{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:URJWRPAHUAG5L7JOUAFSFIJBBB","short_pith_number":"pith:URJWRPAH","canonical_record":{"source":{"id":"1604.08579","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-28T19:53:45Z","cross_cats_sorted":["math.DG","math.GR"],"title_canon_sha256":"63785cef74ae4dea273f5772643d942c3b8e8dddc0fe9ec1f960cf5a0b5417cc","abstract_canon_sha256":"2640fad93f516f1d0a33994d0858296227b73fa7a10881ee329cef359f36df90"},"schema_version":"1.0"},"canonical_sha256":"a45368bc07a00dd5fd2ea00b22a12108600ef5827a058b1e703bea17b1b1fc80","source":{"kind":"arxiv","id":"1604.08579","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08579","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08579v1","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08579","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"URJWRPAHUAG5","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"URJWRPAHUAG5L7JO","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"URJWRPAH","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:URJWRPAHUAG5L7JOUAFSFIJBBB","target":"record","payload":{"canonical_record":{"source":{"id":"1604.08579","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-28T19:53:45Z","cross_cats_sorted":["math.DG","math.GR"],"title_canon_sha256":"63785cef74ae4dea273f5772643d942c3b8e8dddc0fe9ec1f960cf5a0b5417cc","abstract_canon_sha256":"2640fad93f516f1d0a33994d0858296227b73fa7a10881ee329cef359f36df90"},"schema_version":"1.0"},"canonical_sha256":"a45368bc07a00dd5fd2ea00b22a12108600ef5827a058b1e703bea17b1b1fc80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:03.631519Z","signature_b64":"jiHG1vrwYJDFyk0hVtnSUbXnpnJ9UwcCQWIu5PKdecPyWRikYIV+un441Ntb+jOic+04DvItoluGH+nh6mLEBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a45368bc07a00dd5fd2ea00b22a12108600ef5827a058b1e703bea17b1b1fc80","last_reissued_at":"2026-05-18T01:16:03.630886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:03.630886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.08579","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":"2026-05-18T01:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n3jshExqaLou++LxHdh4GpZAAkK8aSyiBYrFrBpW5x2uqFWUPpDDM3885rvdP2Ix5xE31W6LPdb3bcq3ZDISBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T14:33:09.762068Z"},"content_sha256":"1788d42086796464f2f32ee815b98ce692dec890efc8dd1c95d086496017223c","schema_version":"1.0","event_id":"sha256:1788d42086796464f2f32ee815b98ce692dec890efc8dd1c95d086496017223c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:URJWRPAHUAG5L7JOUAFSFIJBBB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A primer on Carnot groups: homogenous groups, CC spaces, and regularity of their isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GR"],"primary_cat":"math.MG","authors_text":"Enrico Le Donne","submitted_at":"2016-04-28T19:53:45Z","abstract_excerpt":"Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Caratheodory spaces and of nilpotent metric Lie groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08579","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":"2026-05-18T01:16:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4qAf7ZJb5E/JzX9aWAYCDVn3ft9wg5y1wH1ViKhKtxClQ426Or0E7SzHcXQeT3RnuPFAKTESoaTbEr9+0G4eAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T14:33:09.762429Z"},"content_sha256":"30ad3c71cae716d42abef1b551ef7300f032490e5e75246d7de66fca66e7c337","schema_version":"1.0","event_id":"sha256:30ad3c71cae716d42abef1b551ef7300f032490e5e75246d7de66fca66e7c337"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/bundle.json","state_url":"https://pith.science/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/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":"2026-07-03T14:33:09Z","links":{"resolver":"https://pith.science/pith/URJWRPAHUAG5L7JOUAFSFIJBBB","bundle":"https://pith.science/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/bundle.json","state":"https://pith.science/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/URJWRPAHUAG5L7JOUAFSFIJBBB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:URJWRPAHUAG5L7JOUAFSFIJBBB","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":"2640fad93f516f1d0a33994d0858296227b73fa7a10881ee329cef359f36df90","cross_cats_sorted":["math.DG","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-28T19:53:45Z","title_canon_sha256":"63785cef74ae4dea273f5772643d942c3b8e8dddc0fe9ec1f960cf5a0b5417cc"},"schema_version":"1.0","source":{"id":"1604.08579","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.08579","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"arxiv_version","alias_value":"1604.08579v1","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.08579","created_at":"2026-05-18T01:16:03Z"},{"alias_kind":"pith_short_12","alias_value":"URJWRPAHUAG5","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"URJWRPAHUAG5L7JO","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"URJWRPAH","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:30ad3c71cae716d42abef1b551ef7300f032490e5e75246d7de66fca66e7c337","target":"graph","created_at":"2026-05-18T01:16:03Z","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":"Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Caratheodory spaces and of nilpotent metric Lie groups.","authors_text":"Enrico Le Donne","cross_cats":["math.DG","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-28T19:53:45Z","title":"A primer on Carnot groups: homogenous groups, CC spaces, and regularity of their isometries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08579","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":"sha256:1788d42086796464f2f32ee815b98ce692dec890efc8dd1c95d086496017223c","target":"record","created_at":"2026-05-18T01:16:03Z","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":"2640fad93f516f1d0a33994d0858296227b73fa7a10881ee329cef359f36df90","cross_cats_sorted":["math.DG","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-04-28T19:53:45Z","title_canon_sha256":"63785cef74ae4dea273f5772643d942c3b8e8dddc0fe9ec1f960cf5a0b5417cc"},"schema_version":"1.0","source":{"id":"1604.08579","kind":"arxiv","version":1}},"canonical_sha256":"a45368bc07a00dd5fd2ea00b22a12108600ef5827a058b1e703bea17b1b1fc80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a45368bc07a00dd5fd2ea00b22a12108600ef5827a058b1e703bea17b1b1fc80","first_computed_at":"2026-05-18T01:16:03.630886Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:03.630886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jiHG1vrwYJDFyk0hVtnSUbXnpnJ9UwcCQWIu5PKdecPyWRikYIV+un441Ntb+jOic+04DvItoluGH+nh6mLEBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:03.631519Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.08579","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1788d42086796464f2f32ee815b98ce692dec890efc8dd1c95d086496017223c","sha256:30ad3c71cae716d42abef1b551ef7300f032490e5e75246d7de66fca66e7c337"],"state_sha256":"8a953a5145e2cc94bb235ea32bfcce5eab1fc0d9226ed6c641ffc8c469cc4908"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zLm3TrjvUCrbMtMlE4kou5LsM3SjaUKA0FK9x9I7abvLW5rJbdzRkKE/YOL7nEKMV4RbNbt60wooQPbFLFj9Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T14:33:09.764347Z","bundle_sha256":"be5e015a6af23f48054607e0965bad739d29a3e79fb2804c446da575597a0693"}}