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We prove that the free energy $F_N = F_{N,\\beta=1}$ of this model has variance\n  \\[\n  \\mathrm{Var}(F_N) = \\frac16 \\log N + O(1)\\,,\n  \\] confirming a physics prediction of Aspelmeier \\cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving\n  \\[\n  \\mathbb{E} \\langle R_{1,2}^2\\rangle \\asymp N^{-2/3}\\,,\n  \\] as conjectured by Talagrand \\cite{talagrand2011mean2}, togethe"},"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":"2607.02172","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-02T13:44:13Z","cross_cats_sorted":["cond-mat.dis-nn","math-ph","math.MP"],"title_canon_sha256":"6028b209f3644e7570819b44450a316c2965958d9def915fa4bf55a8bffe9089","abstract_canon_sha256":"bfa9e67865ca326cdecdeff772ce9f30a2f7de08da9d3d5e4d1c182f56277c31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:43.977681Z","signature_b64":"j4vHRXX5gb+NgHJhqgrF5vLdEwyb3XLrlbhbkTIE8+AAjN6NzBqVeHRUvX9unMWiFrBiFccj2HpFhSIUO3/uBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0846013288ea61d43da807d324928915fb5ce395790848328c41d11a08f2e485","last_reissued_at":"2026-07-03T01:17:43.977255Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:43.977255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fluctuations of the Sherrington--Kirkpatrick free energy at critical temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Brice Huang, Hang Du","submitted_at":"2026-07-02T13:44:13Z","abstract_excerpt":"We consider the Sherrington--Kirkpatrick spin glass model at the critical inverse temperature $\\beta = 1$ with zero external field. 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We also identify the critical two-replica overlap scale, proving\n  \\[\n  \\mathbb{E} \\langle R_{1,2}^2\\rangle \\asymp N^{-2/3}\\,,\n  \\] as conjectured by Talagrand \\cite{talagrand2011mean2}, togethe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02172","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/2607.02172/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":"2607.02172","created_at":"2026-07-03T01:17:43.977321+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.02172v1","created_at":"2026-07-03T01:17:43.977321+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02172","created_at":"2026-07-03T01:17:43.977321+00:00"},{"alias_kind":"pith_short_12","alias_value":"BBDACMUI5JQ5","created_at":"2026-07-03T01:17:43.977321+00:00"},{"alias_kind":"pith_short_16","alias_value":"BBDACMUI5JQ5IPNI","created_at":"2026-07-03T01:17:43.977321+00:00"},{"alias_kind":"pith_short_8","alias_value":"BBDACMUI","created_at":"2026-07-03T01:17:43.977321+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/BBDACMUI5JQ5IPNIA7JSJEUJCX","json":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX.json","graph_json":"https://pith.science/api/pith-number/BBDACMUI5JQ5IPNIA7JSJEUJCX/graph.json","events_json":"https://pith.science/api/pith-number/BBDACMUI5JQ5IPNIA7JSJEUJCX/events.json","paper":"https://pith.science/paper/BBDACMUI"},"agent_actions":{"view_html":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX","download_json":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX.json","view_paper":"https://pith.science/paper/BBDACMUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.02172&json=true","fetch_graph":"https://pith.science/api/pith-number/BBDACMUI5JQ5IPNIA7JSJEUJCX/graph.json","fetch_events":"https://pith.science/api/pith-number/BBDACMUI5JQ5IPNIA7JSJEUJCX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX/action/storage_attestation","attest_author":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX/action/author_attestation","sign_citation":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX/action/citation_signature","submit_replication":"https://pith.science/pith/BBDACMUI5JQ5IPNIA7JSJEUJCX/action/replication_record"}},"created_at":"2026-07-03T01:17:43.977321+00:00","updated_at":"2026-07-03T01:17:43.977321+00:00"}