{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:44ERRALMWNBQVBKAYRICXCS4QO","short_pith_number":"pith:44ERRALM","canonical_record":{"source":{"id":"0806.4473","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-27T09:49:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"98995e5f7082e5023f2099ba8f79e57ff40dfd50e2e341c11be0cfe1fc830b60","abstract_canon_sha256":"e262b4700c5cca2383bd49976dbeadd04002763dc7b73799f7b16cfd1f60d73f"},"schema_version":"1.0"},"canonical_sha256":"e70918816cb3430a8540c4502b8a5c8381fc10d9591c26a45f3a38eea8e3852d","source":{"kind":"arxiv","id":"0806.4473","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.4473","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"0806.4473v1","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.4473","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"44ERRALMWNBQ","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"44ERRALMWNBQVBKA","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"44ERRALM","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:44ERRALMWNBQVBKAYRICXCS4QO","target":"record","payload":{"canonical_record":{"source":{"id":"0806.4473","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-27T09:49:37Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"98995e5f7082e5023f2099ba8f79e57ff40dfd50e2e341c11be0cfe1fc830b60","abstract_canon_sha256":"e262b4700c5cca2383bd49976dbeadd04002763dc7b73799f7b16cfd1f60d73f"},"schema_version":"1.0"},"canonical_sha256":"e70918816cb3430a8540c4502b8a5c8381fc10d9591c26a45f3a38eea8e3852d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.500901Z","signature_b64":"LpRoKHnDCssg+YKrP6PWMXDSSmL1Ox4G54jt61r0gSjdH6bdF8bTL63eYyZ8O3HeYwFDG97qAjh2oPFTMvdDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e70918816cb3430a8540c4502b8a5c8381fc10d9591c26a45f3a38eea8e3852d","last_reissued_at":"2026-05-18T02:28:21.500301Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.500301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0806.4473","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":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cVfJpg7+8g5Nxt35VYdCx4zT0j2mwh4Swchs16mLkClZ9Hjw0J1F1Y3rWSrnJ31IXJVGjt1pyOKnF9IgnyDsCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T23:13:33.153482Z"},"content_sha256":"6a525943fd3187793ea7f537bbef012edd67f16e0b30b7587ec7763fb3284057","schema_version":"1.0","event_id":"sha256:6a525943fd3187793ea7f537bbef012edd67f16e0b30b7587ec7763fb3284057"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:44ERRALMWNBQVBKAYRICXCS4QO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Simultaneous packing and covering in sequence spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Konrad J. Swanepoel","submitted_at":"2008-06-27T09:49:37Z","abstract_excerpt":"We adapt a construction of Klee (1981) to find a packing of unit balls in $\\ell_p$ ($1\\leq p<\\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.4473","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":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OWYOymytyWV7tr45c48B2KQ3Z10StL8ILWHubcs5k4N0xxvOD9hLMuA0t1l9R5W4IhCqSrCNmp6rclbuCqXCCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T23:13:33.154170Z"},"content_sha256":"e60212e5a64f20868b1d527ece60f1cf76c701f7d1f45702ba561cf3ab206cbe","schema_version":"1.0","event_id":"sha256:e60212e5a64f20868b1d527ece60f1cf76c701f7d1f45702ba561cf3ab206cbe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/44ERRALMWNBQVBKAYRICXCS4QO/bundle.json","state_url":"https://pith.science/pith/44ERRALMWNBQVBKAYRICXCS4QO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/44ERRALMWNBQVBKAYRICXCS4QO/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":"2026-05-18T23:13:33Z","links":{"resolver":"https://pith.science/pith/44ERRALMWNBQVBKAYRICXCS4QO","bundle":"https://pith.science/pith/44ERRALMWNBQVBKAYRICXCS4QO/bundle.json","state":"https://pith.science/pith/44ERRALMWNBQVBKAYRICXCS4QO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/44ERRALMWNBQVBKAYRICXCS4QO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:44ERRALMWNBQVBKAYRICXCS4QO","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":"e262b4700c5cca2383bd49976dbeadd04002763dc7b73799f7b16cfd1f60d73f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-27T09:49:37Z","title_canon_sha256":"98995e5f7082e5023f2099ba8f79e57ff40dfd50e2e341c11be0cfe1fc830b60"},"schema_version":"1.0","source":{"id":"0806.4473","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0806.4473","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"0806.4473v1","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.4473","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"44ERRALMWNBQ","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"44ERRALMWNBQVBKA","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"44ERRALM","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:e60212e5a64f20868b1d527ece60f1cf76c701f7d1f45702ba561cf3ab206cbe","target":"graph","created_at":"2026-05-18T02:28:21Z","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 adapt a construction of Klee (1981) to find a packing of unit balls in $\\ell_p$ ($1\\leq p<\\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal.","authors_text":"Konrad J. Swanepoel","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-27T09:49:37Z","title":"Simultaneous packing and covering in sequence spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.4473","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":"sha256:6a525943fd3187793ea7f537bbef012edd67f16e0b30b7587ec7763fb3284057","target":"record","created_at":"2026-05-18T02:28:21Z","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":"e262b4700c5cca2383bd49976dbeadd04002763dc7b73799f7b16cfd1f60d73f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-27T09:49:37Z","title_canon_sha256":"98995e5f7082e5023f2099ba8f79e57ff40dfd50e2e341c11be0cfe1fc830b60"},"schema_version":"1.0","source":{"id":"0806.4473","kind":"arxiv","version":1}},"canonical_sha256":"e70918816cb3430a8540c4502b8a5c8381fc10d9591c26a45f3a38eea8e3852d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e70918816cb3430a8540c4502b8a5c8381fc10d9591c26a45f3a38eea8e3852d","first_computed_at":"2026-05-18T02:28:21.500301Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:21.500301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LpRoKHnDCssg+YKrP6PWMXDSSmL1Ox4G54jt61r0gSjdH6bdF8bTL63eYyZ8O3HeYwFDG97qAjh2oPFTMvdDAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:21.500901Z","signed_message":"canonical_sha256_bytes"},"source_id":"0806.4473","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a525943fd3187793ea7f537bbef012edd67f16e0b30b7587ec7763fb3284057","sha256:e60212e5a64f20868b1d527ece60f1cf76c701f7d1f45702ba561cf3ab206cbe"],"state_sha256":"36ef2994ae225ea5d78f26151eb0b77b33d9cf70b044ed575a545003de4c45fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JG65/Q0RozgrVpGTsOrhK7uRW0Jcfe6a7XFt4RLlzg6GWegB2Hc2zE5eZgTWYr9Z9S4MNvkShhu5nPoTk8hWDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-18T23:13:33.156385Z","bundle_sha256":"6b699ca44861c3c158a3c1f933ec9e864d182a8374d2052e3c7970116ea67c4b"}}