{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:AJETUWRMR3NDAVHBFQNTQDIR4Y","short_pith_number":"pith:AJETUWRM","schema_version":"1.0","canonical_sha256":"02493a5a2c8eda3054e12c1b380d11e6194d0d914430390a9f11d416648d230b","source":{"kind":"arxiv","id":"2606.31961","version":1},"attestation_state":"computed","paper":{"title":"A Beckmann boundary form of Talagrand's conjecture on the discrete cube","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.CA","authors_text":"Haonan Zhang, Paata Ivanisvili, Xinyuan Xie","submitted_at":"2026-06-30T17:02:21Z","abstract_excerpt":"We introduce the Beckmann boundary of a Boolean function \\[\n  \\mathsf{B}(f)=\\inf_{\\operatorname{div} V=Lf}\\mathbb E\\|V(x)\\|_2. \\] Here \\[\n  L=\\sum_iD_i,\\qquad D_i f(x)=\\frac{f(x)-f(x^{\\oplus i})}{2}, \\] and $\\operatorname{div} V(x)=\\sum_i (V_{i}(x)-V_{i}(x^{\\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \\[\n  \\mathsf{B}(f)\\gtrsim \\operatorname{Var}(f)\n  \\sqrt{\\log\\!\\left(1+\\frac{1}{\\sum_i\\operatorname{Inf}_i(f)^2}\\right)}. \\] We al"},"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":"2606.31961","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-30T17:02:21Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"3ae2d4a5ae5271745652d5e63dd9bcb01cdcb2f83dba75b1ac83466bc07073eb","abstract_canon_sha256":"b0767609ed35441f034d093a21532306cea8bf3b3ce5b75f094bb77a41a816ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:18:27.080357Z","signature_b64":"YePTML0ukmgC3FBv2Ag/weaLTEYvoz4yOku/DNfvjGYXEt/K5jhpmrQ7TE7d+Bmppi/h9miVJ7jSd5pM2yMXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02493a5a2c8eda3054e12c1b380d11e6194d0d914430390a9f11d416648d230b","last_reissued_at":"2026-07-01T01:18:27.079926Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:18:27.079926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Beckmann boundary form of Talagrand's conjecture on the discrete cube","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.CA","authors_text":"Haonan Zhang, Paata Ivanisvili, Xinyuan Xie","submitted_at":"2026-06-30T17:02:21Z","abstract_excerpt":"We introduce the Beckmann boundary of a Boolean function \\[\n  \\mathsf{B}(f)=\\inf_{\\operatorname{div} V=Lf}\\mathbb E\\|V(x)\\|_2. \\] Here \\[\n  L=\\sum_iD_i,\\qquad D_i f(x)=\\frac{f(x)-f(x^{\\oplus i})}{2}, \\] and $\\operatorname{div} V(x)=\\sum_i (V_{i}(x)-V_{i}(x^{\\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \\[\n  \\mathsf{B}(f)\\gtrsim \\operatorname{Var}(f)\n  \\sqrt{\\log\\!\\left(1+\\frac{1}{\\sum_i\\operatorname{Inf}_i(f)^2}\\right)}. \\] We al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31961","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/2606.31961/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":"2606.31961","created_at":"2026-07-01T01:18:27.079983+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.31961v1","created_at":"2026-07-01T01:18:27.079983+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.31961","created_at":"2026-07-01T01:18:27.079983+00:00"},{"alias_kind":"pith_short_12","alias_value":"AJETUWRMR3ND","created_at":"2026-07-01T01:18:27.079983+00:00"},{"alias_kind":"pith_short_16","alias_value":"AJETUWRMR3NDAVHB","created_at":"2026-07-01T01:18:27.079983+00:00"},{"alias_kind":"pith_short_8","alias_value":"AJETUWRM","created_at":"2026-07-01T01:18:27.079983+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/AJETUWRMR3NDAVHBFQNTQDIR4Y","json":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y.json","graph_json":"https://pith.science/api/pith-number/AJETUWRMR3NDAVHBFQNTQDIR4Y/graph.json","events_json":"https://pith.science/api/pith-number/AJETUWRMR3NDAVHBFQNTQDIR4Y/events.json","paper":"https://pith.science/paper/AJETUWRM"},"agent_actions":{"view_html":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y","download_json":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y.json","view_paper":"https://pith.science/paper/AJETUWRM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.31961&json=true","fetch_graph":"https://pith.science/api/pith-number/AJETUWRMR3NDAVHBFQNTQDIR4Y/graph.json","fetch_events":"https://pith.science/api/pith-number/AJETUWRMR3NDAVHBFQNTQDIR4Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y/action/storage_attestation","attest_author":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y/action/author_attestation","sign_citation":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y/action/citation_signature","submit_replication":"https://pith.science/pith/AJETUWRMR3NDAVHBFQNTQDIR4Y/action/replication_record"}},"created_at":"2026-07-01T01:18:27.079983+00:00","updated_at":"2026-07-01T01:18:27.079983+00:00"}