{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:GOBQQC2NCPXJKVKCUWJKXNLRWQ","short_pith_number":"pith:GOBQQC2N","canonical_record":{"source":{"id":"1710.11007","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-30T15:20:19Z","cross_cats_sorted":[],"title_canon_sha256":"613959691b3aef9e5fe6a78d579703735af5bd70c892e80f89fe60562358c18f","abstract_canon_sha256":"2d7b6d849577c904748b6b9eed9e0ddfb32a7c6b5a2e2fa2c931e4aec24414e1"},"schema_version":"1.0"},"canonical_sha256":"3383080b4d13ee955542a592abb571b4131f8abd81bbc18d954b775eb76f488b","source":{"kind":"arxiv","id":"1710.11007","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.11007","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"arxiv_version","alias_value":"1710.11007v3","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.11007","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"pith_short_12","alias_value":"GOBQQC2NCPXJ","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GOBQQC2NCPXJKVKC","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GOBQQC2N","created_at":"2026-05-18T12:31:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:GOBQQC2NCPXJKVKCUWJKXNLRWQ","target":"record","payload":{"canonical_record":{"source":{"id":"1710.11007","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-30T15:20:19Z","cross_cats_sorted":[],"title_canon_sha256":"613959691b3aef9e5fe6a78d579703735af5bd70c892e80f89fe60562358c18f","abstract_canon_sha256":"2d7b6d849577c904748b6b9eed9e0ddfb32a7c6b5a2e2fa2c931e4aec24414e1"},"schema_version":"1.0"},"canonical_sha256":"3383080b4d13ee955542a592abb571b4131f8abd81bbc18d954b775eb76f488b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:56.755979Z","signature_b64":"R15VihCLkhyGpg2uPA1YHm3zWXdUpEeUC2eyj3rFTNx0J2tW6cbhu7ZDaazgL9SkF3FEThHQ+uBtvZyXsQkXCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3383080b4d13ee955542a592abb571b4131f8abd81bbc18d954b775eb76f488b","last_reissued_at":"2026-05-18T00:03:56.755269Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:56.755269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.11007","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":"2026-05-18T00:03:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LccR9slxqp3CK/pdw87yKnHroBJKnukSMnTeLvxBV0wZzDBistNaOBr8H9Lz0E7eBgWqBTxuoQ/APtPvG0gABw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:29:33.631221Z"},"content_sha256":"7a18d0b6a63f701ff582a4c9b5a06f8f430372473c9d950dfb5b3e1d5b2cb7b8","schema_version":"1.0","event_id":"sha256:7a18d0b6a63f701ff582a4c9b5a06f8f430372473c9d950dfb5b3e1d5b2cb7b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:GOBQQC2NCPXJKVKCUWJKXNLRWQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kirszbraun-type Theorems For Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Igor Pak, Martin Tassy, Nishant Chandgotia","submitted_at":"2017-10-30T15:20:19Z","abstract_excerpt":"The classical Kirszbraun theorem says that all $1$-Lipschitz functions $f:A\\longrightarrow \\mathbb{R}^n$, $A\\subset \\mathbb{R}^n$, with the Euclidean metric have a $1$-Lipschitz extension to $\\mathbb{R}^n$. For metric spaces $X,Y$ we say that $Y$ is $X$-Kirszbraun if all $1$-Lipschitz functions $f:A\\longrightarrow Y$, $A\\subset X$, have a $1$-Lipschitz extension to~$X$. We analyze the case when $X$ and $Y$ are graphs with the usual path metric. We prove that $\\mathbb{Z}^d$-Kirszbraun graphs are exactly graphs that satisfies a certain Helly property. We also consider complexity aspects of these"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11007","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":"2026-05-18T00:03:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uVnfo10L/OkLg2inXn41g78f5YfzOMO+ZHdLg7sMJ1KHJWjRkcaeJEEVUk2KWtIpnAwEDIqMNAxBudvgpYUKAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:29:33.631589Z"},"content_sha256":"a87b13dbe0ed6a739838af7938246c660c15b4dfebf39302b8ee9abdc58ee327","schema_version":"1.0","event_id":"sha256:a87b13dbe0ed6a739838af7938246c660c15b4dfebf39302b8ee9abdc58ee327"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/bundle.json","state_url":"https://pith.science/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/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":"2026-05-31T00:29:33Z","links":{"resolver":"https://pith.science/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ","bundle":"https://pith.science/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/bundle.json","state":"https://pith.science/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GOBQQC2NCPXJKVKCUWJKXNLRWQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:GOBQQC2NCPXJKVKCUWJKXNLRWQ","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":"2d7b6d849577c904748b6b9eed9e0ddfb32a7c6b5a2e2fa2c931e4aec24414e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-30T15:20:19Z","title_canon_sha256":"613959691b3aef9e5fe6a78d579703735af5bd70c892e80f89fe60562358c18f"},"schema_version":"1.0","source":{"id":"1710.11007","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.11007","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"arxiv_version","alias_value":"1710.11007v3","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.11007","created_at":"2026-05-18T00:03:56Z"},{"alias_kind":"pith_short_12","alias_value":"GOBQQC2NCPXJ","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GOBQQC2NCPXJKVKC","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GOBQQC2N","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:a87b13dbe0ed6a739838af7938246c660c15b4dfebf39302b8ee9abdc58ee327","target":"graph","created_at":"2026-05-18T00:03:56Z","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":"The classical Kirszbraun theorem says that all $1$-Lipschitz functions $f:A\\longrightarrow \\mathbb{R}^n$, $A\\subset \\mathbb{R}^n$, with the Euclidean metric have a $1$-Lipschitz extension to $\\mathbb{R}^n$. For metric spaces $X,Y$ we say that $Y$ is $X$-Kirszbraun if all $1$-Lipschitz functions $f:A\\longrightarrow Y$, $A\\subset X$, have a $1$-Lipschitz extension to~$X$. We analyze the case when $X$ and $Y$ are graphs with the usual path metric. We prove that $\\mathbb{Z}^d$-Kirszbraun graphs are exactly graphs that satisfies a certain Helly property. We also consider complexity aspects of these","authors_text":"Igor Pak, Martin Tassy, Nishant Chandgotia","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-30T15:20:19Z","title":"Kirszbraun-type Theorems For Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.11007","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":"sha256:7a18d0b6a63f701ff582a4c9b5a06f8f430372473c9d950dfb5b3e1d5b2cb7b8","target":"record","created_at":"2026-05-18T00:03:56Z","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":"2d7b6d849577c904748b6b9eed9e0ddfb32a7c6b5a2e2fa2c931e4aec24414e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-30T15:20:19Z","title_canon_sha256":"613959691b3aef9e5fe6a78d579703735af5bd70c892e80f89fe60562358c18f"},"schema_version":"1.0","source":{"id":"1710.11007","kind":"arxiv","version":3}},"canonical_sha256":"3383080b4d13ee955542a592abb571b4131f8abd81bbc18d954b775eb76f488b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3383080b4d13ee955542a592abb571b4131f8abd81bbc18d954b775eb76f488b","first_computed_at":"2026-05-18T00:03:56.755269Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:56.755269Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R15VihCLkhyGpg2uPA1YHm3zWXdUpEeUC2eyj3rFTNx0J2tW6cbhu7ZDaazgL9SkF3FEThHQ+uBtvZyXsQkXCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:56.755979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.11007","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7a18d0b6a63f701ff582a4c9b5a06f8f430372473c9d950dfb5b3e1d5b2cb7b8","sha256:a87b13dbe0ed6a739838af7938246c660c15b4dfebf39302b8ee9abdc58ee327"],"state_sha256":"6ba065bd57d0d055686c9b00c071f3fc8da6e37ec6d09f59a2455728fb7f8e71"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4X65zb95UbN2aWEJY2j3enx0NhNK05hAo5hbdhgNsQpP7oZNDliYWFmZ6t+NDbNddMMMKqvLmEi6Hgc/7Th1CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:29:33.633555Z","bundle_sha256":"95aba518cd1b68a3b711641aa3225a19817eb087af0eea67047bf8607cb907b0"}}