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Suppose $M$ is a function of $N$ satisfying $M\\geq N/(\\ln N)^{\\alpha}$ for some $\\alpha\\in[0,1)$. In particular, one may take $M=N$. We prove that \\begin{equation*} \\frac{E_{M,N}+4\\sqrt{2}M N-(4/\\pi)N\\ln N}{\\sqrt{(32/\\pi)MN\\ln N}} \\end{equation*} converges weakly to a standard Gaussian distribution as $N\\rightarrow\\infty$."},"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":"1908.06704","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-08-19T11:25:24Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"fcc667eda6e3954f004a270a7651235ea92910ed9a624972f9d418a61814616c","abstract_canon_sha256":"49ceccb4971e4976389eac33a05d4dda931dea30b5193e43fe50540b79ec87b7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:58:18.649614Z","signature_b64":"Y81zDiTcCM8EFeub+IPys35z+s2ei8oQk6aqemanfal4KBz1R7lEUGcLZBizJ29EJAoCZTuAJN6Hur2+Dbx0CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf7bc9e52ae218fddb8497083d7a00f28776966bb886456521949a0a9a6cc1ff","last_reissued_at":"2026-07-04T23:58:18.649261Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:58:18.649261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A CLT for the total energy of the two-dimensional critical Ising model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jianping Jiang","submitted_at":"2019-08-19T11:25:24Z","abstract_excerpt":"Consider the Ising model on $([1,2N]\\times[1,2M])\\cap\\mathbb{Z}^2$ at critical temperature with periodic boundary condition in the horizontal direction and free boundary condition in the vertical direction. Let $E_{M,N}$ be its total energy (or Hamiltonian). Suppose $M$ is a function of $N$ satisfying $M\\geq N/(\\ln N)^{\\alpha}$ for some $\\alpha\\in[0,1)$. In particular, one may take $M=N$. We prove that \\begin{equation*} \\frac{E_{M,N}+4\\sqrt{2}M N-(4/\\pi)N\\ln N}{\\sqrt{(32/\\pi)MN\\ln N}} \\end{equation*} converges weakly to a standard Gaussian distribution as $N\\rightarrow\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06704","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.06704/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.06704","created_at":"2026-07-04T23:58:18.649324+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.06704v1","created_at":"2026-07-04T23:58:18.649324+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.06704","created_at":"2026-07-04T23:58:18.649324+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z554TZJK4IMP","created_at":"2026-07-04T23:58:18.649324+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z554TZJK4IMP3W4E","created_at":"2026-07-04T23:58:18.649324+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z554TZJK","created_at":"2026-07-04T23:58:18.649324+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/Z554TZJK4IMP3W4ES4ED26QA6K","json":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K.json","graph_json":"https://pith.science/api/pith-number/Z554TZJK4IMP3W4ES4ED26QA6K/graph.json","events_json":"https://pith.science/api/pith-number/Z554TZJK4IMP3W4ES4ED26QA6K/events.json","paper":"https://pith.science/paper/Z554TZJK"},"agent_actions":{"view_html":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K","download_json":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K.json","view_paper":"https://pith.science/paper/Z554TZJK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.06704&json=true","fetch_graph":"https://pith.science/api/pith-number/Z554TZJK4IMP3W4ES4ED26QA6K/graph.json","fetch_events":"https://pith.science/api/pith-number/Z554TZJK4IMP3W4ES4ED26QA6K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K/action/storage_attestation","attest_author":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K/action/author_attestation","sign_citation":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K/action/citation_signature","submit_replication":"https://pith.science/pith/Z554TZJK4IMP3W4ES4ED26QA6K/action/replication_record"}},"created_at":"2026-07-04T23:58:18.649324+00:00","updated_at":"2026-07-04T23:58:18.649324+00:00"}