{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Q7VXSIP2S7BWKUYOIM5VMTZYXK","short_pith_number":"pith:Q7VXSIP2","canonical_record":{"source":{"id":"1509.02104","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-07T16:07:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a8f06e030342b1cb7f4459ddc5a098fccb2f036a7ccc040362be963e410d54fe","abstract_canon_sha256":"041c163660f6989223495c492ffc9e263e88535746c261946d710cb60d68b1ca"},"schema_version":"1.0"},"canonical_sha256":"87eb7921fa97c365530e433b564f38ba9a909f1b49293508e8fe51ab5e5a4b2d","source":{"kind":"arxiv","id":"1509.02104","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02104","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02104v4","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02104","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"pith_short_12","alias_value":"Q7VXSIP2S7BW","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q7VXSIP2S7BWKUYO","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q7VXSIP2","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Q7VXSIP2S7BWKUYOIM5VMTZYXK","target":"record","payload":{"canonical_record":{"source":{"id":"1509.02104","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-07T16:07:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a8f06e030342b1cb7f4459ddc5a098fccb2f036a7ccc040362be963e410d54fe","abstract_canon_sha256":"041c163660f6989223495c492ffc9e263e88535746c261946d710cb60d68b1ca"},"schema_version":"1.0"},"canonical_sha256":"87eb7921fa97c365530e433b564f38ba9a909f1b49293508e8fe51ab5e5a4b2d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:14.250616Z","signature_b64":"TdZ00/YYJtVQX8vWZHRBi7d8duP0mDuyRLw16gpwCKXRUbruxJHzFYAGzVSI3X+JTGiI38Fesvcsbd/7D7bpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87eb7921fa97c365530e433b564f38ba9a909f1b49293508e8fe51ab5e5a4b2d","last_reissued_at":"2026-05-18T01:24:14.249930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:14.249930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.02104","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bNh3DrIjcYvz9x0efFKYI/utiHuDx9Ihs51nFvQocOCkoJB9anheKo32L1F9Yf19RZQJKkjn2CvtxZHRiNkrBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T08:08:02.148046Z"},"content_sha256":"b2b6c2c98ee7dfe5f1eddcd5beb6689d32bfaed1d401497056d942d2b72cba76","schema_version":"1.0","event_id":"sha256:b2b6c2c98ee7dfe5f1eddcd5beb6689d32bfaed1d401497056d942d2b72cba76"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Q7VXSIP2S7BWKUYOIM5VMTZYXK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Power graphs of (non)orientable genus two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Gary L. Walls, Kaishun Wang, Xuanlong Ma","submitted_at":"2015-09-07T16:07:43Z","abstract_excerpt":"The power graph $\\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02104","kind":"arxiv","version":4},"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":"2026-05-18T01:24:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NlBJEEg594XzEsV7blnCQOT1tupgg8CTiwxNQv1CeyIoCIFdPruHZvd6a01euH/GRpxzIjL0AxCbIMXwZh8EBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T08:08:02.148408Z"},"content_sha256":"6a89a0fdb81ff29581e5375ffd4f190584a0b0d00d3eaba82193dc949b6c83a7","schema_version":"1.0","event_id":"sha256:6a89a0fdb81ff29581e5375ffd4f190584a0b0d00d3eaba82193dc949b6c83a7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/bundle.json","state_url":"https://pith.science/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/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":"2026-06-04T08:08:02Z","links":{"resolver":"https://pith.science/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK","bundle":"https://pith.science/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/bundle.json","state":"https://pith.science/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q7VXSIP2S7BWKUYOIM5VMTZYXK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Q7VXSIP2S7BWKUYOIM5VMTZYXK","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":"041c163660f6989223495c492ffc9e263e88535746c261946d710cb60d68b1ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-07T16:07:43Z","title_canon_sha256":"a8f06e030342b1cb7f4459ddc5a098fccb2f036a7ccc040362be963e410d54fe"},"schema_version":"1.0","source":{"id":"1509.02104","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02104","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02104v4","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02104","created_at":"2026-05-18T01:24:14Z"},{"alias_kind":"pith_short_12","alias_value":"Q7VXSIP2S7BW","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q7VXSIP2S7BWKUYO","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q7VXSIP2","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:6a89a0fdb81ff29581e5375ffd4f190584a0b0d00d3eaba82193dc949b6c83a7","target":"graph","created_at":"2026-05-18T01:24:14Z","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":"The power graph $\\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two.","authors_text":"Gary L. Walls, Kaishun Wang, Xuanlong Ma","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-07T16:07:43Z","title":"Power graphs of (non)orientable genus two"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02104","kind":"arxiv","version":4},"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:b2b6c2c98ee7dfe5f1eddcd5beb6689d32bfaed1d401497056d942d2b72cba76","target":"record","created_at":"2026-05-18T01:24:14Z","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":"041c163660f6989223495c492ffc9e263e88535746c261946d710cb60d68b1ca","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-07T16:07:43Z","title_canon_sha256":"a8f06e030342b1cb7f4459ddc5a098fccb2f036a7ccc040362be963e410d54fe"},"schema_version":"1.0","source":{"id":"1509.02104","kind":"arxiv","version":4}},"canonical_sha256":"87eb7921fa97c365530e433b564f38ba9a909f1b49293508e8fe51ab5e5a4b2d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87eb7921fa97c365530e433b564f38ba9a909f1b49293508e8fe51ab5e5a4b2d","first_computed_at":"2026-05-18T01:24:14.249930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:14.249930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TdZ00/YYJtVQX8vWZHRBi7d8duP0mDuyRLw16gpwCKXRUbruxJHzFYAGzVSI3X+JTGiI38Fesvcsbd/7D7bpAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:14.250616Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.02104","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b2b6c2c98ee7dfe5f1eddcd5beb6689d32bfaed1d401497056d942d2b72cba76","sha256:6a89a0fdb81ff29581e5375ffd4f190584a0b0d00d3eaba82193dc949b6c83a7"],"state_sha256":"3df7fe3e9f3f2a6999acc3f39bce653b6148774693e0a3ef4d3513e5e1e09aa9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2oe+714JTP8vMebqSm67jnpqG45B688GUGvnju112MIbU8ke48KoKblw9GHf6RNK/H1lhrj5LEU//DYVhbhsBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T08:08:02.150726Z","bundle_sha256":"50c53004eee7a6ad4f27e55764ca9492882c97aedd92a68ebf42b33d697d6488"}}