{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:JXIOVH3KB5KQZVTDGNIMJUBUBV","short_pith_number":"pith:JXIOVH3K","canonical_record":{"source":{"id":"1608.01915","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T15:23:58Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2ae08075195d22f722a626d59b8ba29ab136b1e8d553173a2de9b3a79db2dd71","abstract_canon_sha256":"273180b4630d26b3f5198bdb65340ef6a57301765660ae93c1d4a8730229b7a3"},"schema_version":"1.0"},"canonical_sha256":"4dd0ea9f6a0f550cd6633350c4d0340d5ea4b15a46c1391a3b7c12f03f3e6ed6","source":{"kind":"arxiv","id":"1608.01915","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01915","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01915v1","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01915","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"JXIOVH3KB5KQ","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JXIOVH3KB5KQZVTD","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JXIOVH3K","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:JXIOVH3KB5KQZVTDGNIMJUBUBV","target":"record","payload":{"canonical_record":{"source":{"id":"1608.01915","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T15:23:58Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"2ae08075195d22f722a626d59b8ba29ab136b1e8d553173a2de9b3a79db2dd71","abstract_canon_sha256":"273180b4630d26b3f5198bdb65340ef6a57301765660ae93c1d4a8730229b7a3"},"schema_version":"1.0"},"canonical_sha256":"4dd0ea9f6a0f550cd6633350c4d0340d5ea4b15a46c1391a3b7c12f03f3e6ed6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:44.275818Z","signature_b64":"MNNRFwP045OWpdzXudyA/lsUG/NgY0LD6aZjrvhUi+ln0bT33sE/NQnJFaLbDpJmzEQIUT0+MXV4p2MQHw5dDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4dd0ea9f6a0f550cd6633350c4d0340d5ea4b15a46c1391a3b7c12f03f3e6ed6","last_reissued_at":"2026-05-18T01:09:44.275254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:44.275254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.01915","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":"2026-05-18T01:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V44+FHtHMul81tudoPd/sei89zvTTlrjN3OBQkOMmTiTj9QYcfrEeFZ/y9sBG9KPJSegK9EtWqxRFEgayUCzCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T13:53:06.477202Z"},"content_sha256":"4d9028d7457911e264feea7043ff3bff1b358e58cf6c0b52855e68c9cfff5b65","schema_version":"1.0","event_id":"sha256:4d9028d7457911e264feea7043ff3bff1b358e58cf6c0b52855e68c9cfff5b65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:JXIOVH3KB5KQZVTDGNIMJUBUBV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Heat flow and quantitative differentiation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Assaf Naor, Tuomas Hyt\\\"onen","submitted_at":"2016-08-05T15:23:58Z","abstract_excerpt":"For every Banach space $(Y,\\|\\cdot\\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\\in (0,\\infty)$ with the following property. Suppose that $n\\in \\mathbb{N}$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\\to Y$ and for every $\\varepsilon\\in (0,1/2]$ there exists a radius $r\\ge\\exp(-1/\\varepsilon^{cn})$, a point $x\\in B_X$ with $x+rB_X\\subset B_X$, and an affine mapping $\\Lambda:X\\to Y$ such that $\\|f(y)-\\Lambda(y)\\|_Y\\le \\varepsilon r$ for every $y\\in x+rB_X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01915","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":"2026-05-18T01:09:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ymq6hlwFcuTCeidjL4HLETxrOzTsHneV3lbq/xU31ewz75QuCVDennnXcaippOxSaKOFqkFyxO4N4tLYxhNBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T13:53:06.477607Z"},"content_sha256":"d51d771ef40d79fbc6bba460d4268e1dd7264dea1a657978f4db3a0abb25350d","schema_version":"1.0","event_id":"sha256:d51d771ef40d79fbc6bba460d4268e1dd7264dea1a657978f4db3a0abb25350d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/bundle.json","state_url":"https://pith.science/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/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":"2026-05-28T13:53:06Z","links":{"resolver":"https://pith.science/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV","bundle":"https://pith.science/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/bundle.json","state":"https://pith.science/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JXIOVH3KB5KQZVTDGNIMJUBUBV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:JXIOVH3KB5KQZVTDGNIMJUBUBV","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":"273180b4630d26b3f5198bdb65340ef6a57301765660ae93c1d4a8730229b7a3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T15:23:58Z","title_canon_sha256":"2ae08075195d22f722a626d59b8ba29ab136b1e8d553173a2de9b3a79db2dd71"},"schema_version":"1.0","source":{"id":"1608.01915","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.01915","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"arxiv_version","alias_value":"1608.01915v1","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01915","created_at":"2026-05-18T01:09:44Z"},{"alias_kind":"pith_short_12","alias_value":"JXIOVH3KB5KQ","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"JXIOVH3KB5KQZVTD","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"JXIOVH3K","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:d51d771ef40d79fbc6bba460d4268e1dd7264dea1a657978f4db3a0abb25350d","target":"graph","created_at":"2026-05-18T01:09:44Z","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":"For every Banach space $(Y,\\|\\cdot\\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\\in (0,\\infty)$ with the following property. Suppose that $n\\in \\mathbb{N}$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\\to Y$ and for every $\\varepsilon\\in (0,1/2]$ there exists a radius $r\\ge\\exp(-1/\\varepsilon^{cn})$, a point $x\\in B_X$ with $x+rB_X\\subset B_X$, and an affine mapping $\\Lambda:X\\to Y$ such that $\\|f(y)-\\Lambda(y)\\|_Y\\le \\varepsilon r$ for every $y\\in x+rB_X$.","authors_text":"Assaf Naor, Tuomas Hyt\\\"onen","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T15:23:58Z","title":"Heat flow and quantitative differentiation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01915","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":"sha256:4d9028d7457911e264feea7043ff3bff1b358e58cf6c0b52855e68c9cfff5b65","target":"record","created_at":"2026-05-18T01:09:44Z","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":"273180b4630d26b3f5198bdb65340ef6a57301765660ae93c1d4a8730229b7a3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-08-05T15:23:58Z","title_canon_sha256":"2ae08075195d22f722a626d59b8ba29ab136b1e8d553173a2de9b3a79db2dd71"},"schema_version":"1.0","source":{"id":"1608.01915","kind":"arxiv","version":1}},"canonical_sha256":"4dd0ea9f6a0f550cd6633350c4d0340d5ea4b15a46c1391a3b7c12f03f3e6ed6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4dd0ea9f6a0f550cd6633350c4d0340d5ea4b15a46c1391a3b7c12f03f3e6ed6","first_computed_at":"2026-05-18T01:09:44.275254Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:09:44.275254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MNNRFwP045OWpdzXudyA/lsUG/NgY0LD6aZjrvhUi+ln0bT33sE/NQnJFaLbDpJmzEQIUT0+MXV4p2MQHw5dDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:09:44.275818Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.01915","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d9028d7457911e264feea7043ff3bff1b358e58cf6c0b52855e68c9cfff5b65","sha256:d51d771ef40d79fbc6bba460d4268e1dd7264dea1a657978f4db3a0abb25350d"],"state_sha256":"c9a09775f354dac4839d467ba090ffaba59fac9e088b222f41206cd27611b67c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0M2aM2PIRxJWjMW4L3dGRB6xPg2nNcNRW2wB1cF8sp5PK1y9l6B1n/R/kAdxGZdjTGOCNH44etq+IjbqDiwyCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T13:53:06.479543Z","bundle_sha256":"efcb391821d7412cc26f4cb33aec440b136a920dc51388ddfcd3b5a9dfd1af44"}}