{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IGFPJQXDVQS57CQNYONCC7MZOT","short_pith_number":"pith:IGFPJQXD","canonical_record":{"source":{"id":"1503.01020","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-03T17:29:40Z","cross_cats_sorted":["math.CA","math.FA"],"title_canon_sha256":"daa52716d8c3b81757240a3473d8561e1d95d31d5e12883cac9b8969fbc12a7f","abstract_canon_sha256":"18056444c1655223f179b047e0fc99b23d1bf6a9483b01e2a69b119428ebc343"},"schema_version":"1.0"},"canonical_sha256":"418af4c2e3ac25df8a0dc39a217d9974cff03ea700b2265d3448b893ab9b0c07","source":{"kind":"arxiv","id":"1503.01020","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01020","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01020v2","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01020","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"pith_short_12","alias_value":"IGFPJQXDVQS5","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IGFPJQXDVQS57CQN","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IGFPJQXD","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IGFPJQXDVQS57CQNYONCC7MZOT","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01020","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-03T17:29:40Z","cross_cats_sorted":["math.CA","math.FA"],"title_canon_sha256":"daa52716d8c3b81757240a3473d8561e1d95d31d5e12883cac9b8969fbc12a7f","abstract_canon_sha256":"18056444c1655223f179b047e0fc99b23d1bf6a9483b01e2a69b119428ebc343"},"schema_version":"1.0"},"canonical_sha256":"418af4c2e3ac25df8a0dc39a217d9974cff03ea700b2265d3448b893ab9b0c07","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:33.093739Z","signature_b64":"Z00hoKU4b1bWavwaumKt/0yEQ9xv2HfGVAGx1Bk+w2Cc3gXDYUbHJl5G4BUVrF55cr2neuEmEoYkFgCBZSTeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"418af4c2e3ac25df8a0dc39a217d9974cff03ea700b2265d3448b893ab9b0c07","last_reissued_at":"2026-05-18T02:17:33.093143Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:33.093143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01020","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:17:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0K/gLJddFfbMxm46J7vZ0AYyziS+ApKtqRk+d1FEQOXkC2oXS9s2IydaGzzlnZFvpQjZifYnw7LLq2udLsIcCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T03:22:03.516697Z"},"content_sha256":"25d4fe962c312660e7a0c1b00d4d97082da7251f210c3e27c4920d3efca6d45c","schema_version":"1.0","event_id":"sha256:25d4fe962c312660e7a0c1b00d4d97082da7251f210c3e27c4920d3efca6d45c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IGFPJQXDVQS57CQNYONCC7MZOT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tangent lines and Lipschitz differentiability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.MG","authors_text":"Fabio Cavalletti, Tapio Rajala","submitted_at":"2015-03-03T17:29:40Z","abstract_excerpt":"We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line.\n  Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01020","kind":"arxiv","version":2},"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":"2026-05-18T02:17:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k3/3GYaK4sIuLirUd2e3/licu5Jv8D9sVya/3FbpzdDBBlhcfDlpAft9DNwJ0gF1P8QCeDlwnR4fVaM7DDubCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T03:22:03.517058Z"},"content_sha256":"4105e7febc64de65357765901378ed0c0c6f9afed595acbbc4ba29a3a8b6c570","schema_version":"1.0","event_id":"sha256:4105e7febc64de65357765901378ed0c0c6f9afed595acbbc4ba29a3a8b6c570"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IGFPJQXDVQS57CQNYONCC7MZOT/bundle.json","state_url":"https://pith.science/pith/IGFPJQXDVQS57CQNYONCC7MZOT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IGFPJQXDVQS57CQNYONCC7MZOT/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":"2026-05-25T03:22:03Z","links":{"resolver":"https://pith.science/pith/IGFPJQXDVQS57CQNYONCC7MZOT","bundle":"https://pith.science/pith/IGFPJQXDVQS57CQNYONCC7MZOT/bundle.json","state":"https://pith.science/pith/IGFPJQXDVQS57CQNYONCC7MZOT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IGFPJQXDVQS57CQNYONCC7MZOT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IGFPJQXDVQS57CQNYONCC7MZOT","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":"18056444c1655223f179b047e0fc99b23d1bf6a9483b01e2a69b119428ebc343","cross_cats_sorted":["math.CA","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-03T17:29:40Z","title_canon_sha256":"daa52716d8c3b81757240a3473d8561e1d95d31d5e12883cac9b8969fbc12a7f"},"schema_version":"1.0","source":{"id":"1503.01020","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01020","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01020v2","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01020","created_at":"2026-05-18T02:17:33Z"},{"alias_kind":"pith_short_12","alias_value":"IGFPJQXDVQS5","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IGFPJQXDVQS57CQN","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IGFPJQXD","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:4105e7febc64de65357765901378ed0c0c6f9afed595acbbc4ba29a3a8b6c570","target":"graph","created_at":"2026-05-18T02:17:33Z","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 the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line.\n  Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability spa","authors_text":"Fabio Cavalletti, Tapio Rajala","cross_cats":["math.CA","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-03T17:29:40Z","title":"Tangent lines and Lipschitz differentiability spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01020","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":"sha256:25d4fe962c312660e7a0c1b00d4d97082da7251f210c3e27c4920d3efca6d45c","target":"record","created_at":"2026-05-18T02:17:33Z","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":"18056444c1655223f179b047e0fc99b23d1bf6a9483b01e2a69b119428ebc343","cross_cats_sorted":["math.CA","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-03-03T17:29:40Z","title_canon_sha256":"daa52716d8c3b81757240a3473d8561e1d95d31d5e12883cac9b8969fbc12a7f"},"schema_version":"1.0","source":{"id":"1503.01020","kind":"arxiv","version":2}},"canonical_sha256":"418af4c2e3ac25df8a0dc39a217d9974cff03ea700b2265d3448b893ab9b0c07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"418af4c2e3ac25df8a0dc39a217d9974cff03ea700b2265d3448b893ab9b0c07","first_computed_at":"2026-05-18T02:17:33.093143Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:33.093143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z00hoKU4b1bWavwaumKt/0yEQ9xv2HfGVAGx1Bk+w2Cc3gXDYUbHJl5G4BUVrF55cr2neuEmEoYkFgCBZSTeCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:33.093739Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01020","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25d4fe962c312660e7a0c1b00d4d97082da7251f210c3e27c4920d3efca6d45c","sha256:4105e7febc64de65357765901378ed0c0c6f9afed595acbbc4ba29a3a8b6c570"],"state_sha256":"90f9f4521a1d74df1ca20ea88b4110e74546d963a03a0d3073e45959e21f9916"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I1MnyQFRMUX3dxKrR8C9hr7hTaCnGqe4zLqlJL48MVmTd+Ylohcklk4dOX3BMu+nKVJFDkVehPv0Z5ngW0cLDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T03:22:03.519611Z","bundle_sha256":"f42b9a1858a7159cf37a130c8c952402eb949de3ec2c8d2b2f9537611661c84d"}}