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We prove a version of their theorem for the Grassmann scheme over $\\mathbb{F}_2$. More precisely, we prove if a function $f\\colon \\genfrac{[}{]}{0pt}{}{\\mathbb{F}_2^n}{\\ell}\\to\\{0,1\\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \\sum_{x\\in\\mathcal{X}}1_{x\\in L}+\\sum_{W\\in\\mathcal{W}}1_{L\\"},"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":"2608.11320","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-11T18:12:14Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bb2e2d25e3b5e6efe53984ec0de901101eeae7a890b6ff30dbd1e4474691b7fc","abstract_canon_sha256":"db7b00cb84cd0adc58000776b6cecfec47a81e9893b188b4a2f6a6113c2e1add"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-13T00:21:35.343126Z","signature_b64":"0Y343RRUHMYKkmUwjk25thrmDiXMeH7TDRaFKTTsmXZjySjC6mUcYBFkWSH2afEdQdHHTMf8umnv1zJqwYMUCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"193a926ca8dac4548d269684263c08d3e17c34646f4bdc3f86cb807646a1c85c","last_reissued_at":"2026-08-13T00:21:35.341458Z","signature_status":"signed_v1","first_computed_at":"2026-08-13T00:21:35.341458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An FKN Theorem for the Binary Grassmann Scheme","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CC","authors_text":"Anqi Li, Dor Minzer, Yuval Filmus","submitted_at":"2026-08-11T18:12:14Z","abstract_excerpt":"A classical theorem due to Friedgut, Kalai and Naor asserts that if a function $f\\colon \\{0,1\\}^n\\to\\{-1,1\\}$ close to a degree $1$ function, then either $f$ or $-f$ is close to either the all $1$ function, or to $(-1)^{x_i}$ for some $i\\in [n]$. We prove a version of their theorem for the Grassmann scheme over $\\mathbb{F}_2$. More precisely, we prove if a function $f\\colon \\genfrac{[}{]}{0pt}{}{\\mathbb{F}_2^n}{\\ell}\\to\\{0,1\\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \\sum_{x\\in\\mathcal{X}}1_{x\\in L}+\\sum_{W\\in\\mathcal{W}}1_{L\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11320","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/2608.11320/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":"2608.11320","created_at":"2026-08-13T00:21:35.341820+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.11320v1","created_at":"2026-08-13T00:21:35.341820+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.11320","created_at":"2026-08-13T00:21:35.341820+00:00"},{"alias_kind":"pith_short_12","alias_value":"DE5JE3FI3LCF","created_at":"2026-08-13T00:21:35.341820+00:00"},{"alias_kind":"pith_short_16","alias_value":"DE5JE3FI3LCFJDJG","created_at":"2026-08-13T00:21:35.341820+00:00"},{"alias_kind":"pith_short_8","alias_value":"DE5JE3FI","created_at":"2026-08-13T00:21:35.341820+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/DE5JE3FI3LCFJDJGS2CCMPAI2P","json":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P.json","graph_json":"https://pith.science/api/pith-number/DE5JE3FI3LCFJDJGS2CCMPAI2P/graph.json","events_json":"https://pith.science/api/pith-number/DE5JE3FI3LCFJDJGS2CCMPAI2P/events.json","paper":"https://pith.science/paper/DE5JE3FI"},"agent_actions":{"view_html":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P","download_json":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P.json","view_paper":"https://pith.science/paper/DE5JE3FI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.11320&json=true","fetch_graph":"https://pith.science/api/pith-number/DE5JE3FI3LCFJDJGS2CCMPAI2P/graph.json","fetch_events":"https://pith.science/api/pith-number/DE5JE3FI3LCFJDJGS2CCMPAI2P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P/action/storage_attestation","attest_author":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P/action/author_attestation","sign_citation":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P/action/citation_signature","submit_replication":"https://pith.science/pith/DE5JE3FI3LCFJDJGS2CCMPAI2P/action/replication_record"}},"created_at":"2026-08-13T00:21:35.341820+00:00","updated_at":"2026-08-13T00:21:35.341820+00:00"}