{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NGB3IDD5PSKDBYTRVUPNHUKOX7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bad6c1d7d8d45442788a691cf24a00393c59dd29cca202a1b2a027c265bf9d75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T00:56:21Z","title_canon_sha256":"c15b13c49ae5155ff801bc16eb0c8a41794620247768bcf2b224012cda83482c"},"schema_version":"1.0","source":{"id":"2608.01568","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.01568","created_at":"2026-08-04T02:06:00Z"},{"alias_kind":"arxiv_version","alias_value":"2608.01568v1","created_at":"2026-08-04T02:06:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.01568","created_at":"2026-08-04T02:06:00Z"},{"alias_kind":"pith_short_12","alias_value":"NGB3IDD5PSKD","created_at":"2026-08-04T02:06:00Z"},{"alias_kind":"pith_short_16","alias_value":"NGB3IDD5PSKDBYTR","created_at":"2026-08-04T02:06:00Z"},{"alias_kind":"pith_short_8","alias_value":"NGB3IDD5","created_at":"2026-08-04T02:06:00Z"}],"graph_snapshots":[{"event_id":"sha256:06546a7e7c539cb69d9701775d621d5cee20c3b209bab15ef1a0b072c844d30d","target":"graph","created_at":"2026-08-04T02:06:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.01568/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 $","authors_text":"Mahesh Ramani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T00:56:21Z","title":"Projective-Plane Sharp States and Repulsion in Linear Crown-Free Hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01568","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e963244302f6c6c8b2e99f874bc68dcc5b798978aa46b0bc41a23d68d31d0736","target":"record","created_at":"2026-08-04T02:06:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bad6c1d7d8d45442788a691cf24a00393c59dd29cca202a1b2a027c265bf9d75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-08-03T00:56:21Z","title_canon_sha256":"c15b13c49ae5155ff801bc16eb0c8a41794620247768bcf2b224012cda83482c"},"schema_version":"1.0","source":{"id":"2608.01568","kind":"arxiv","version":1}},"canonical_sha256":"6983b40c7d7c9430e271ad1ed3d14ebfc2a424f6ea2f570ab54a23a4bb2ca2d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6983b40c7d7c9430e271ad1ed3d14ebfc2a424f6ea2f570ab54a23a4bb2ca2d7","first_computed_at":"2026-08-04T02:06:00.488163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T02:06:00.488163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cilTA620aMRbBz6zGdgIlsVPxXyFwc03cQ+rLfzBUV956d0oVMlHvJ0/H8FUr30ggg1kqmPqJ6jgFQzHbMAiDA==","signature_status":"signed_v1","signed_at":"2026-08-04T02:06:00.489726Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.01568","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e963244302f6c6c8b2e99f874bc68dcc5b798978aa46b0bc41a23d68d31d0736","sha256:06546a7e7c539cb69d9701775d621d5cee20c3b209bab15ef1a0b072c844d30d"],"state_sha256":"d93a91b3f2ee7ac7406cfc4b19c213d3bda8187c3f9c420dbcc890098cbf331c"}