{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:V65537UZHKOC6BT6G6FCOLF3WA","short_pith_number":"pith:V65537UZ","canonical_record":{"source":{"id":"1202.1539","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","cross_cats_sorted":[],"title_canon_sha256":"01b36490dbb09b263a8baa5ed43a44a8d112987c923e26077ad4fbd1f84363ce","abstract_canon_sha256":"5cdcfba56e73ec82f97e21afa5fffd17a13ca45bb8ad36da71bace7ef92c8217"},"schema_version":"1.0"},"canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","source":{"kind":"arxiv","id":"1202.1539","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1539","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1539v3","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1539","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"V65537UZHKOC","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"V65537UZHKOC6BT6","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"V65537UZ","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:V65537UZHKOC6BT6G6FCOLF3WA","target":"record","payload":{"canonical_record":{"source":{"id":"1202.1539","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","cross_cats_sorted":[],"title_canon_sha256":"01b36490dbb09b263a8baa5ed43a44a8d112987c923e26077ad4fbd1f84363ce","abstract_canon_sha256":"5cdcfba56e73ec82f97e21afa5fffd17a13ca45bb8ad36da71bace7ef92c8217"},"schema_version":"1.0"},"canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:26.421562Z","signature_b64":"18BmQ1JXIJImiln/WAAl2JLFnMMpW6o249/OOKbyclrxS3eSciIYSH3qdkQ+8KuBop63gihmViIgVIAKtaiCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","last_reissued_at":"2026-05-18T02:41:26.421150Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:26.421150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.1539","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":"2026-05-18T02:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a4cmUQj9Wi4JNzjsJQYRFF6MnpOl7lfimBxk8izli1ifzgqHFi45jWl72v0TRG/UjeJaNMUz5TlQQGkOBVPCBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:17:29.227769Z"},"content_sha256":"4cf442de4b4e8202f073ff6b79c5604b05fa0e720e46404a9dcfa74a1340f1b9","schema_version":"1.0","event_id":"sha256:4cf442de4b4e8202f073ff6b79c5604b05fa0e720e46404a9dcfa74a1340f1b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:V65537UZHKOC6BT6G6FCOLF3WA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Metric spaces admitting only trivial weak contractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Rich\\'ard Balka","submitted_at":"2012-02-07T21:15:25Z","abstract_excerpt":"If $(X,d)$ is a metric space then the map $f\\colon X\\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\\in X$, $x\\neq y$. We determine the simplest non-closed sets $X\\subseteq \\mathbb{R}^n$ in the sense of descriptive set theoretic complexity such that every weak contraction $f\\colon X\\to X$ is constant. In order to do so, we prove that there exists a non-closed $F_{\\sigma}$ set $F\\subseteq \\mathbb{R}$ such that every weak contraction $f\\colon F\\to F$ is constant. Similarly, there exists a non-closed $G_{\\delta}$ set $G\\subseteq \\mathbb{R}$ such that every weak con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1539","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":"2026-05-18T02:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UNkdf6AWTmCZQCedjU8nN5toCgsasMmN/ouL1LjQqNVo96iXZmCI84lBRd/Y2lq5C7cKHuUhI6co0pVmy3tICw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:17:29.228454Z"},"content_sha256":"486f8754b50dda1dc4409b129ccb3374124d9599e4d7b114c1aa044f0844d396","schema_version":"1.0","event_id":"sha256:486f8754b50dda1dc4409b129ccb3374124d9599e4d7b114c1aa044f0844d396"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V65537UZHKOC6BT6G6FCOLF3WA/bundle.json","state_url":"https://pith.science/pith/V65537UZHKOC6BT6G6FCOLF3WA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V65537UZHKOC6BT6G6FCOLF3WA/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":"2026-06-22T14:17:29Z","links":{"resolver":"https://pith.science/pith/V65537UZHKOC6BT6G6FCOLF3WA","bundle":"https://pith.science/pith/V65537UZHKOC6BT6G6FCOLF3WA/bundle.json","state":"https://pith.science/pith/V65537UZHKOC6BT6G6FCOLF3WA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V65537UZHKOC6BT6G6FCOLF3WA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:V65537UZHKOC6BT6G6FCOLF3WA","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":"5cdcfba56e73ec82f97e21afa5fffd17a13ca45bb8ad36da71bace7ef92c8217","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","title_canon_sha256":"01b36490dbb09b263a8baa5ed43a44a8d112987c923e26077ad4fbd1f84363ce"},"schema_version":"1.0","source":{"id":"1202.1539","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1539","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1539v3","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1539","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"V65537UZHKOC","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"V65537UZHKOC6BT6","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"V65537UZ","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:486f8754b50dda1dc4409b129ccb3374124d9599e4d7b114c1aa044f0844d396","target":"graph","created_at":"2026-05-18T02:41:26Z","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":"If $(X,d)$ is a metric space then the map $f\\colon X\\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\\in X$, $x\\neq y$. We determine the simplest non-closed sets $X\\subseteq \\mathbb{R}^n$ in the sense of descriptive set theoretic complexity such that every weak contraction $f\\colon X\\to X$ is constant. In order to do so, we prove that there exists a non-closed $F_{\\sigma}$ set $F\\subseteq \\mathbb{R}$ such that every weak contraction $f\\colon F\\to F$ is constant. Similarly, there exists a non-closed $G_{\\delta}$ set $G\\subseteq \\mathbb{R}$ such that every weak con","authors_text":"Rich\\'ard Balka","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","title":"Metric spaces admitting only trivial weak contractions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1539","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":"sha256:4cf442de4b4e8202f073ff6b79c5604b05fa0e720e46404a9dcfa74a1340f1b9","target":"record","created_at":"2026-05-18T02:41:26Z","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":"5cdcfba56e73ec82f97e21afa5fffd17a13ca45bb8ad36da71bace7ef92c8217","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-02-07T21:15:25Z","title_canon_sha256":"01b36490dbb09b263a8baa5ed43a44a8d112987c923e26077ad4fbd1f84363ce"},"schema_version":"1.0","source":{"id":"1202.1539","kind":"arxiv","version":3}},"canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"afbbddfe993a9c2f067e378a272cbbb02e8d68d5425f830c25a72c82c8f59a44","first_computed_at":"2026-05-18T02:41:26.421150Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:26.421150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"18BmQ1JXIJImiln/WAAl2JLFnMMpW6o249/OOKbyclrxS3eSciIYSH3qdkQ+8KuBop63gihmViIgVIAKtaiCCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:26.421562Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.1539","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4cf442de4b4e8202f073ff6b79c5604b05fa0e720e46404a9dcfa74a1340f1b9","sha256:486f8754b50dda1dc4409b129ccb3374124d9599e4d7b114c1aa044f0844d396"],"state_sha256":"582873d4eac209212d771f460bf794079a877b0baf29ec193b5eea627cffeef2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Ub7BRoNFJfNk1TFTcb1xYZM8S7uy6ii9hSsbTIIpEgApssPgQAjrfDcrVwkEINRxty66diyVnUrji5/DTDEAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T14:17:29.231455Z","bundle_sha256":"f877f8b251bb6e144f8174c35e3794e7fb8503b38d66b052dd3924f039a78b58"}}