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We study the weighted independent-set polynomial $W(n)=\\sum_F\\prod_{S\\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\\cap(n)\\sim n\\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\\log_2(Z_\\cap(n)/R(n))=2^{n-1}-2+\\log_2("},"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":"2607.16040","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T15:15:13Z","cross_cats_sorted":[],"title_canon_sha256":"69e8beb8d3e3d5999d45379a3136b10a10f68fbe3c136f691ff6db8f20aaba29","abstract_canon_sha256":"03dc93257a3e9756ae62d40b6bd905532950ac4d62a20b5efca5774073474eba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T02:19:27.769300Z","signature_b64":"kQr8KJdwTwwD0pT6a/f3E55viQkqzHhzIlUOf8UptxgRJXdDUelq9BCq4BjpNvB+6fcjOfsDzUFEbwJ45eJjCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a6b1e746ac55294d748a792c4f728f5ce0415fb5e15d13588b991c62a49077b","last_reissued_at":"2026-07-20T02:19:27.768439Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T02:19:27.768439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A gap theorem for non-trivial maximal intersecting families and an exact weighted asymptotic","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"El'mira Yu. Kalimulina","submitted_at":"2026-07-17T15:15:13Z","abstract_excerpt":"Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\\sum_F\\prod_{S\\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\\cap(n)\\sim n\\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\\log_2(Z_\\cap(n)/R(n))=2^{n-1}-2+\\log_2("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16040","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/2607.16040/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":"2607.16040","created_at":"2026-07-20T02:19:27.768876+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.16040v1","created_at":"2026-07-20T02:19:27.768876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16040","created_at":"2026-07-20T02:19:27.768876+00:00"},{"alias_kind":"pith_short_12","alias_value":"PJVR45DKYVJJ","created_at":"2026-07-20T02:19:27.768876+00:00"},{"alias_kind":"pith_short_16","alias_value":"PJVR45DKYVJJJV2I","created_at":"2026-07-20T02:19:27.768876+00:00"},{"alias_kind":"pith_short_8","alias_value":"PJVR45DK","created_at":"2026-07-20T02:19:27.768876+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/PJVR45DKYVJJJV2IU6JMJ5ZI6X","json":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X.json","graph_json":"https://pith.science/api/pith-number/PJVR45DKYVJJJV2IU6JMJ5ZI6X/graph.json","events_json":"https://pith.science/api/pith-number/PJVR45DKYVJJJV2IU6JMJ5ZI6X/events.json","paper":"https://pith.science/paper/PJVR45DK"},"agent_actions":{"view_html":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X","download_json":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X.json","view_paper":"https://pith.science/paper/PJVR45DK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.16040&json=true","fetch_graph":"https://pith.science/api/pith-number/PJVR45DKYVJJJV2IU6JMJ5ZI6X/graph.json","fetch_events":"https://pith.science/api/pith-number/PJVR45DKYVJJJV2IU6JMJ5ZI6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X/action/storage_attestation","attest_author":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X/action/author_attestation","sign_citation":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X/action/citation_signature","submit_replication":"https://pith.science/pith/PJVR45DKYVJJJV2IU6JMJ5ZI6X/action/replication_record"}},"created_at":"2026-07-20T02:19:27.768876+00:00","updated_at":"2026-07-20T02:19:27.768876+00:00"}