{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XXKOCARKGNUL5C6IDDFFQUCENG","short_pith_number":"pith:XXKOCARK","canonical_record":{"source":{"id":"1311.4584","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-18T23:02:22Z","cross_cats_sorted":[],"title_canon_sha256":"82e4287a655d044fdd38d5ed2406305cc4302b9a7c177fa5c73f0cff55c0c7a5","abstract_canon_sha256":"e0269c5e5f4be37e23e360f85a34b0669868b00dd927e8985720e156e08bad68"},"schema_version":"1.0"},"canonical_sha256":"bdd4e1022a3368be8bc818ca58504469b790d40fc23575f8da8323c857b27624","source":{"kind":"arxiv","id":"1311.4584","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4584","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4584v3","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4584","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"XXKOCARKGNUL","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XXKOCARKGNUL5C6I","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XXKOCARK","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XXKOCARKGNUL5C6IDDFFQUCENG","target":"record","payload":{"canonical_record":{"source":{"id":"1311.4584","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-18T23:02:22Z","cross_cats_sorted":[],"title_canon_sha256":"82e4287a655d044fdd38d5ed2406305cc4302b9a7c177fa5c73f0cff55c0c7a5","abstract_canon_sha256":"e0269c5e5f4be37e23e360f85a34b0669868b00dd927e8985720e156e08bad68"},"schema_version":"1.0"},"canonical_sha256":"bdd4e1022a3368be8bc818ca58504469b790d40fc23575f8da8323c857b27624","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:58.171237Z","signature_b64":"uaLSwxbd5R3/wo7S25zLg/7G7ZPrEehPCzqTK0x60iZCN3lvotAvGgeC/O7nj5nfxsjGLCO2XScedBFpiyxLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdd4e1022a3368be8bc818ca58504469b790d40fc23575f8da8323c857b27624","last_reissued_at":"2026-05-18T02:45:58.170777Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:58.170777Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.4584","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0rNvoyvxNmE9JH/k90/Eyo1Cv2/5CyHPeGNAVEezQdwosA/gjmv9+dTSPsOJgd/xe+gE8F+YssnVHYrDtwcuCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T09:28:13.811009Z"},"content_sha256":"bfb97c0e47ba1145728566e4ddb0353b499febb74cd98b83cf2f0ab15bc0f25d","schema_version":"1.0","event_id":"sha256:bfb97c0e47ba1145728566e4ddb0353b499febb74cd98b83cf2f0ab15bc0f25d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XXKOCARKGNUL5C6IDDFFQUCENG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Low distortion embeddings into Asplund Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anton\\'in Proch\\'azka, Luis S\\'anchez-Gonz\\'alez","submitted_at":"2013-11-18T23:02:22Z","abstract_excerpt":"We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then $X$ contains an isomorphic copy of $\\ell_1$. We also show that the space $M$ does not embed with distortion strictly less than $2$ into $\\ell_1$ itself but it does embed isometrically into a space that is isomorphic to $\\ell_1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4584","kind":"arxiv","version":3},"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":"2026-05-18T02:45:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EO+Z+r14KtTo7jf/DfeqB+WpYdziqodJxC7dqt++pHRrlbPc3gLPaWxdGS6/6/9Yj/czkdH6nMx9HspPt3ONBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T09:28:13.811363Z"},"content_sha256":"bb5735bb31cc0c42b479ab5a1fe5e8b6fb6c8f85714a1f09f6a2fb2867b3cad6","schema_version":"1.0","event_id":"sha256:bb5735bb31cc0c42b479ab5a1fe5e8b6fb6c8f85714a1f09f6a2fb2867b3cad6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XXKOCARKGNUL5C6IDDFFQUCENG/bundle.json","state_url":"https://pith.science/pith/XXKOCARKGNUL5C6IDDFFQUCENG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XXKOCARKGNUL5C6IDDFFQUCENG/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":"2026-06-20T09:28:13Z","links":{"resolver":"https://pith.science/pith/XXKOCARKGNUL5C6IDDFFQUCENG","bundle":"https://pith.science/pith/XXKOCARKGNUL5C6IDDFFQUCENG/bundle.json","state":"https://pith.science/pith/XXKOCARKGNUL5C6IDDFFQUCENG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XXKOCARKGNUL5C6IDDFFQUCENG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XXKOCARKGNUL5C6IDDFFQUCENG","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":"e0269c5e5f4be37e23e360f85a34b0669868b00dd927e8985720e156e08bad68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-18T23:02:22Z","title_canon_sha256":"82e4287a655d044fdd38d5ed2406305cc4302b9a7c177fa5c73f0cff55c0c7a5"},"schema_version":"1.0","source":{"id":"1311.4584","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4584","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4584v3","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4584","created_at":"2026-05-18T02:45:58Z"},{"alias_kind":"pith_short_12","alias_value":"XXKOCARKGNUL","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XXKOCARKGNUL5C6I","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XXKOCARK","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:bb5735bb31cc0c42b479ab5a1fe5e8b6fb6c8f85714a1f09f6a2fb2867b3cad6","target":"graph","created_at":"2026-05-18T02:45:58Z","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 give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then $X$ contains an isomorphic copy of $\\ell_1$. We also show that the space $M$ does not embed with distortion strictly less than $2$ into $\\ell_1$ itself but it does embed isometrically into a space that is isomorphic to $\\ell_1$.","authors_text":"Anton\\'in Proch\\'azka, Luis S\\'anchez-Gonz\\'alez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-18T23:02:22Z","title":"Low distortion embeddings into Asplund Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4584","kind":"arxiv","version":3},"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:bfb97c0e47ba1145728566e4ddb0353b499febb74cd98b83cf2f0ab15bc0f25d","target":"record","created_at":"2026-05-18T02:45:58Z","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":"e0269c5e5f4be37e23e360f85a34b0669868b00dd927e8985720e156e08bad68","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-18T23:02:22Z","title_canon_sha256":"82e4287a655d044fdd38d5ed2406305cc4302b9a7c177fa5c73f0cff55c0c7a5"},"schema_version":"1.0","source":{"id":"1311.4584","kind":"arxiv","version":3}},"canonical_sha256":"bdd4e1022a3368be8bc818ca58504469b790d40fc23575f8da8323c857b27624","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdd4e1022a3368be8bc818ca58504469b790d40fc23575f8da8323c857b27624","first_computed_at":"2026-05-18T02:45:58.170777Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:58.170777Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uaLSwxbd5R3/wo7S25zLg/7G7ZPrEehPCzqTK0x60iZCN3lvotAvGgeC/O7nj5nfxsjGLCO2XScedBFpiyxLBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:58.171237Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.4584","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bfb97c0e47ba1145728566e4ddb0353b499febb74cd98b83cf2f0ab15bc0f25d","sha256:bb5735bb31cc0c42b479ab5a1fe5e8b6fb6c8f85714a1f09f6a2fb2867b3cad6"],"state_sha256":"72c74181b2fa13267915b8336040bdc3d6111d68d10747677d29a4d293ba6647"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Neb3ZuEDFw/66sREvx+Xp4DvoHo+IqgOC84HO+SZ6hhAJfg9HwDBxvdw1AncF4ITbE5+wCQ1wr94toDJJWYFAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T09:28:13.813172Z","bundle_sha256":"ecc3e297af292ed71f10fb7f25443afc8afc62d1e9af35c5b7893686345274e2"}}