{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:3O6UNRZAA5YXPY7VAMGKVQH7LS","short_pith_number":"pith:3O6UNRZA","canonical_record":{"source":{"id":"1112.5860","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2011-12-26T14:43:32Z","cross_cats_sorted":["math.AG","math.DS"],"title_canon_sha256":"1ec8da9055f915b72c5c80f13f5d5349fcf89caaf34604d292987161c1458167","abstract_canon_sha256":"a73af7c0086883b433e6595a71fd9e8f65c764992a54fcb484bdbeaa2c81f5b3"},"schema_version":"1.0"},"canonical_sha256":"dbbd46c720077177e3f5030caac0ff5cade8db6570c0b64de9f0853752be65f4","source":{"kind":"arxiv","id":"1112.5860","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5860","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5860v2","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5860","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"pith_short_12","alias_value":"3O6UNRZAA5YX","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3O6UNRZAA5YXPY7V","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3O6UNRZA","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:3O6UNRZAA5YXPY7VAMGKVQH7LS","target":"record","payload":{"canonical_record":{"source":{"id":"1112.5860","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2011-12-26T14:43:32Z","cross_cats_sorted":["math.AG","math.DS"],"title_canon_sha256":"1ec8da9055f915b72c5c80f13f5d5349fcf89caaf34604d292987161c1458167","abstract_canon_sha256":"a73af7c0086883b433e6595a71fd9e8f65c764992a54fcb484bdbeaa2c81f5b3"},"schema_version":"1.0"},"canonical_sha256":"dbbd46c720077177e3f5030caac0ff5cade8db6570c0b64de9f0853752be65f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:41.552834Z","signature_b64":"jttJV5BukBEOVO0l3yTRiwFETXCEMF5LKnHwEwEUL2qj9DCw0KpGlCvUWaBjfRBu1eO4gY8nCm0fAQO5nDVaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbbd46c720077177e3f5030caac0ff5cade8db6570c0b64de9f0853752be65f4","last_reissued_at":"2026-05-18T03:37:41.552338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:41.552338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.5860","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:37:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e2K4fJVrOmOxR8kisx9/v+huVzzMwS8p/C4VDhza4tWKEN9YHK/rNQ2Oxjlik2KLZKUgwMBM36sXw/srCpJLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T14:13:58.194091Z"},"content_sha256":"53d45ad78b9f5eae98966b1bf9c0740e36ad443a0dcde586423e8b0439c617fd","schema_version":"1.0","event_id":"sha256:53d45ad78b9f5eae98966b1bf9c0740e36ad443a0dcde586423e8b0439c617fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:3O6UNRZAA5YXPY7VAMGKVQH7LS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Billiard algebra, integrable line congruences, and double reflection nets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.DS"],"primary_cat":"nlin.SI","authors_text":"Milena Radnovic, Vladimir Dragovic","submitted_at":"2011-12-26T14:43:32Z","abstract_excerpt":"The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5860","kind":"arxiv","version":2},"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":"2026-05-18T03:37:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lpzjIbE1GDHTG59kXgEanA1XAF2g7MjuUvyspRS13G55/XIQ1eIizrK+s50zL2CePFdoXA29WdhQxvx2bWvJCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T14:13:58.194506Z"},"content_sha256":"aebd358795c97e261508588accf409cb15f864afc87935867b2749d89f97367d","schema_version":"1.0","event_id":"sha256:aebd358795c97e261508588accf409cb15f864afc87935867b2749d89f97367d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/bundle.json","state_url":"https://pith.science/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/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":"2026-06-04T14:13:58Z","links":{"resolver":"https://pith.science/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS","bundle":"https://pith.science/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/bundle.json","state":"https://pith.science/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3O6UNRZAA5YXPY7VAMGKVQH7LS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3O6UNRZAA5YXPY7VAMGKVQH7LS","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":"a73af7c0086883b433e6595a71fd9e8f65c764992a54fcb484bdbeaa2c81f5b3","cross_cats_sorted":["math.AG","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2011-12-26T14:43:32Z","title_canon_sha256":"1ec8da9055f915b72c5c80f13f5d5349fcf89caaf34604d292987161c1458167"},"schema_version":"1.0","source":{"id":"1112.5860","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5860","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5860v2","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5860","created_at":"2026-05-18T03:37:41Z"},{"alias_kind":"pith_short_12","alias_value":"3O6UNRZAA5YX","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3O6UNRZAA5YXPY7V","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3O6UNRZA","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:aebd358795c97e261508588accf409cb15f864afc87935867b2749d89f97367d","target":"graph","created_at":"2026-05-18T03:37:41Z","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":"The billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, are incorporated into the theory of integrable quad-graphs. An initial observation is that the Six-pointed star theorem, as the operational consistency for the billiard algebra, is equivalent to an integrabilty condition of a line congruence. A new notion of the double-reflection nets as a subclass of dual Darboux nets associated with pencils of quadrics is introduced, basic properies and several examples are presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics are","authors_text":"Milena Radnovic, Vladimir Dragovic","cross_cats":["math.AG","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2011-12-26T14:43:32Z","title":"Billiard algebra, integrable line congruences, and double reflection nets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5860","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":"sha256:53d45ad78b9f5eae98966b1bf9c0740e36ad443a0dcde586423e8b0439c617fd","target":"record","created_at":"2026-05-18T03:37:41Z","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":"a73af7c0086883b433e6595a71fd9e8f65c764992a54fcb484bdbeaa2c81f5b3","cross_cats_sorted":["math.AG","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2011-12-26T14:43:32Z","title_canon_sha256":"1ec8da9055f915b72c5c80f13f5d5349fcf89caaf34604d292987161c1458167"},"schema_version":"1.0","source":{"id":"1112.5860","kind":"arxiv","version":2}},"canonical_sha256":"dbbd46c720077177e3f5030caac0ff5cade8db6570c0b64de9f0853752be65f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dbbd46c720077177e3f5030caac0ff5cade8db6570c0b64de9f0853752be65f4","first_computed_at":"2026-05-18T03:37:41.552338Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:41.552338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jttJV5BukBEOVO0l3yTRiwFETXCEMF5LKnHwEwEUL2qj9DCw0KpGlCvUWaBjfRBu1eO4gY8nCm0fAQO5nDVaCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:41.552834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.5860","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53d45ad78b9f5eae98966b1bf9c0740e36ad443a0dcde586423e8b0439c617fd","sha256:aebd358795c97e261508588accf409cb15f864afc87935867b2749d89f97367d"],"state_sha256":"7eb5ec511975c2df446f7b72e0a1a019652003c318fecd85fa32a89f946c78af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eyJbaxEeeYj/5n7wZ5HH8mgS/gHaelnIgwEJ9Q9Amp5ccZnQXTDQCmZuFxfMO00OJTvY3YnxwGLVKdyOLrpwAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T14:13:58.196717Z","bundle_sha256":"9c7196dcd4724cb498f7346d2c10dbd1fa44d11cbc6e69e80568a3d83af83d16"}}