{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:YTZH7TTLWAYNETVVR62D3EZCW5","short_pith_number":"pith:YTZH7TTL","canonical_record":{"source":{"id":"1012.1059","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"adabfdd72a91064945bd517e5a73f21e6cf711e404cbb174c966e43a33fc2e4c","abstract_canon_sha256":"418bcad166b0fb8c8d111721e89cf87ce98a4793e03e5a08657c1d71daf94382"},"schema_version":"1.0"},"canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","source":{"kind":"arxiv","id":"1012.1059","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1059","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1059v5","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1059","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"pith_short_12","alias_value":"YTZH7TTLWAYN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YTZH7TTLWAYNETVV","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YTZH7TTL","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:YTZH7TTLWAYNETVVR62D3EZCW5","target":"record","payload":{"canonical_record":{"source":{"id":"1012.1059","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"adabfdd72a91064945bd517e5a73f21e6cf711e404cbb174c966e43a33fc2e4c","abstract_canon_sha256":"418bcad166b0fb8c8d111721e89cf87ce98a4793e03e5a08657c1d71daf94382"},"schema_version":"1.0"},"canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:12.016376Z","signature_b64":"I5rIxeIgrzxG5FGYENBGXV7f4NoMiYhYVFGPXa6CMhhWGxG2LyDIpFfxEKyuQnF41CYLucETkUD3x1/F5kSqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","last_reissued_at":"2026-05-18T04:02:12.015745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:12.015745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.1059","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:02:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xvo2PsbvlR59nDLfEHKPLA6zz9IsWOd406J9kI0xI2wssqD+W9Ts8ujy9bcTY+lkmiUE/JGZ4lmeb2DyS4jgBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:15:41.120709Z"},"content_sha256":"52a43786284c9e9345871f6451bd6f7f34a304ab28749e65252694dca7e96a0b","schema_version":"1.0","event_id":"sha256:52a43786284c9e9345871f6451bd6f7f34a304ab28749e65252694dca7e96a0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:YTZH7TTLWAYNETVVR62D3EZCW5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Circular planar nearrings: geometrical and combinatorial aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Achille Frigeri, Anna Benini, Fiorenza Morini","submitted_at":"2010-12-06T00:44:28Z","abstract_excerpt":"Let $(N,\\Phi)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\\mathcal{D}(a;b)$, as \\[\\mathcal{D}(a;b)=\\{x\\in \\Phi(r)+c\\mid r\\neq 0,\\ b\\in \\Phi(r)+c,\\ |(\\Phi(r)+c)\\cap (\\Phi(a)+b)|=1\\}.\\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if $\\mathcal{B}^{\\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,\\mathcal{B}^{\\mathcal{D}},\\in)$ is actually a balanced incomplete block design."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1059","kind":"arxiv","version":5},"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":"2026-05-18T04:02:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r5dRyqEyhl2prUOSkNlB/WOCLbzLkEEmPEgkolG4CZTHj2TfiPIwO9GhLXy0+uZwzCmc2jcsktZs0BHIUF/KDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T06:15:41.121060Z"},"content_sha256":"d9252a9cbcbc04123ac4d4227de646e1a97b549470aa5d9fb42e72a2a25b1a8b","schema_version":"1.0","event_id":"sha256:d9252a9cbcbc04123ac4d4227de646e1a97b549470aa5d9fb42e72a2a25b1a8b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/bundle.json","state_url":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YTZH7TTLWAYNETVVR62D3EZCW5/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":"2026-06-01T06:15:41Z","links":{"resolver":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5","bundle":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/bundle.json","state":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YTZH7TTLWAYNETVVR62D3EZCW5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YTZH7TTLWAYNETVVR62D3EZCW5","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":"418bcad166b0fb8c8d111721e89cf87ce98a4793e03e5a08657c1d71daf94382","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","title_canon_sha256":"adabfdd72a91064945bd517e5a73f21e6cf711e404cbb174c966e43a33fc2e4c"},"schema_version":"1.0","source":{"id":"1012.1059","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1059","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1059v5","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1059","created_at":"2026-05-18T04:02:12Z"},{"alias_kind":"pith_short_12","alias_value":"YTZH7TTLWAYN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YTZH7TTLWAYNETVV","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YTZH7TTL","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:d9252a9cbcbc04123ac4d4227de646e1a97b549470aa5d9fb42e72a2a25b1a8b","target":"graph","created_at":"2026-05-18T04:02:12Z","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":"Let $(N,\\Phi)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\\mathcal{D}(a;b)$, as \\[\\mathcal{D}(a;b)=\\{x\\in \\Phi(r)+c\\mid r\\neq 0,\\ b\\in \\Phi(r)+c,\\ |(\\Phi(r)+c)\\cap (\\Phi(a)+b)|=1\\}.\\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if $\\mathcal{B}^{\\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,\\mathcal{B}^{\\mathcal{D}},\\in)$ is actually a balanced incomplete block design.","authors_text":"Achille Frigeri, Anna Benini, Fiorenza Morini","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","title":"Circular planar nearrings: geometrical and combinatorial aspects"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1059","kind":"arxiv","version":5},"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":"sha256:52a43786284c9e9345871f6451bd6f7f34a304ab28749e65252694dca7e96a0b","target":"record","created_at":"2026-05-18T04:02:12Z","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":"418bcad166b0fb8c8d111721e89cf87ce98a4793e03e5a08657c1d71daf94382","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","title_canon_sha256":"adabfdd72a91064945bd517e5a73f21e6cf711e404cbb174c966e43a33fc2e4c"},"schema_version":"1.0","source":{"id":"1012.1059","kind":"arxiv","version":5}},"canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","first_computed_at":"2026-05-18T04:02:12.015745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:02:12.015745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I5rIxeIgrzxG5FGYENBGXV7f4NoMiYhYVFGPXa6CMhhWGxG2LyDIpFfxEKyuQnF41CYLucETkUD3x1/F5kSqDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:02:12.016376Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.1059","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52a43786284c9e9345871f6451bd6f7f34a304ab28749e65252694dca7e96a0b","sha256:d9252a9cbcbc04123ac4d4227de646e1a97b549470aa5d9fb42e72a2a25b1a8b"],"state_sha256":"736cf44913a79d8640077f43dcf6026cd1bb67b6a571ad12f2a458400e24fb91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YxGUhH7Hp9puvO0PKuUmv9mfIvhO6JvanEz/53Va9XIH3YqN3TssEZjzg30ncRpcEX1ckqfFflA6EAEhKTHiBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T06:15:41.123115Z","bundle_sha256":"d5a133aee756b01a80bccf4b56bf76a3a2f51b226cb078773f1f1040709ef867"}}