{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:CYNXYR5VSJV7YBRMH4XTIPSCSE","short_pith_number":"pith:CYNXYR5V","canonical_record":{"source":{"id":"1512.05948","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-18T13:38:15Z","cross_cats_sorted":["math.DS","math.GR"],"title_canon_sha256":"2f09ffc1fe5008128854002bc18de09b92f4b6c5835c20a7b9f507b490445bcd","abstract_canon_sha256":"a63058b8553a6c5aef6b07d76c0339fb7f8b73dd44e932306f03074e68ed3ab5"},"schema_version":"1.0"},"canonical_sha256":"161b7c47b5926bfc062c3f2f343e42913b5ccbd88844361ba60c3eb13957b3e5","source":{"kind":"arxiv","id":"1512.05948","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05948","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05948v3","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05948","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"CYNXYR5VSJV7","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CYNXYR5VSJV7YBRM","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CYNXYR5V","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:CYNXYR5VSJV7YBRMH4XTIPSCSE","target":"record","payload":{"canonical_record":{"source":{"id":"1512.05948","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-18T13:38:15Z","cross_cats_sorted":["math.DS","math.GR"],"title_canon_sha256":"2f09ffc1fe5008128854002bc18de09b92f4b6c5835c20a7b9f507b490445bcd","abstract_canon_sha256":"a63058b8553a6c5aef6b07d76c0339fb7f8b73dd44e932306f03074e68ed3ab5"},"schema_version":"1.0"},"canonical_sha256":"161b7c47b5926bfc062c3f2f343e42913b5ccbd88844361ba60c3eb13957b3e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:13.340666Z","signature_b64":"NbxNgS09fIg7cfFPYU330Z+qJTHIXjLtd86HNNvJERBPVp8Dl1wsX2w3hNdj4rcJEPfaDT8YhYsX9q0LhU0kCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"161b7c47b5926bfc062c3f2f343e42913b5ccbd88844361ba60c3eb13957b3e5","last_reissued_at":"2026-05-18T00:42:13.339978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:13.339978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.05948","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1U29Jg6wfW5cOvpzvtOEjGCXVZZn1C4VqJKuiCPpho3KzV7Ace0Lfb/mYFMJq9J365r35xLRV5JMK2LYnWFuDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:27:15.162427Z"},"content_sha256":"16a9a7c5406611ebbdf53c6cdfa430f07e9e8dd5088b6d8b58d789a07aaa9bde","schema_version":"1.0","event_id":"sha256:16a9a7c5406611ebbdf53c6cdfa430f07e9e8dd5088b6d8b58d789a07aaa9bde"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:CYNXYR5VSJV7YBRMH4XTIPSCSE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algorithmic aspects of branched coverings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR"],"primary_cat":"cs.CC","authors_text":"Dzmitry Dudko, Laurent Bartholdi","submitted_at":"2015-12-18T13:38:15Z","abstract_excerpt":"This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets.\n  We introduce a canonical \"Levy\" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps.\n  As an outcome, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05948","kind":"arxiv","version":3},"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":"2026-05-18T00:42:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HN/TOTM7BBcQZdPq1Pc1lE7muFnSoPqcSYiL4jrrGbJDXGxACpe2eZali0X3tOXMsqv5XJ2h5+01hQAw50VzBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:27:15.163011Z"},"content_sha256":"c7646e56366a752e5815f8c04d7989dacc62ca572c7494906b2f2b8bc4772393","schema_version":"1.0","event_id":"sha256:c7646e56366a752e5815f8c04d7989dacc62ca572c7494906b2f2b8bc4772393"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/bundle.json","state_url":"https://pith.science/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/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":"2026-06-02T20:27:15Z","links":{"resolver":"https://pith.science/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE","bundle":"https://pith.science/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/bundle.json","state":"https://pith.science/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CYNXYR5VSJV7YBRMH4XTIPSCSE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CYNXYR5VSJV7YBRMH4XTIPSCSE","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":"a63058b8553a6c5aef6b07d76c0339fb7f8b73dd44e932306f03074e68ed3ab5","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-18T13:38:15Z","title_canon_sha256":"2f09ffc1fe5008128854002bc18de09b92f4b6c5835c20a7b9f507b490445bcd"},"schema_version":"1.0","source":{"id":"1512.05948","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05948","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05948v3","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05948","created_at":"2026-05-18T00:42:13Z"},{"alias_kind":"pith_short_12","alias_value":"CYNXYR5VSJV7","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CYNXYR5VSJV7YBRM","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CYNXYR5V","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:c7646e56366a752e5815f8c04d7989dacc62ca572c7494906b2f2b8bc4772393","target":"graph","created_at":"2026-05-18T00:42:13Z","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":"This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets.\n  We introduce a canonical \"Levy\" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps.\n  As an outcome, we","authors_text":"Dzmitry Dudko, Laurent Bartholdi","cross_cats":["math.DS","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-18T13:38:15Z","title":"Algorithmic aspects of branched coverings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05948","kind":"arxiv","version":3},"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":"sha256:16a9a7c5406611ebbdf53c6cdfa430f07e9e8dd5088b6d8b58d789a07aaa9bde","target":"record","created_at":"2026-05-18T00:42:13Z","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":"a63058b8553a6c5aef6b07d76c0339fb7f8b73dd44e932306f03074e68ed3ab5","cross_cats_sorted":["math.DS","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-18T13:38:15Z","title_canon_sha256":"2f09ffc1fe5008128854002bc18de09b92f4b6c5835c20a7b9f507b490445bcd"},"schema_version":"1.0","source":{"id":"1512.05948","kind":"arxiv","version":3}},"canonical_sha256":"161b7c47b5926bfc062c3f2f343e42913b5ccbd88844361ba60c3eb13957b3e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"161b7c47b5926bfc062c3f2f343e42913b5ccbd88844361ba60c3eb13957b3e5","first_computed_at":"2026-05-18T00:42:13.339978Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:13.339978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NbxNgS09fIg7cfFPYU330Z+qJTHIXjLtd86HNNvJERBPVp8Dl1wsX2w3hNdj4rcJEPfaDT8YhYsX9q0LhU0kCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:13.340666Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.05948","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16a9a7c5406611ebbdf53c6cdfa430f07e9e8dd5088b6d8b58d789a07aaa9bde","sha256:c7646e56366a752e5815f8c04d7989dacc62ca572c7494906b2f2b8bc4772393"],"state_sha256":"843f3d8cee16d2530a1ac2f81ab314cfa524fe80930a154ccab44c273fa52a45"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cMG89zzojVXwtn+rZG0ckXfHT7C/68QqfZyxIEPhgAZ4rUUz/hY6hu24ckI3vg0QA8kJGCw/XetWF1BISfrCBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:27:15.166512Z","bundle_sha256":"710c21ec2c5d684f1bee54d0b79a73583c3cfca41a70277b5b90229b222e2204"}}