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For the cyclic case we prove that the partition function has the form $Z(\\Lambda,L_y \\times L_x,q,v,w)=\\sum_{d=0}^{L_y} \\tilde c^{(d)} Tr[(T_{Z,\\Lambda,L_y,d})^m]$, where $\\Lambda$ denotes the lattice type, $\\tilde c^{(d)}"},"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":"0907.0925","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-07-06T07:13:42Z","cross_cats_sorted":[],"title_canon_sha256":"df1bbb49765e897d24b3e67548f26aefc98b73ab3f92e7721886ee7d159e940a","abstract_canon_sha256":"6fd26d3c483e56a92d9c6517b966f2eb71ab28b6b941072869eb3ba720f02c60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:13:05.118599Z","signature_b64":"s/tvNfHFPbWforLvgTfqlEwYrOGka8Y8OHomoXMerY3mi9DhfTrnfhQS4RiWL3MbkJgdro75VE91ripNdIu0Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"122f7f387e82c94f449cbe9dfa669b86eb38c1e5db6b4183e8e96fa27aa125ca","last_reissued_at":"2026-05-18T02:13:05.117879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:13:05.117879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Robert Shrock, Shu-Chiuan Chang","submitted_at":"2009-07-06T07:13:42Z","abstract_excerpt":"We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, M\\\"obius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(\\Lambda,L_y \\times L_x,q,v,w)=\\sum_{d=0}^{L_y} \\tilde c^{(d)} Tr[(T_{Z,\\Lambda,L_y,d})^m]$, where $\\Lambda$ denotes the lattice type, $\\tilde c^{(d)}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.0925","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":"0907.0925","created_at":"2026-05-18T02:13:05.117993+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.0925v2","created_at":"2026-05-18T02:13:05.117993+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.0925","created_at":"2026-05-18T02:13:05.117993+00:00"},{"alias_kind":"pith_short_12","alias_value":"CIXX6OD6QLEU","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"CIXX6OD6QLEU6RE4","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"CIXX6OD6","created_at":"2026-05-18T12:25:59.703012+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/CIXX6OD6QLEU6RE4X2O7UZU3Q3","json":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3.json","graph_json":"https://pith.science/api/pith-number/CIXX6OD6QLEU6RE4X2O7UZU3Q3/graph.json","events_json":"https://pith.science/api/pith-number/CIXX6OD6QLEU6RE4X2O7UZU3Q3/events.json","paper":"https://pith.science/paper/CIXX6OD6"},"agent_actions":{"view_html":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3","download_json":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3.json","view_paper":"https://pith.science/paper/CIXX6OD6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.0925&json=true","fetch_graph":"https://pith.science/api/pith-number/CIXX6OD6QLEU6RE4X2O7UZU3Q3/graph.json","fetch_events":"https://pith.science/api/pith-number/CIXX6OD6QLEU6RE4X2O7UZU3Q3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3/action/storage_attestation","attest_author":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3/action/author_attestation","sign_citation":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3/action/citation_signature","submit_replication":"https://pith.science/pith/CIXX6OD6QLEU6RE4X2O7UZU3Q3/action/replication_record"}},"created_at":"2026-05-18T02:13:05.117993+00:00","updated_at":"2026-05-18T02:13:05.117993+00:00"}