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We classify equality: $e$ is sharp if and only if $e$ and all edges meeting it form $t$ projective planes of order $q$ glued along the common line $e$. This equality state is repulsive. If $f\\ne e$ meets a sharp edge $e$, then at least $q-t+2$ vertices of $f\\setminus e$ have degree at most $D^-$. It follows that sharp edges are pairwise disjoint, every edge meets at most $"},"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":"2608.01568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T00:56:21Z","cross_cats_sorted":[],"title_canon_sha256":"c15b13c49ae5155ff801bc16eb0c8a41794620247768bcf2b224012cda83482c","abstract_canon_sha256":"bad6c1d7d8d45442788a691cf24a00393c59dd29cca202a1b2a027c265bf9d75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:06:00.489726Z","signature_b64":"cilTA620aMRbBz6zGdgIlsVPxXyFwc03cQ+rLfzBUV956d0oVMlHvJ0/H8FUr30ggg1kqmPqJ6jgFQzHbMAiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6983b40c7d7c9430e271ad1ed3d14ebfc2a424f6ea2f570ab54a23a4bb2ca2d7","last_reissued_at":"2026-08-04T02:06:00.488163Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:06:00.488163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Projective-Plane Sharp States and Repulsion in Linear Crown-Free Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mahesh Ramani","submitted_at":"2026-08-03T00:56:21Z","abstract_excerpt":"Let $3 \\le k \\le r$, and let $C^r_{1,k}$ be the $r$-uniform $k$-crown. Set $q=r-1$, $t=k-1$, $D=tq+1$, and $D^-=(t-1)q+1$. For every edge $e$ of a linear $C^r_{1,k}$-free $r$-graph, $\\Phi_H(e):=\\sum_{v\\in e}1/d_H(v)\\ge r/D$. We classify equality: $e$ is sharp if and only if $e$ and all edges meeting it form $t$ projective planes of order $q$ glued along the common line $e$. This equality state is repulsive. If $f\\ne e$ meets a sharp edge $e$, then at least $q-t+2$ vertices of $f\\setminus e$ have degree at most $D^-$. It follows that sharp edges are pairwise disjoint, every edge meets at most $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01568","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/2608.01568/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":"2608.01568","created_at":"2026-08-04T02:06:00.489635+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.01568v1","created_at":"2026-08-04T02:06:00.489635+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01568","created_at":"2026-08-04T02:06:00.489635+00:00"},{"alias_kind":"pith_short_12","alias_value":"NGB3IDD5PSKD","created_at":"2026-08-04T02:06:00.489635+00:00"},{"alias_kind":"pith_short_16","alias_value":"NGB3IDD5PSKDBYTR","created_at":"2026-08-04T02:06:00.489635+00:00"},{"alias_kind":"pith_short_8","alias_value":"NGB3IDD5","created_at":"2026-08-04T02:06:00.489635+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/NGB3IDD5PSKDBYTRVUPNHUKOX7","json":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7.json","graph_json":"https://pith.science/api/pith-number/NGB3IDD5PSKDBYTRVUPNHUKOX7/graph.json","events_json":"https://pith.science/api/pith-number/NGB3IDD5PSKDBYTRVUPNHUKOX7/events.json","paper":"https://pith.science/paper/NGB3IDD5"},"agent_actions":{"view_html":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7","download_json":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7.json","view_paper":"https://pith.science/paper/NGB3IDD5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.01568&json=true","fetch_graph":"https://pith.science/api/pith-number/NGB3IDD5PSKDBYTRVUPNHUKOX7/graph.json","fetch_events":"https://pith.science/api/pith-number/NGB3IDD5PSKDBYTRVUPNHUKOX7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7/action/storage_attestation","attest_author":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7/action/author_attestation","sign_citation":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7/action/citation_signature","submit_replication":"https://pith.science/pith/NGB3IDD5PSKDBYTRVUPNHUKOX7/action/replication_record"}},"created_at":"2026-08-04T02:06:00.489635+00:00","updated_at":"2026-08-04T02:06:00.489635+00:00"}