{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GTMIUAYGLY6BYB3CCQU3SMINQN","short_pith_number":"pith:GTMIUAYG","canonical_record":{"source":{"id":"1212.2817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-12T13:54:17Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"55ceb472e7204483de160f3e1ad4e05546135af15b389fecdc3392bf1a136970","abstract_canon_sha256":"806c85f4d08afd94eb6f4348fa1449c1ecab0692bb2d7868c7e72bf3ad46475b"},"schema_version":"1.0"},"canonical_sha256":"34d88a03065e3c1c07621429b9310d83527dff530c60a085966114aa54063377","source":{"kind":"arxiv","id":"1212.2817","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2817","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2817v1","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2817","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"pith_short_12","alias_value":"GTMIUAYGLY6B","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GTMIUAYGLY6BYB3C","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GTMIUAYG","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GTMIUAYGLY6BYB3CCQU3SMINQN","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2817","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-12T13:54:17Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"55ceb472e7204483de160f3e1ad4e05546135af15b389fecdc3392bf1a136970","abstract_canon_sha256":"806c85f4d08afd94eb6f4348fa1449c1ecab0692bb2d7868c7e72bf3ad46475b"},"schema_version":"1.0"},"canonical_sha256":"34d88a03065e3c1c07621429b9310d83527dff530c60a085966114aa54063377","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:35.293008Z","signature_b64":"woC5+KJhX0qCbMskycYcd7yWXQEa5gw0+ct2tFInEphApZPzmDnXW3eG9f8lru7HnHIuJFT5uQDcNywyW88zBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34d88a03065e3c1c07621429b9310d83527dff530c60a085966114aa54063377","last_reissued_at":"2026-05-18T03:38:35.292300Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:35.292300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2817","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":"2026-05-18T03:38:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"buVrsqX0T8dAXxLmAhwkwpbt69H+jRvPdkOi1+Uz6HRLY+wywroU2RzIVddUU7z0RAhW+wm4ciLkBQqKFHtwBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:59:34.370803Z"},"content_sha256":"0f55e50ac64c59fd5b0b33c7b42c3bc81fad1e5145cf19ae969c29fa20f912f7","schema_version":"1.0","event_id":"sha256:0f55e50ac64c59fd5b0b33c7b42c3bc81fad1e5145cf19ae969c29fa20f912f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GTMIUAYGLY6BYB3CCQU3SMINQN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Covering $L^p$ spaces by balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"Clemente Zanco, Michael Levin, Vladimir P. Fonf","submitted_at":"2012-12-12T13:54:17Z","abstract_excerpt":"We prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2817","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":"2026-05-18T03:38:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FDLHqyxCASEI+09g5bGiAneVAF7TQOqlqKvQk+Fpo+eHnPiWWhMC2BzUOBUIkQXZS/ChFDXI6tP/eZeaMwGFCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:59:34.371148Z"},"content_sha256":"f9282ce055cfa7fbf709d943f20970440f172fbf8a656e201fb4cdc1c96173f0","schema_version":"1.0","event_id":"sha256:f9282ce055cfa7fbf709d943f20970440f172fbf8a656e201fb4cdc1c96173f0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/bundle.json","state_url":"https://pith.science/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/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":"2026-06-02T21:59:34Z","links":{"resolver":"https://pith.science/pith/GTMIUAYGLY6BYB3CCQU3SMINQN","bundle":"https://pith.science/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/bundle.json","state":"https://pith.science/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GTMIUAYGLY6BYB3CCQU3SMINQN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GTMIUAYGLY6BYB3CCQU3SMINQN","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":"806c85f4d08afd94eb6f4348fa1449c1ecab0692bb2d7868c7e72bf3ad46475b","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-12T13:54:17Z","title_canon_sha256":"55ceb472e7204483de160f3e1ad4e05546135af15b389fecdc3392bf1a136970"},"schema_version":"1.0","source":{"id":"1212.2817","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2817","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2817v1","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2817","created_at":"2026-05-18T03:38:35Z"},{"alias_kind":"pith_short_12","alias_value":"GTMIUAYGLY6B","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GTMIUAYGLY6BYB3C","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GTMIUAYG","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:f9282ce055cfa7fbf709d943f20970440f172fbf8a656e201fb4cdc1c96173f0","target":"graph","created_at":"2026-05-18T03:38:35Z","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 prove that, given any covering of any separable infinite-dimensional uniformly rotund and uniformly smooth Banach space $X$ by closed balls each of positive radius, some point exists in $X$ which belongs to infinitely many balls.","authors_text":"Clemente Zanco, Michael Levin, Vladimir P. Fonf","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-12T13:54:17Z","title":"Covering $L^p$ spaces by balls"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2817","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":"sha256:0f55e50ac64c59fd5b0b33c7b42c3bc81fad1e5145cf19ae969c29fa20f912f7","target":"record","created_at":"2026-05-18T03:38:35Z","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":"806c85f4d08afd94eb6f4348fa1449c1ecab0692bb2d7868c7e72bf3ad46475b","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-12T13:54:17Z","title_canon_sha256":"55ceb472e7204483de160f3e1ad4e05546135af15b389fecdc3392bf1a136970"},"schema_version":"1.0","source":{"id":"1212.2817","kind":"arxiv","version":1}},"canonical_sha256":"34d88a03065e3c1c07621429b9310d83527dff530c60a085966114aa54063377","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34d88a03065e3c1c07621429b9310d83527dff530c60a085966114aa54063377","first_computed_at":"2026-05-18T03:38:35.292300Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:35.292300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"woC5+KJhX0qCbMskycYcd7yWXQEa5gw0+ct2tFInEphApZPzmDnXW3eG9f8lru7HnHIuJFT5uQDcNywyW88zBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:35.293008Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2817","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f55e50ac64c59fd5b0b33c7b42c3bc81fad1e5145cf19ae969c29fa20f912f7","sha256:f9282ce055cfa7fbf709d943f20970440f172fbf8a656e201fb4cdc1c96173f0"],"state_sha256":"56e8887f4b9500f052a3f5284628569c49778a1318b9f96a1949e28c462924ba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vMQ5AoK8U1SLtn2EJk8nFa+4LvPpHqlohIC0g4vPaWQK+Ms8yrldye693/oBP8viGkWXU/lLFrLgB2TZepqQBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T21:59:34.373141Z","bundle_sha256":"cf99a1224100b2fcf7c88f47a2257ea650fdf8437ae7522a4ba89c3fd388f4c9"}}