{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HON7BTEP455VOYERRGPBQ4QYR7","short_pith_number":"pith:HON7BTEP","canonical_record":{"source":{"id":"1604.02159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-04-07T20:05:04Z","cross_cats_sorted":[],"title_canon_sha256":"a01196cc791b73402e3d437be73be8dbfc60d56bbefc05ce29ba33b90fcf7879","abstract_canon_sha256":"f880fcf1905339f6d9943bd688bca05b613c431c32e396059d3301c198cedfcc"},"schema_version":"1.0"},"canonical_sha256":"3b9bf0cc8fe77b576091899e1872188fd28d79381b6285f6558f0b255fb44fbc","source":{"kind":"arxiv","id":"1604.02159","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02159","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02159v1","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02159","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"pith_short_12","alias_value":"HON7BTEP455V","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HON7BTEP455VOYER","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HON7BTEP","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HON7BTEP455VOYERRGPBQ4QYR7","target":"record","payload":{"canonical_record":{"source":{"id":"1604.02159","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-04-07T20:05:04Z","cross_cats_sorted":[],"title_canon_sha256":"a01196cc791b73402e3d437be73be8dbfc60d56bbefc05ce29ba33b90fcf7879","abstract_canon_sha256":"f880fcf1905339f6d9943bd688bca05b613c431c32e396059d3301c198cedfcc"},"schema_version":"1.0"},"canonical_sha256":"3b9bf0cc8fe77b576091899e1872188fd28d79381b6285f6558f0b255fb44fbc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:28.306291Z","signature_b64":"gnwYmwETTsCTSWz5DY94SRDo0w7S5yIKTAK06lsmmn1o0ogHSJET4yCb8t/b3wyYk4Z2d6b6xSBPIaku5nTVCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b9bf0cc8fe77b576091899e1872188fd28d79381b6285f6558f0b255fb44fbc","last_reissued_at":"2026-05-18T01:17:28.305687Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:28.305687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.02159","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":"2026-05-18T01:17:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VNms/Tgx0zx6F4yfZD1dStnDEe1pJVGyKjvRcjMcgnGjBQAOe7AHaytYsD83QJa2pGzPqR+BgxaJgpCt6kCoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:44:04.466032Z"},"content_sha256":"0362c8458859526e00a4d736d42bf53f5a86b5df3a3f69a00390da6d5fcd6955","schema_version":"1.0","event_id":"sha256:0362c8458859526e00a4d736d42bf53f5a86b5df3a3f69a00390da6d5fcd6955"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HON7BTEP455VOYERRGPBQ4QYR7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Involution and commutator length for complex hyperbolic isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Julien Paupert, Pierre Will","submitted_at":"2016-04-07T20:05:04Z","abstract_excerpt":"We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \\geqslant 3$ PU($n$,1) has involution length at most 8."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02159","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":"2026-05-18T01:17:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Eun/eCXT9wOncKok80m7lkXuQsK3At9RJ2Yz4kpcLe+ryuS4w+/ubKhRLjHFTLPhbAbO5I5OfzsC3Ovp48ZmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:44:04.466381Z"},"content_sha256":"96ba2060f033c469462eebf78c189146ceae77e329b62c26ebf895795f0a2328","schema_version":"1.0","event_id":"sha256:96ba2060f033c469462eebf78c189146ceae77e329b62c26ebf895795f0a2328"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HON7BTEP455VOYERRGPBQ4QYR7/bundle.json","state_url":"https://pith.science/pith/HON7BTEP455VOYERRGPBQ4QYR7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HON7BTEP455VOYERRGPBQ4QYR7/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":"2026-06-27T13:44:04Z","links":{"resolver":"https://pith.science/pith/HON7BTEP455VOYERRGPBQ4QYR7","bundle":"https://pith.science/pith/HON7BTEP455VOYERRGPBQ4QYR7/bundle.json","state":"https://pith.science/pith/HON7BTEP455VOYERRGPBQ4QYR7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HON7BTEP455VOYERRGPBQ4QYR7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HON7BTEP455VOYERRGPBQ4QYR7","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":"f880fcf1905339f6d9943bd688bca05b613c431c32e396059d3301c198cedfcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-04-07T20:05:04Z","title_canon_sha256":"a01196cc791b73402e3d437be73be8dbfc60d56bbefc05ce29ba33b90fcf7879"},"schema_version":"1.0","source":{"id":"1604.02159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.02159","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"arxiv_version","alias_value":"1604.02159v1","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.02159","created_at":"2026-05-18T01:17:28Z"},{"alias_kind":"pith_short_12","alias_value":"HON7BTEP455V","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HON7BTEP455VOYER","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HON7BTEP","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:96ba2060f033c469462eebf78c189146ceae77e329b62c26ebf895795f0a2328","target":"graph","created_at":"2026-05-18T01:17:28Z","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 study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \\geqslant 3$ PU($n$,1) has involution length at most 8.","authors_text":"Julien Paupert, Pierre Will","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-04-07T20:05:04Z","title":"Involution and commutator length for complex hyperbolic isometries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.02159","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":"sha256:0362c8458859526e00a4d736d42bf53f5a86b5df3a3f69a00390da6d5fcd6955","target":"record","created_at":"2026-05-18T01:17:28Z","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":"f880fcf1905339f6d9943bd688bca05b613c431c32e396059d3301c198cedfcc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-04-07T20:05:04Z","title_canon_sha256":"a01196cc791b73402e3d437be73be8dbfc60d56bbefc05ce29ba33b90fcf7879"},"schema_version":"1.0","source":{"id":"1604.02159","kind":"arxiv","version":1}},"canonical_sha256":"3b9bf0cc8fe77b576091899e1872188fd28d79381b6285f6558f0b255fb44fbc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b9bf0cc8fe77b576091899e1872188fd28d79381b6285f6558f0b255fb44fbc","first_computed_at":"2026-05-18T01:17:28.305687Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:28.305687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gnwYmwETTsCTSWz5DY94SRDo0w7S5yIKTAK06lsmmn1o0ogHSJET4yCb8t/b3wyYk4Z2d6b6xSBPIaku5nTVCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:28.306291Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.02159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0362c8458859526e00a4d736d42bf53f5a86b5df3a3f69a00390da6d5fcd6955","sha256:96ba2060f033c469462eebf78c189146ceae77e329b62c26ebf895795f0a2328"],"state_sha256":"44d49c65530d8304b82f648a1d201aad9945060e322ab2d348c27df0134233ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CtJzv+wvKRnm1sx5rWCHuORYqJRvVHVok5lzXsBQL4eI1qEUBNnqwsDvrTw5VmvhCgSB1Ex2r4duj3AjmeXuCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T13:44:04.468208Z","bundle_sha256":"d5fe4491836ec5a639281785ad90c9ba8b1f2632ceebd529baf99bf53477d680"}}