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Let $\\phi_i: \\{0, 1\\}^n \\longrightarrow {\\Bbb C}$ be $1$-Lipschitz functions in the Hamming metric, such that each $\\phi_i$ depends on at most $r$ coordinates of $x \\in \\{0, 1\\}^n$, where $rp \\geq 12$. For $j=1, \\ldots, n$, let $I_j $ be the set of indices $i$ such that $\\phi_i$ depends on the $j$-th coordinate. We prove that $E\\thinspace \\exp\\left\\{ \\sum_{i=1}^m \\lambda_i \\phi_i \\right\\} \\ne 0$ provided $\\lambda_i \\in {\\Bbb C}$ satisfy $\\sum_{i"},"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":"2608.03687","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-08-04T13:56:28Z","cross_cats_sorted":["math.CO","math.MP","math.PR"],"title_canon_sha256":"4d8994d590304f52072802972ff10e6d43fbf1c6642b48491d3662231e33e896","abstract_canon_sha256":"3662957e99cf8df713bfdafecedca9447e4341f7dede9d36efa7056e7d1606e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-05T01:36:53.627478Z","signature_b64":"f0fASBOzKnfa64l6j0Dx8W6orlCrdoBSiZZ2vpHbhX8sQ7Ff+DfRhMWn1fY2HgxFQVm/lZPt0Tp94WBFbaTFBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d5e89268f737796ab9bfad9d5669a13c4114aa24e0efc365e922944c5885890a","last_reissued_at":"2026-08-05T01:36:53.626011Z","signature_status":"signed_v1","first_computed_at":"2026-08-05T01:36:53.626011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dependence of the zero-free region of a partition function on the external field","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Alexander Barvinok","submitted_at":"2026-08-04T13:56:28Z","abstract_excerpt":"Let $\\{0, 1\\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\\Bbb P}(1)=p$ and ${\\Bbb P}(0)=q$ with $0 < p \\leq q=1-p$. Let $\\phi_i: \\{0, 1\\}^n \\longrightarrow {\\Bbb C}$ be $1$-Lipschitz functions in the Hamming metric, such that each $\\phi_i$ depends on at most $r$ coordinates of $x \\in \\{0, 1\\}^n$, where $rp \\geq 12$. For $j=1, \\ldots, n$, let $I_j $ be the set of indices $i$ such that $\\phi_i$ depends on the $j$-th coordinate. We prove that $E\\thinspace \\exp\\left\\{ \\sum_{i=1}^m \\lambda_i \\phi_i \\right\\} \\ne 0$ provided $\\lambda_i \\in {\\Bbb C}$ satisfy $\\sum_{i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.03687","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/2608.03687/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":"2608.03687","created_at":"2026-08-05T01:36:53.626528+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.03687v1","created_at":"2026-08-05T01:36:53.626528+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.03687","created_at":"2026-08-05T01:36:53.626528+00:00"},{"alias_kind":"pith_short_12","alias_value":"2XUJE2HXG54W","created_at":"2026-08-05T01:36:53.626528+00:00"},{"alias_kind":"pith_short_16","alias_value":"2XUJE2HXG54WVON7","created_at":"2026-08-05T01:36:53.626528+00:00"},{"alias_kind":"pith_short_8","alias_value":"2XUJE2HX","created_at":"2026-08-05T01:36:53.626528+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/2XUJE2HXG54WVON7VWOVM2NBHR","json":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR.json","graph_json":"https://pith.science/api/pith-number/2XUJE2HXG54WVON7VWOVM2NBHR/graph.json","events_json":"https://pith.science/api/pith-number/2XUJE2HXG54WVON7VWOVM2NBHR/events.json","paper":"https://pith.science/paper/2XUJE2HX"},"agent_actions":{"view_html":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR","download_json":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR.json","view_paper":"https://pith.science/paper/2XUJE2HX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.03687&json=true","fetch_graph":"https://pith.science/api/pith-number/2XUJE2HXG54WVON7VWOVM2NBHR/graph.json","fetch_events":"https://pith.science/api/pith-number/2XUJE2HXG54WVON7VWOVM2NBHR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR/action/storage_attestation","attest_author":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR/action/author_attestation","sign_citation":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR/action/citation_signature","submit_replication":"https://pith.science/pith/2XUJE2HXG54WVON7VWOVM2NBHR/action/replication_record"}},"created_at":"2026-08-05T01:36:53.626528+00:00","updated_at":"2026-08-05T01:36:53.626528+00:00"}