{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:EGZ5SS2LODRPOULJE5EOOSBMVC","short_pith_number":"pith:EGZ5SS2L","canonical_record":{"source":{"id":"2212.11231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-12-21T17:57:37Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"5972bbd1d1a41438b76378debb9abff655940908d87dfdfecaf34bebf9926a1f","abstract_canon_sha256":"23eae784fecc0a0d380e4837384423e1ee488962f305e0bc6fc9ef0b72adc7aa"},"schema_version":"1.0"},"canonical_sha256":"21b3d94b4b70e2f751692748e7482ca88d8f94d0471ab5dd4c12beaacee78af3","source":{"kind":"arxiv","id":"2212.11231","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2212.11231","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"arxiv_version","alias_value":"2212.11231v2","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.11231","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_12","alias_value":"EGZ5SS2LODRP","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_16","alias_value":"EGZ5SS2LODRPOULJ","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_8","alias_value":"EGZ5SS2L","created_at":"2026-07-05T09:43:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:EGZ5SS2LODRPOULJE5EOOSBMVC","target":"record","payload":{"canonical_record":{"source":{"id":"2212.11231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-12-21T17:57:37Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"5972bbd1d1a41438b76378debb9abff655940908d87dfdfecaf34bebf9926a1f","abstract_canon_sha256":"23eae784fecc0a0d380e4837384423e1ee488962f305e0bc6fc9ef0b72adc7aa"},"schema_version":"1.0"},"canonical_sha256":"21b3d94b4b70e2f751692748e7482ca88d8f94d0471ab5dd4c12beaacee78af3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:43:21.900687Z","signature_b64":"MoerBXRa3hJrm7j2/cooGp+J5yg6uFKb8vZPmRimja7yF2CelLzbQqkD2uLcOxORm9d1qc4OSPVwV2mbXWbNAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21b3d94b4b70e2f751692748e7482ca88d8f94d0471ab5dd4c12beaacee78af3","last_reissued_at":"2026-07-05T09:43:21.900266Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:43:21.900266Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2212.11231","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":"2026-07-05T09:43:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W5zoyV3b3DiEjEAMUNNh6X7AxDsSFjpsEesnqkaGX0Nod6JHZsOT5W66gn1vBd5DQdxy2o3WzM2mWRKS9qLBBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:10:23.875712Z"},"content_sha256":"457a307a38568d5fdc81faa8168db31ff148d5b2dfec0a3ab98128f5bccbcabc","schema_version":"1.0","event_id":"sha256:457a307a38568d5fdc81faa8168db31ff148d5b2dfec0a3ab98128f5bccbcabc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:EGZ5SS2LODRPOULJE5EOOSBMVC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complete bipartite graphs flexible in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AG","authors_text":"M. D. Kovalev, S. Yu. Orevkov","submitted_at":"2022-12-21T17:57:37Z","abstract_excerpt":"A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by different authors, the question of flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. In the present paper, we solve it for all complete bipartite graphs in the Euclidean plane as well as in the sphere and in the hyperbolic plane. We give independent self-contained proofs without extensive computa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.11231","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2212.11231/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:43:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cFcRJHKktL5ZY+3DkYiZ08JfAR+rks13keV/Vfx+IIg3GR8vkOXQzt4e+8Ni6tMQnQRh7JNzmRblDkFxTbwMAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:10:23.876206Z"},"content_sha256":"5113bdc1ddfad534861a0b571c1b3d5bc8068d9513523ccc1b8eedc2149a19e3","schema_version":"1.0","event_id":"sha256:5113bdc1ddfad534861a0b571c1b3d5bc8068d9513523ccc1b8eedc2149a19e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/bundle.json","state_url":"https://pith.science/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/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":"2026-08-16T13:10:23Z","links":{"resolver":"https://pith.science/pith/EGZ5SS2LODRPOULJE5EOOSBMVC","bundle":"https://pith.science/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/bundle.json","state":"https://pith.science/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EGZ5SS2LODRPOULJE5EOOSBMVC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:EGZ5SS2LODRPOULJE5EOOSBMVC","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":"23eae784fecc0a0d380e4837384423e1ee488962f305e0bc6fc9ef0b72adc7aa","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-12-21T17:57:37Z","title_canon_sha256":"5972bbd1d1a41438b76378debb9abff655940908d87dfdfecaf34bebf9926a1f"},"schema_version":"1.0","source":{"id":"2212.11231","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2212.11231","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"arxiv_version","alias_value":"2212.11231v2","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2212.11231","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_12","alias_value":"EGZ5SS2LODRP","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_16","alias_value":"EGZ5SS2LODRPOULJ","created_at":"2026-07-05T09:43:21Z"},{"alias_kind":"pith_short_8","alias_value":"EGZ5SS2L","created_at":"2026-07-05T09:43:21Z"}],"graph_snapshots":[{"event_id":"sha256:5113bdc1ddfad534861a0b571c1b3d5bc8068d9513523ccc1b8eedc2149a19e3","target":"graph","created_at":"2026-07-05T09:43:21Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2212.11231/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by Dixon in 1899. Later on, in a series of papers by different authors, the question of flexibility of $K_{m,n}$ was solved for almost all pairs $(m,n)$. In the present paper, we solve it for all complete bipartite graphs in the Euclidean plane as well as in the sphere and in the hyperbolic plane. We give independent self-contained proofs without extensive computa","authors_text":"M. D. Kovalev, S. Yu. Orevkov","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-12-21T17:57:37Z","title":"Complete bipartite graphs flexible in the plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.11231","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":"sha256:457a307a38568d5fdc81faa8168db31ff148d5b2dfec0a3ab98128f5bccbcabc","target":"record","created_at":"2026-07-05T09:43:21Z","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":"23eae784fecc0a0d380e4837384423e1ee488962f305e0bc6fc9ef0b72adc7aa","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-12-21T17:57:37Z","title_canon_sha256":"5972bbd1d1a41438b76378debb9abff655940908d87dfdfecaf34bebf9926a1f"},"schema_version":"1.0","source":{"id":"2212.11231","kind":"arxiv","version":2}},"canonical_sha256":"21b3d94b4b70e2f751692748e7482ca88d8f94d0471ab5dd4c12beaacee78af3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21b3d94b4b70e2f751692748e7482ca88d8f94d0471ab5dd4c12beaacee78af3","first_computed_at":"2026-07-05T09:43:21.900266Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:43:21.900266Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MoerBXRa3hJrm7j2/cooGp+J5yg6uFKb8vZPmRimja7yF2CelLzbQqkD2uLcOxORm9d1qc4OSPVwV2mbXWbNAA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:43:21.900687Z","signed_message":"canonical_sha256_bytes"},"source_id":"2212.11231","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:457a307a38568d5fdc81faa8168db31ff148d5b2dfec0a3ab98128f5bccbcabc","sha256:5113bdc1ddfad534861a0b571c1b3d5bc8068d9513523ccc1b8eedc2149a19e3"],"state_sha256":"c446690cfc12fd81498b34316fe46291e28dbd2ecbe795c7fb9b5b624baf2201"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5UjiCpYe86YsLG9sJzLBsq/2L0Us1Y82dGxfuJbFmVFnT8zhfWXjQlJkZk5VBoG2zScWNOoajMMh83zVsVW0Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:10:23.881312Z","bundle_sha256":"2cc78693a0be30fbf0f9e5f4cc97a054e5ab199cdfa771295ce40eceb8becaea"}}