{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:YKMZ45VS2T3RSW5V3BLGIS5HIF","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":"06c12d45f549521177c8e757da9dfef622d4c98fa5f51c2ce37920717ca0fbdc","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2005-07-26T14:25:41Z","title_canon_sha256":"b22e8285e96f4602e3235c7f45d10c5a30f75462d801476d191e43a885e99f0c"},"schema_version":"1.0","source":{"id":"math/0507536","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0507536","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"math/0507536v2","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507536","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"YKMZ45VS2T3R","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"YKMZ45VS2T3RSW5V","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"YKMZ45VS","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:d545c5b772f2889106f731e0fdeefbd96967f714363cfeaabd44203d984bde17","target":"graph","created_at":"2026-05-18T03:26:32Z","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 introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths,","authors_text":"Ben Hambly, Terry Lyons","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2005-07-26T14:25:41Z","title":"Uniqueness for the signature of a path of bounded variation and the reduced path group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507536","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":"sha256:0ae060c941a94ff0640b8d3e0e2517bc0f5f1ba26d22470ac9ec3a15db0a20ad","target":"record","created_at":"2026-05-18T03:26:32Z","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":"06c12d45f549521177c8e757da9dfef622d4c98fa5f51c2ce37920717ca0fbdc","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2005-07-26T14:25:41Z","title_canon_sha256":"b22e8285e96f4602e3235c7f45d10c5a30f75462d801476d191e43a885e99f0c"},"schema_version":"1.0","source":{"id":"math/0507536","kind":"arxiv","version":2}},"canonical_sha256":"c2999e76b2d4f7195bb5d856644ba7416f30c1dfd765bac7f177b61821b85ef3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2999e76b2d4f7195bb5d856644ba7416f30c1dfd765bac7f177b61821b85ef3","first_computed_at":"2026-05-18T03:26:32.842240Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.842240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qql7iqWCtCue8aA7IO/+R4sGDyvhMqEyHusATL8mEAD4e96LRo/3wBuRWGyJAG6t4k3TlaLBjsOOLO7n0OtmAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.842610Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0507536","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ae060c941a94ff0640b8d3e0e2517bc0f5f1ba26d22470ac9ec3a15db0a20ad","sha256:d545c5b772f2889106f731e0fdeefbd96967f714363cfeaabd44203d984bde17"],"state_sha256":"d7f138f2d8bd4e58d1babf472bb2add56f53df973343477182b3640d0f17fcfb"}