{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:WUETK75ZDKNMSO5UY3MRD5NKCC","short_pith_number":"pith:WUETK75Z","canonical_record":{"source":{"id":"2003.01439","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-03-03T10:48:13Z","cross_cats_sorted":[],"title_canon_sha256":"6f9ef81503c902e384e0a80979389e24760e3d8633d5593ea62d30aacc0cc1be","abstract_canon_sha256":"0ffa6070d9f3594a713baa7b03057e84b66074cc4dacb9997c256485d9556fb3"},"schema_version":"1.0"},"canonical_sha256":"b509357fb91a9ac93bb4c6d911f5aa108dd0400f83a95da20831fc2eff36378f","source":{"kind":"arxiv","id":"2003.01439","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.01439","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"arxiv_version","alias_value":"2003.01439v1","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.01439","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_12","alias_value":"WUETK75ZDKNM","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_16","alias_value":"WUETK75ZDKNMSO5U","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_8","alias_value":"WUETK75Z","created_at":"2026-07-05T04:05:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:WUETK75ZDKNMSO5UY3MRD5NKCC","target":"record","payload":{"canonical_record":{"source":{"id":"2003.01439","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-03-03T10:48:13Z","cross_cats_sorted":[],"title_canon_sha256":"6f9ef81503c902e384e0a80979389e24760e3d8633d5593ea62d30aacc0cc1be","abstract_canon_sha256":"0ffa6070d9f3594a713baa7b03057e84b66074cc4dacb9997c256485d9556fb3"},"schema_version":"1.0"},"canonical_sha256":"b509357fb91a9ac93bb4c6d911f5aa108dd0400f83a95da20831fc2eff36378f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:05:09.381404Z","signature_b64":"NmtiYwAkRKjJKaO1wK3Uj/QjwyjNwfMW8OGn9H6OxsltdKeJ20eAt04b66qamPWZfyYldPSXLijV8V/PCiaSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b509357fb91a9ac93bb4c6d911f5aa108dd0400f83a95da20831fc2eff36378f","last_reissued_at":"2026-07-05T04:05:09.381013Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:05:09.381013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2003.01439","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":"2026-07-05T04:05:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CcC6zyHqQLqFe5Cw1dADqazaJlDm74QpAq5VQhbpOliNzRWdZCgXpwh90D7V6NJIZJnTpm+Q+1gMN8C0MamxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T13:33:19.757539Z"},"content_sha256":"a89b34f853da03f5d09141fd1fae6c9dea61cab43f5fe87f45ed646782c9d4bb","schema_version":"1.0","event_id":"sha256:a89b34f853da03f5d09141fd1fae6c9dea61cab43f5fe87f45ed646782c9d4bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:WUETK75ZDKNMSO5UY3MRD5NKCC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Points of differentiability of the norm in Lipschitz-free spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Abraham Rueda Zoca, Ram\\'on J. Aliaga","submitted_at":"2020-03-03T10:48:13Z","abstract_excerpt":"We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $\\mu=\\sum_n \\lambda_n \\frac{\\delta_{x_n}-\\delta_{y_n}}{d(x_n,y_n)}$ such that $\\|\\mu\\|=\\sum_n |\\lambda_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ of the underlying metric space, and determine when they are points of G\\^ateaux differentiability of the norm. In particular, we show that G\\^ateaux and Fr\\'echet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.01439","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.01439/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T04:05:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nagy0h2VG88LwbG428wo+XbyjJ950P+hg/AIyNsJpxSiD0ukZwHau4mKH+pZGm6btc8WVV+DncWIIxe+yujsAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T13:33:19.758011Z"},"content_sha256":"0b9af4fdd8bd09e0397d0037bf1f01b87840f46a47a4269358f4d7255dcbea15","schema_version":"1.0","event_id":"sha256:0b9af4fdd8bd09e0397d0037bf1f01b87840f46a47a4269358f4d7255dcbea15"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/bundle.json","state_url":"https://pith.science/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/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":"2026-08-14T13:33:19Z","links":{"resolver":"https://pith.science/pith/WUETK75ZDKNMSO5UY3MRD5NKCC","bundle":"https://pith.science/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/bundle.json","state":"https://pith.science/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WUETK75ZDKNMSO5UY3MRD5NKCC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:WUETK75ZDKNMSO5UY3MRD5NKCC","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":"0ffa6070d9f3594a713baa7b03057e84b66074cc4dacb9997c256485d9556fb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-03-03T10:48:13Z","title_canon_sha256":"6f9ef81503c902e384e0a80979389e24760e3d8633d5593ea62d30aacc0cc1be"},"schema_version":"1.0","source":{"id":"2003.01439","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2003.01439","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"arxiv_version","alias_value":"2003.01439v1","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2003.01439","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_12","alias_value":"WUETK75ZDKNM","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_16","alias_value":"WUETK75ZDKNMSO5U","created_at":"2026-07-05T04:05:09Z"},{"alias_kind":"pith_short_8","alias_value":"WUETK75Z","created_at":"2026-07-05T04:05:09Z"}],"graph_snapshots":[{"event_id":"sha256:0b9af4fdd8bd09e0397d0037bf1f01b87840f46a47a4269358f4d7255dcbea15","target":"graph","created_at":"2026-07-05T04:05:09Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2003.01439/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form $\\mu=\\sum_n \\lambda_n \\frac{\\delta_{x_n}-\\delta_{y_n}}{d(x_n,y_n)}$ such that $\\|\\mu\\|=\\sum_n |\\lambda_n |$. We characterise these elements in terms of geometric conditions on the points $x_n$, $y_n$ of the underlying metric space, and determine when they are points of G\\^ateaux differentiability of the norm. In particular, we show that G\\^ateaux and Fr\\'echet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and t","authors_text":"Abraham Rueda Zoca, Ram\\'on J. Aliaga","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-03-03T10:48:13Z","title":"Points of differentiability of the norm in Lipschitz-free spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.01439","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:a89b34f853da03f5d09141fd1fae6c9dea61cab43f5fe87f45ed646782c9d4bb","target":"record","created_at":"2026-07-05T04:05:09Z","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":"0ffa6070d9f3594a713baa7b03057e84b66074cc4dacb9997c256485d9556fb3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-03-03T10:48:13Z","title_canon_sha256":"6f9ef81503c902e384e0a80979389e24760e3d8633d5593ea62d30aacc0cc1be"},"schema_version":"1.0","source":{"id":"2003.01439","kind":"arxiv","version":1}},"canonical_sha256":"b509357fb91a9ac93bb4c6d911f5aa108dd0400f83a95da20831fc2eff36378f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b509357fb91a9ac93bb4c6d911f5aa108dd0400f83a95da20831fc2eff36378f","first_computed_at":"2026-07-05T04:05:09.381013Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:05:09.381013Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NmtiYwAkRKjJKaO1wK3Uj/QjwyjNwfMW8OGn9H6OxsltdKeJ20eAt04b66qamPWZfyYldPSXLijV8V/PCiaSAw==","signature_status":"signed_v1","signed_at":"2026-07-05T04:05:09.381404Z","signed_message":"canonical_sha256_bytes"},"source_id":"2003.01439","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a89b34f853da03f5d09141fd1fae6c9dea61cab43f5fe87f45ed646782c9d4bb","sha256:0b9af4fdd8bd09e0397d0037bf1f01b87840f46a47a4269358f4d7255dcbea15"],"state_sha256":"1481b312706feb753d9238c1dfd5db0cb3e01d12713951d2429737445349b00d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"65Oet8KctmoPrzd46RdIjheKQQPmtF2yoISFes20N+uj0gyxxT3WGRNCNYny9XlsyNw+bQbCtMw4GhtOPCP2Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T13:33:19.762627Z","bundle_sha256":"9ee5c4e4d59a7aa4cb6e50919794b812ca09966c9efa69e0cd09e18f59cfe6e0"}}