{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FK3EVSX7HFBXY4PKLX64SV45F7","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":"bd68ff749e011f5ba6cd164f482988cea453417c12dab4eaf1189f02ab7e393a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-29T14:27:09Z","title_canon_sha256":"84aaa28d89f6a067bc25120aafbd3a3825df3e30e94744c5f11bf66535c40a5b"},"schema_version":"1.0","source":{"id":"1611.09645","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.09645","created_at":"2026-05-18T00:56:17Z"},{"alias_kind":"arxiv_version","alias_value":"1611.09645v1","created_at":"2026-05-18T00:56:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.09645","created_at":"2026-05-18T00:56:17Z"},{"alias_kind":"pith_short_12","alias_value":"FK3EVSX7HFBX","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FK3EVSX7HFBXY4PK","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FK3EVSX7","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:618eb894aff763bd08a2e481bc9a9ce2c422a341f7ea8e1bb0ad9afcf42b926a","target":"graph","created_at":"2026-05-18T00:56:17Z","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 prove that for a suitable class of metric measure spaces, the abstract notion of tangent module as defined by the first author can be isometrically identified with the space of $L^2$-sections of the `Gromov-Hausdorff tangent bundle'.\n  The class of spaces $({\\rm X},{\\sf d},{\\mathfrak m})$ we consider are PI spaces that for every $\\varepsilon>0$ admit a countable collection of Borel sets $(U_i)$ covering ${\\mathfrak m}$-a.e.\\ ${\\rm X}$ and corresponding $(1+\\varepsilon)$-biLipschitz maps $\\varphi_i:U_i\\to{\\mathbb R}^{k_i}$ such that $(\\varphi_i)_*{\\mathfrak m}\\lower3pt\\hbox{$|_{U_i}$}\\ll\\mat","authors_text":"Enrico Pasqualetto, Nicola Gigli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-29T14:27:09Z","title":"Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09645","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:fd40db2b02d9a1b50bb9b17dd591957ec6ac8df8b6c4c9be503d6ed111951933","target":"record","created_at":"2026-05-18T00:56:17Z","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":"bd68ff749e011f5ba6cd164f482988cea453417c12dab4eaf1189f02ab7e393a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-11-29T14:27:09Z","title_canon_sha256":"84aaa28d89f6a067bc25120aafbd3a3825df3e30e94744c5f11bf66535c40a5b"},"schema_version":"1.0","source":{"id":"1611.09645","kind":"arxiv","version":1}},"canonical_sha256":"2ab64acaff39437c71ea5dfdc9579d2fd4a0bfec0361345a378ef4c0ffc9ed64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ab64acaff39437c71ea5dfdc9579d2fd4a0bfec0361345a378ef4c0ffc9ed64","first_computed_at":"2026-05-18T00:56:17.814739Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:17.814739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sJbGDSmPp3ef/56pLiaynGWdcecvQrqNiMl4s7QgnlqqmlVJkesS61pu2fckKju2y+Jhp2DCehutrKx4AlyBDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:17.815443Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.09645","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd40db2b02d9a1b50bb9b17dd591957ec6ac8df8b6c4c9be503d6ed111951933","sha256:618eb894aff763bd08a2e481bc9a9ce2c422a341f7ea8e1bb0ad9afcf42b926a"],"state_sha256":"b5a07695cfaae933077c8a8ef4da1c38f36c779b7cd329240d1f0411ee99f517"}