{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:XILBQJMF3B2WPNIOAA3UJ7D2CV","short_pith_number":"pith:XILBQJMF","schema_version":"1.0","canonical_sha256":"ba16182585d87567b50e003744fc7a15676e203a8fd3bcd2c21366491092e01b","source":{"kind":"arxiv","id":"1907.04208","version":2},"attestation_state":"computed","paper":{"title":"A Direct Construction of Primitive Formally Dual Pairs Having Subsets with Unequal Sizes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Alexander Pott, Shuxing Li","submitted_at":"2019-07-05T22:22:17Z","abstract_excerpt":"The concept of formal duality was proposed by Cohn, Kumar and Sch\\\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\\\"urmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and Pott (arXiv:1810.05433v3), we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.04208","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-05T22:22:17Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"653777360d814928c8a6a6357e89bcfefa3010e8c1f700445a96e61d8fed8c5c","abstract_canon_sha256":"2fd737c7d0969339c67356e1c45987d59148c6c6977128562c74f164ac98b0bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:48.549579Z","signature_b64":"Vh3gMzncIiZmoaNOSKi8ue4u7YzZCOLOtPJhH1bgOA3teurmbbrzEKQP87Dy/ctiQ+t8db6NjI9HBVirMJrfCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba16182585d87567b50e003744fc7a15676e203a8fd3bcd2c21366491092e01b","last_reissued_at":"2026-05-17T23:40:48.548898Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:48.548898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Direct Construction of Primitive Formally Dual Pairs Having Subsets with Unequal Sizes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Alexander Pott, Shuxing Li","submitted_at":"2019-07-05T22:22:17Z","abstract_excerpt":"The concept of formal duality was proposed by Cohn, Kumar and Sch\\\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\\\"urmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and Pott (arXiv:1810.05433v3), we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04208","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1907.04208","created_at":"2026-05-17T23:40:48.549013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.04208v2","created_at":"2026-05-17T23:40:48.549013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04208","created_at":"2026-05-17T23:40:48.549013+00:00"},{"alias_kind":"pith_short_12","alias_value":"XILBQJMF3B2W","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"XILBQJMF3B2WPNIO","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"XILBQJMF","created_at":"2026-05-18T12:33:33.725879+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV","json":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV.json","graph_json":"https://pith.science/api/pith-number/XILBQJMF3B2WPNIOAA3UJ7D2CV/graph.json","events_json":"https://pith.science/api/pith-number/XILBQJMF3B2WPNIOAA3UJ7D2CV/events.json","paper":"https://pith.science/paper/XILBQJMF"},"agent_actions":{"view_html":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV","download_json":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV.json","view_paper":"https://pith.science/paper/XILBQJMF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.04208&json=true","fetch_graph":"https://pith.science/api/pith-number/XILBQJMF3B2WPNIOAA3UJ7D2CV/graph.json","fetch_events":"https://pith.science/api/pith-number/XILBQJMF3B2WPNIOAA3UJ7D2CV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV/action/storage_attestation","attest_author":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV/action/author_attestation","sign_citation":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV/action/citation_signature","submit_replication":"https://pith.science/pith/XILBQJMF3B2WPNIOAA3UJ7D2CV/action/replication_record"}},"created_at":"2026-05-17T23:40:48.549013+00:00","updated_at":"2026-05-17T23:40:48.549013+00:00"}