{"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"}