU6QLJMMTH","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MMJC7CWU","created_at":"2026-05-18T12:27:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:MMJC7CWU6QLJMMTHVB7VVQBVQQ","target":"record","payload":{"canonical_record":{"source":{"id":"1302.1813","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","cross_cats_sorted":[],"title_canon_sha256":"dc3feed24c7dba6acfbe83b058078e91687e8ad9df76ba32dbf77633aa4a7b1f","abstract_canon_sha256":"470b410c86a4b5eaf1e69f711fdf903f17cc134ddc186edaad91a1b362a3fd54"},"schema_version":"1.0"},"canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:14.546730Z","signature_b64":"Pk7rFM79ttFEtSd6ukh5dloF+6MipCkO9HxBN/DPvak3CSWWn4P8hp18B85peQAzNTeXOaxyMsZEdlDtY+dpBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","last_reissued_at":"2026-05-18T03:34:14.545130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:14.545130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.1813","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":"2026-05-18T03:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rFjN5Nm0jyEjfXYT9IrVWGuRe4K7yQyBLBsX+EzaLkcglmAI+dgqmQJFqjdnYQ9HbDoAQQtYtI7xX2a123XKBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:14:10.522964Z"},"content_sha256":"f0293c6252b472cd7f6c7bb84a93c0828b70e06d0835e847d31671b9b8326246","schema_version":"1.0","event_id":"sha256:f0293c6252b472cd7f6c7bb84a93c0828b70e06d0835e847d31671b9b8326246"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:MMJC7CWU6QLJMMTHVB7VVQBVQQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polarit\\'es d\\'efinies par un triangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Beno\\^it Kloeckner (IF)","submitted_at":"2013-02-07T17:49:30Z","abstract_excerpt":"A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1813","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":"2026-05-18T03:34:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fhqHXzYIJFoLv+9Bx7jKsnWwnp3nw8vqxpxoGJj3rRotXG9bptG177yJd9DKOiCc+jtJhQ4XlOMs4S7pyTRMDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T08:14:10.523308Z"},"content_sha256":"bc5cbcba644f96f9fe6f344188166172ff2609e4b723289837c27aed8d9e902a","schema_version":"1.0","event_id":"sha256:bc5cbcba644f96f9fe6f344188166172ff2609e4b723289837c27aed8d9e902a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/bundle.json","state_url":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/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":"2026-06-05T08:14:10Z","links":{"resolver":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ","bundle":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/bundle.json","state":"https://pith.science/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MMJC7CWU6QLJMMTHVB7VVQBVQQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:MMJC7CWU6QLJMMTHVB7VVQBVQQ","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":"470b410c86a4b5eaf1e69f711fdf903f17cc134ddc186edaad91a1b362a3fd54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","title_canon_sha256":"dc3feed24c7dba6acfbe83b058078e91687e8ad9df76ba32dbf77633aa4a7b1f"},"schema_version":"1.0","source":{"id":"1302.1813","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.1813","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"arxiv_version","alias_value":"1302.1813v1","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.1813","created_at":"2026-05-18T03:34:14Z"},{"alias_kind":"pith_short_12","alias_value":"MMJC7CWU6QLJ","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"MMJC7CWU6QLJMMTH","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"MMJC7CWU","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:bc5cbcba644f96f9fe6f344188166172ff2609e4b723289837c27aed8d9e902a","target":"graph","created_at":"2026-05-18T03:34:14Z","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":"A polarity of a projective plane is a map, often assumed to be involutive, mapping a generic point to a generic line and reciprocally. The most classical polarity is the polarity with respect to a conic, but other exist: the harmonic polarity with respect to a triangle, the polarities with respect to high-degree algebraic curve, the polarities with respect to a convex set. In this article, we introduce a polarity with respect to a triangle motivated by a question on duality of projective frames. We show that the four polarities above apply to a triangle and in fact coincide. This result is an ","authors_text":"Beno\\^it Kloeckner (IF)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","title":"Polarit\\'es d\\'efinies par un triangle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1813","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:f0293c6252b472cd7f6c7bb84a93c0828b70e06d0835e847d31671b9b8326246","target":"record","created_at":"2026-05-18T03:34:14Z","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":"470b410c86a4b5eaf1e69f711fdf903f17cc134ddc186edaad91a1b362a3fd54","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-07T17:49:30Z","title_canon_sha256":"dc3feed24c7dba6acfbe83b058078e91687e8ad9df76ba32dbf77633aa4a7b1f"},"schema_version":"1.0","source":{"id":"1302.1813","kind":"arxiv","version":1}},"canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63122f8ad4f416963267a87f5ac035842bb723a6147c0df8ec3042ea769b21a8","first_computed_at":"2026-05-18T03:34:14.545130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:14.545130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Pk7rFM79ttFEtSd6ukh5dloF+6MipCkO9HxBN/DPvak3CSWWn4P8hp18B85peQAzNTeXOaxyMsZEdlDtY+dpBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:14.546730Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.1813","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0293c6252b472cd7f6c7bb84a93c0828b70e06d0835e847d31671b9b8326246","sha256:bc5cbcba644f96f9fe6f344188166172ff2609e4b723289837c27aed8d9e902a"],"state_sha256":"b78a117f035c9b72b448d3981ecb1a938c15371d67c7108fb6fc9a098a647c41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hCUPGyQLZvy4g8s0WGSwyy3GEgVm2bpik0N1uZXq1x3YDcHWqp1GJVhB/Gelv+8XehFeQP8qVTmP5mIGPSI2AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T08:14:10.526091Z","bundle_sha256":"8897eb74d80ed624914df30426f921fa91706d1f06b2e8a201aeb5ef898d1eb9"}}