{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:IQKBNYXF7GQTJAXVNPWWIIVGHS","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":"42c3af8357e7aef4792fc83291196ebc1790d0ac8710df6651286471dcda9d92","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2007-01-03T05:11:31Z","title_canon_sha256":"60ebb4656a7b0adc0939c0a4985351390bdc9a741822e7f3739b5940ad8949f2"},"schema_version":"1.0","source":{"id":"math/0701080","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701080","created_at":"2026-05-18T02:42:39Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701080v1","created_at":"2026-05-18T02:42:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701080","created_at":"2026-05-18T02:42:39Z"},{"alias_kind":"pith_short_12","alias_value":"IQKBNYXF7GQT","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"IQKBNYXF7GQTJAXV","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"IQKBNYXF","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:dbcc68037931be4279ba838827a3af8597584fb87e83b490d88f706fab1a9c57","target":"graph","created_at":"2026-05-18T02:42:39Z","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 mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices Z, A_2 and (presumably) D_4, E_8 and the Leech lattice in being simultaneously optimal with respect to all three criteria.","authors_text":"J. H. Conway, N. J. A. Sloane","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2007-01-03T05:11:31Z","title":"The Optimal Isodual Lattice Quantizer in Three Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701080","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":"sha256:34c7afe709205254e2705fe157f92af12a569bb75d3d71f7ee9ed13cfaa25d7d","target":"record","created_at":"2026-05-18T02:42:39Z","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":"42c3af8357e7aef4792fc83291196ebc1790d0ac8710df6651286471dcda9d92","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2007-01-03T05:11:31Z","title_canon_sha256":"60ebb4656a7b0adc0939c0a4985351390bdc9a741822e7f3739b5940ad8949f2"},"schema_version":"1.0","source":{"id":"math/0701080","kind":"arxiv","version":1}},"canonical_sha256":"441416e2e5f9a13482f56bed6422a63ca5b9e294046a1eda2bba522dcf089d4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"441416e2e5f9a13482f56bed6422a63ca5b9e294046a1eda2bba522dcf089d4c","first_computed_at":"2026-05-18T02:42:39.957781Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:39.957781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tszJGvRbM/UoybTgvdyqyO9+e1rUlf19fRPHwEwP6tzZzpNDdNE1zGhd0icGzVvkCFnLbi6kl/Y3zQuWYKV+Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:39.958461Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701080","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34c7afe709205254e2705fe157f92af12a569bb75d3d71f7ee9ed13cfaa25d7d","sha256:dbcc68037931be4279ba838827a3af8597584fb87e83b490d88f706fab1a9c57"],"state_sha256":"7b2b09b2b56a9fa5229378947fa9a7ced6ab340bc58d9cbb679c235fa96e227a"}