{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XRRBKPYPXVKWRDK5XYGWISWHFO","short_pith_number":"pith:XRRBKPYP","canonical_record":{"source":{"id":"1604.06062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-20T19:03:41Z","cross_cats_sorted":["hep-th","math.AG","math.GR","math.MP"],"title_canon_sha256":"485819e0bf11d9bd7dd8d7df294da0b41314d8dc98e2f2d60c7fc8f1cf41a555","abstract_canon_sha256":"cda92f2cf86e038a81298f04ea39dd09dce94b0d6d795ab9229e309d179dee20"},"schema_version":"1.0"},"canonical_sha256":"bc62153f0fbd55688d5dbe0d644ac72b976396ead1e2a157800f50726be625ac","source":{"kind":"arxiv","id":"1604.06062","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06062","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06062v1","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06062","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"pith_short_12","alias_value":"XRRBKPYPXVKW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XRRBKPYPXVKWRDK5","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XRRBKPYP","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XRRBKPYPXVKWRDK5XYGWISWHFO","target":"record","payload":{"canonical_record":{"source":{"id":"1604.06062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-20T19:03:41Z","cross_cats_sorted":["hep-th","math.AG","math.GR","math.MP"],"title_canon_sha256":"485819e0bf11d9bd7dd8d7df294da0b41314d8dc98e2f2d60c7fc8f1cf41a555","abstract_canon_sha256":"cda92f2cf86e038a81298f04ea39dd09dce94b0d6d795ab9229e309d179dee20"},"schema_version":"1.0"},"canonical_sha256":"bc62153f0fbd55688d5dbe0d644ac72b976396ead1e2a157800f50726be625ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:34.316579Z","signature_b64":"BcX+00hvGg95da8guQwMLolxnltaCLi2DpVTQ1nmLwDc5qNwHwU05plLmYDVlBTd+13EXkggZfsZPiqoLI45Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc62153f0fbd55688d5dbe0d644ac72b976396ead1e2a157800f50726be625ac","last_reissued_at":"2026-05-18T00:53:34.316102Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:34.316102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.06062","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":"2026-05-18T00:53:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xeFOzuMS3rD39MCKUtgIQdinQ1YbINwqPGR2XPFgIbd9Qpq4/Te8+JvT420VsMKKXaYr2sXVyUHY5sAi9DD0DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:08:56.532648Z"},"content_sha256":"2c0a8d28b5a114f554c4995a86e34a687dd5701b04b63649ec0e53c50465e1c7","schema_version":"1.0","event_id":"sha256:2c0a8d28b5a114f554c4995a86e34a687dd5701b04b63649ec0e53c50465e1c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XRRBKPYPXVKWRDK5XYGWISWHFO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.AG","math.GR","math.MP"],"primary_cat":"math-ph","authors_text":"H.M.Khudaverdian, R.L.Mkrtchyan","submitted_at":"2016-04-20T19:03:41Z","abstract_excerpt":"We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes \"regular polyhedrons\" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $\\chi=\\pm 2$. In the case of negative $m$ this equation corresponds to $\\chi=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06062","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":"2026-05-18T00:53:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GrfgLalY+vGRIstD7LNh5wVnhvrHJmEvY7TE+mpaixaM6cxmBLP2cJv+9UZQPWnBxmCp3dl70/or/lUkTto8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:08:56.533013Z"},"content_sha256":"b7db1c560362384b982ead113700c6f59b7735b3098fb0e3770875fcb1b0c8b6","schema_version":"1.0","event_id":"sha256:b7db1c560362384b982ead113700c6f59b7735b3098fb0e3770875fcb1b0c8b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/bundle.json","state_url":"https://pith.science/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/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":"2026-05-26T05:08:56Z","links":{"resolver":"https://pith.science/pith/XRRBKPYPXVKWRDK5XYGWISWHFO","bundle":"https://pith.science/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/bundle.json","state":"https://pith.science/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XRRBKPYPXVKWRDK5XYGWISWHFO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XRRBKPYPXVKWRDK5XYGWISWHFO","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":"cda92f2cf86e038a81298f04ea39dd09dce94b0d6d795ab9229e309d179dee20","cross_cats_sorted":["hep-th","math.AG","math.GR","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-20T19:03:41Z","title_canon_sha256":"485819e0bf11d9bd7dd8d7df294da0b41314d8dc98e2f2d60c7fc8f1cf41a555"},"schema_version":"1.0","source":{"id":"1604.06062","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06062","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06062v1","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06062","created_at":"2026-05-18T00:53:34Z"},{"alias_kind":"pith_short_12","alias_value":"XRRBKPYPXVKW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XRRBKPYPXVKWRDK5","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XRRBKPYP","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:b7db1c560362384b982ead113700c6f59b7735b3098fb0e3770875fcb1b0c8b6","target":"graph","created_at":"2026-05-18T00:53:34Z","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 notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes \"regular polyhedrons\" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $\\chi=\\pm 2$. In the case of negative $m$ this equation corresponds to $\\chi=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic ","authors_text":"H.M.Khudaverdian, R.L.Mkrtchyan","cross_cats":["hep-th","math.AG","math.GR","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-20T19:03:41Z","title":"Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel's universality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06062","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":"sha256:2c0a8d28b5a114f554c4995a86e34a687dd5701b04b63649ec0e53c50465e1c7","target":"record","created_at":"2026-05-18T00:53:34Z","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":"cda92f2cf86e038a81298f04ea39dd09dce94b0d6d795ab9229e309d179dee20","cross_cats_sorted":["hep-th","math.AG","math.GR","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-20T19:03:41Z","title_canon_sha256":"485819e0bf11d9bd7dd8d7df294da0b41314d8dc98e2f2d60c7fc8f1cf41a555"},"schema_version":"1.0","source":{"id":"1604.06062","kind":"arxiv","version":1}},"canonical_sha256":"bc62153f0fbd55688d5dbe0d644ac72b976396ead1e2a157800f50726be625ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc62153f0fbd55688d5dbe0d644ac72b976396ead1e2a157800f50726be625ac","first_computed_at":"2026-05-18T00:53:34.316102Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:34.316102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BcX+00hvGg95da8guQwMLolxnltaCLi2DpVTQ1nmLwDc5qNwHwU05plLmYDVlBTd+13EXkggZfsZPiqoLI45Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:34.316579Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.06062","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c0a8d28b5a114f554c4995a86e34a687dd5701b04b63649ec0e53c50465e1c7","sha256:b7db1c560362384b982ead113700c6f59b7735b3098fb0e3770875fcb1b0c8b6"],"state_sha256":"13e316816e48d5d5cc05eaab63bbbbdcd58fb0f8ac8e9730d20f41e46e902dd5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L6M66hAEua2vxWlQcpnHpeE4xyyZoA9yvUiX6JA/891NCjpHEGHf5Va06oNnltpwOJa/mrQmisgp4aLq3ZsjDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:08:56.535679Z","bundle_sha256":"036dfe3ec7426bc891c36c081b83a13f43ce6103622e90f5b257cd2e27db5ffa"}}