{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:OGX3PJ25W3HYXP3DMN2SH22AZH","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":"2243e20fba264a8e70f95934151a988ac07487907137be5b0d6022e9a83f8b24","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-12T15:45:09Z","title_canon_sha256":"68795c17e182c0d95238c22fcbfc7189ddd5f2d93d1014678c09dfce0e2e8778"},"schema_version":"1.0","source":{"id":"2505.07679","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.07679","created_at":"2026-07-05T11:29:59Z"},{"alias_kind":"arxiv_version","alias_value":"2505.07679v2","created_at":"2026-07-05T11:29:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.07679","created_at":"2026-07-05T11:29:59Z"},{"alias_kind":"pith_short_12","alias_value":"OGX3PJ25W3HY","created_at":"2026-07-05T11:29:59Z"},{"alias_kind":"pith_short_16","alias_value":"OGX3PJ25W3HYXP3D","created_at":"2026-07-05T11:29:59Z"},{"alias_kind":"pith_short_8","alias_value":"OGX3PJ25","created_at":"2026-07-05T11:29:59Z"}],"graph_snapshots":[{"event_id":"sha256:dd21001cd833f4e106609c23690358d97d08b98fd44ec455ee8f822e518d00f7","target":"graph","created_at":"2026-07-05T11:29:59Z","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/2505.07679/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The set $\\mathcal{R}_{G}(h,k)$ consists of all possible sizes for the $h$-fold sumset of sets containing $k$ elements from an additive abelian group $G$. The exact makeup of this set is still unknown, but there has been progress towards determining which integers are present. We know that $\\mathcal{R}_{G}(h,k)\\subseteq\\left[hk-h+1,\\binom{h+k-1}{h}\\right]$, where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. We will prove that the integers in $\\left[hk-h+2,hk-1\\right]$ are not possible sizes for the $h$-fold sumset of a set cont","authors_text":"Vincent Schinina","cross_cats":["math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-12T15:45:09Z","title":"On the Sumset of Sets of Size $k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07679","kind":"arxiv","version":2},"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:e690a06910d045238de684738050fcfe343cc4c95594e3656993a32e7e4d23bd","target":"record","created_at":"2026-07-05T11:29:59Z","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":"2243e20fba264a8e70f95934151a988ac07487907137be5b0d6022e9a83f8b24","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-12T15:45:09Z","title_canon_sha256":"68795c17e182c0d95238c22fcbfc7189ddd5f2d93d1014678c09dfce0e2e8778"},"schema_version":"1.0","source":{"id":"2505.07679","kind":"arxiv","version":2}},"canonical_sha256":"71afb7a75db6cf8bbf63637523eb40c9eabab0419a06fb12dad93cf1a11eb4ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71afb7a75db6cf8bbf63637523eb40c9eabab0419a06fb12dad93cf1a11eb4ca","first_computed_at":"2026-07-05T11:29:59.986236Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:29:59.986236Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gnSIJ558PKcQnKu1QJNPj4A4beBkMH7phzlaWIv3fs8jSVTuqk7dMGH4Ik0jqWnGXRtgbFnFW3n/rCbyZ65UCw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:29:59.986714Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.07679","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e690a06910d045238de684738050fcfe343cc4c95594e3656993a32e7e4d23bd","sha256:dd21001cd833f4e106609c23690358d97d08b98fd44ec455ee8f822e518d00f7"],"state_sha256":"fe801af8f78aa2f5dde579c1cb4daf016b8dba609a5e0192ddff8bb312531e06"}