{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:LCRQTL54HTHC2HYNHLSFHR7UGZ","short_pith_number":"pith:LCRQTL54","canonical_record":{"source":{"id":"1009.1443","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-09-08T02:37:55Z","cross_cats_sorted":["cond-mat.soft","math.MG"],"title_canon_sha256":"2a213e6da4f94ee2ad62510dde90c8c7162db2a03b3998e85576839d3c4379c4","abstract_canon_sha256":"2bd0fa1833e67d38020716fcea209addcbcfd10fda67d4680fb2b18708897f83"},"schema_version":"1.0"},"canonical_sha256":"58a309afbc3cce2d1f0d3ae453c7f436443a6aaa7f1b0884841e9ae7f9b3253c","source":{"kind":"arxiv","id":"1009.1443","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.1443","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1009.1443v2","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.1443","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"LCRQTL54HTHC","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LCRQTL54HTHC2HYN","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LCRQTL54","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:LCRQTL54HTHC2HYNHLSFHR7UGZ","target":"record","payload":{"canonical_record":{"source":{"id":"1009.1443","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-09-08T02:37:55Z","cross_cats_sorted":["cond-mat.soft","math.MG"],"title_canon_sha256":"2a213e6da4f94ee2ad62510dde90c8c7162db2a03b3998e85576839d3c4379c4","abstract_canon_sha256":"2bd0fa1833e67d38020716fcea209addcbcfd10fda67d4680fb2b18708897f83"},"schema_version":"1.0"},"canonical_sha256":"58a309afbc3cce2d1f0d3ae453c7f436443a6aaa7f1b0884841e9ae7f9b3253c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:05:35.266647Z","signature_b64":"uq5SCzdJY0xX49w/ZnrnTaHLD6FjEvkqV6pUGQ1sQFDgZy6/qi8OOa6nuGpgsAKQ43sJZV8FaivzwUQ5NNSBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58a309afbc3cce2d1f0d3ae453c7f436443a6aaa7f1b0884841e9ae7f9b3253c","last_reissued_at":"2026-05-18T02:05:35.265822Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:05:35.265822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.1443","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O3JY2tEBMSihcoSOh3BfRLsv1GGko5RShbTuGueCmBndjxuDGb+70teVvEqR3ilVg59zGX/tOqwOAC8PRFNhAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:19:34.316236Z"},"content_sha256":"26204ea1c6dab451c2298c0607386c1796f71ae741b77a3984fc45a3e59fe54d","schema_version":"1.0","event_id":"sha256:26204ea1c6dab451c2298c0607386c1796f71ae741b77a3984fc45a3e59fe54d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:LCRQTL54HTHC2HYNHLSFHR7UGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.soft","math.MG"],"primary_cat":"cond-mat.stat-mech","authors_text":"S. Torquato","submitted_at":"2010-09-08T02:37:55Z","abstract_excerpt":"We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in $\\mathbb{R}^d$ that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the \"void\" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization technique"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1443","kind":"arxiv","version":2},"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":"2026-05-18T02:05:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fKj77VbHSWcXOSyBjieDj8Vye8Y9jnQ9OZCFZ6x0PIGurini+3wwYW4Wya1qNEP3sg/F90AhjGeAdt4fFVYxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:19:34.316607Z"},"content_sha256":"b453768b483f37d60c06d7409a3d3ae30b41418652e024ab82fb16f4dd2981a8","schema_version":"1.0","event_id":"sha256:b453768b483f37d60c06d7409a3d3ae30b41418652e024ab82fb16f4dd2981a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/bundle.json","state_url":"https://pith.science/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/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":"2026-06-01T10:19:34Z","links":{"resolver":"https://pith.science/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ","bundle":"https://pith.science/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/bundle.json","state":"https://pith.science/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LCRQTL54HTHC2HYNHLSFHR7UGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:LCRQTL54HTHC2HYNHLSFHR7UGZ","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":"2bd0fa1833e67d38020716fcea209addcbcfd10fda67d4680fb2b18708897f83","cross_cats_sorted":["cond-mat.soft","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-09-08T02:37:55Z","title_canon_sha256":"2a213e6da4f94ee2ad62510dde90c8c7162db2a03b3998e85576839d3c4379c4"},"schema_version":"1.0","source":{"id":"1009.1443","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.1443","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"arxiv_version","alias_value":"1009.1443v2","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.1443","created_at":"2026-05-18T02:05:35Z"},{"alias_kind":"pith_short_12","alias_value":"LCRQTL54HTHC","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LCRQTL54HTHC2HYN","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LCRQTL54","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:b453768b483f37d60c06d7409a3d3ae30b41418652e024ab82fb16f4dd2981a8","target":"graph","created_at":"2026-05-18T02:05:35Z","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 reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in $\\mathbb{R}^d$ that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the \"void\" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization technique","authors_text":"S. Torquato","cross_cats":["cond-mat.soft","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-09-08T02:37:55Z","title":"Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1443","kind":"arxiv","version":2},"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":"sha256:26204ea1c6dab451c2298c0607386c1796f71ae741b77a3984fc45a3e59fe54d","target":"record","created_at":"2026-05-18T02:05:35Z","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":"2bd0fa1833e67d38020716fcea209addcbcfd10fda67d4680fb2b18708897f83","cross_cats_sorted":["cond-mat.soft","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-09-08T02:37:55Z","title_canon_sha256":"2a213e6da4f94ee2ad62510dde90c8c7162db2a03b3998e85576839d3c4379c4"},"schema_version":"1.0","source":{"id":"1009.1443","kind":"arxiv","version":2}},"canonical_sha256":"58a309afbc3cce2d1f0d3ae453c7f436443a6aaa7f1b0884841e9ae7f9b3253c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58a309afbc3cce2d1f0d3ae453c7f436443a6aaa7f1b0884841e9ae7f9b3253c","first_computed_at":"2026-05-18T02:05:35.265822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:05:35.265822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uq5SCzdJY0xX49w/ZnrnTaHLD6FjEvkqV6pUGQ1sQFDgZy6/qi8OOa6nuGpgsAKQ43sJZV8FaivzwUQ5NNSBBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:05:35.266647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.1443","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26204ea1c6dab451c2298c0607386c1796f71ae741b77a3984fc45a3e59fe54d","sha256:b453768b483f37d60c06d7409a3d3ae30b41418652e024ab82fb16f4dd2981a8"],"state_sha256":"f3efd6bc4432b6973832f5262c962a5d2fd0ead6587d08be114f5a9e7da49ca2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xzm12VVIwlEWNpnHyE/dvKnax0AaMy7jb07Rop+tmOXKpa3dpVvQ1Rf7GjmUo5NMfLOvUITD3O1xly7bLsIXDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T10:19:34.318583Z","bundle_sha256":"7399a5b4a085d20b9579a45be4aaa9f3a83a870ec0f45658ce1c389305e47614"}}