{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:C56OK6IHPPP2JJ2SJZXMVGYBCX","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":"2391ccf8427e6e44537c272af8f44e006a3c35d157199cec41a33144b622d863","cross_cats_sorted":["math.CA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-29T08:43:48Z","title_canon_sha256":"4ad60d91937af0e24381018f5cfff1174106e9672e24cedd12bb7851dc32237a"},"schema_version":"1.0","source":{"id":"2606.29965","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.29965","created_at":"2026-06-30T02:17:42Z"},{"alias_kind":"arxiv_version","alias_value":"2606.29965v1","created_at":"2026-06-30T02:17:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.29965","created_at":"2026-06-30T02:17:42Z"},{"alias_kind":"pith_short_12","alias_value":"C56OK6IHPPP2","created_at":"2026-06-30T02:17:42Z"},{"alias_kind":"pith_short_16","alias_value":"C56OK6IHPPP2JJ2S","created_at":"2026-06-30T02:17:42Z"},{"alias_kind":"pith_short_8","alias_value":"C56OK6IH","created_at":"2026-06-30T02:17:42Z"}],"graph_snapshots":[{"event_id":"sha256:ebf6244cf9cc2c9d57aee6d73c1e6616b1675d8183e2017b5fb785779fda9319","target":"graph","created_at":"2026-06-30T02:17:42Z","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/2606.29965/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a Delsarte linear programming approach to the finite field Erd\\H{o}s--Falconer distance problem. Let \\(q\\) be an odd prime power, let \\(n\\) be even, and let \\(Q\\) be a non-degenerate quadratic form on \\(\\mathbb{F}_q^n\\). For \\(E\\subset \\mathbb{F}_q^n\\), define\n  \\[\n  \\Delta_Q(E)=\\{Q(x-y):\\ x,y\\in E\\}.\n  \\]\n  We prove that, for every fixed \\(0<\\alpha<\\frac{1}{2}\\), there exist constants \\(C_\\alpha>0\\) and \\(q_\\alpha\\) such that if \\(q\\ge q_\\alpha\\) and $|E|\\ge C_\\alpha q^{\\frac n2+\\frac13},$\n  then\n  \\[\n  |\\Delta_Q(E)|>1+\\alpha(q-1).\n  \\]\n  In particular, \\(\\Delta_Q(E)\\) contains a","authors_text":"Tao Zhang","cross_cats":["math.CA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-29T08:43:48Z","title":"A Delsarte Linear Programming Approach to the Erd\\H{o}s--Falconer Distance Problem over Finite Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29965","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":"sha256:3cdbd70ad2255c292a999887aaca2d1173ae07a10df5395f4843d25e89276713","target":"record","created_at":"2026-06-30T02:17:42Z","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":"2391ccf8427e6e44537c272af8f44e006a3c35d157199cec41a33144b622d863","cross_cats_sorted":["math.CA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-29T08:43:48Z","title_canon_sha256":"4ad60d91937af0e24381018f5cfff1174106e9672e24cedd12bb7851dc32237a"},"schema_version":"1.0","source":{"id":"2606.29965","kind":"arxiv","version":1}},"canonical_sha256":"177ce579077bdfa4a7524e6eca9b0115d45b3ffc8a01962ea62db1a7c7e35498","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"177ce579077bdfa4a7524e6eca9b0115d45b3ffc8a01962ea62db1a7c7e35498","first_computed_at":"2026-06-30T02:17:42.988860Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-30T02:17:42.988860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u/RAmr67xktxxja/0PVX40deZory3AFQjsNNfaZnSmnOu3Y2MMXOXHTaHqnmQfQUZFRYETn4/X2EgZM9v0n9CA==","signature_status":"signed_v1","signed_at":"2026-06-30T02:17:42.989376Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.29965","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cdbd70ad2255c292a999887aaca2d1173ae07a10df5395f4843d25e89276713","sha256:ebf6244cf9cc2c9d57aee6d73c1e6616b1675d8183e2017b5fb785779fda9319"],"state_sha256":"ad3bce08ef088a05d3a56de44a2244432baba4aee3c552d40752d938ef17bbbc"}