{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IAAPXCOXNDT5Y2NODWLBQLSD7J","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":"32bb321b4508c4102d50ddf81532f045bb4901c4ff2fe51d21591bad35e18baa","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-04-21T12:35:40Z","title_canon_sha256":"4ec064539a17e8148b399603ddfc945a1bf100085426c8000e288d39c575d70b"},"schema_version":"1.0","source":{"id":"2604.19408","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.19408","created_at":"2026-06-09T01:05:17Z"},{"alias_kind":"arxiv_version","alias_value":"2604.19408v3","created_at":"2026-06-09T01:05:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.19408","created_at":"2026-06-09T01:05:17Z"},{"alias_kind":"pith_short_12","alias_value":"IAAPXCOXNDT5","created_at":"2026-06-09T01:05:17Z"},{"alias_kind":"pith_short_16","alias_value":"IAAPXCOXNDT5Y2NO","created_at":"2026-06-09T01:05:17Z"},{"alias_kind":"pith_short_8","alias_value":"IAAPXCOX","created_at":"2026-06-09T01:05:17Z"}],"graph_snapshots":[{"event_id":"sha256:652d9f65a2ee987c5b4f7b74e78b4b615a440f301930a3b0c332202d6685f432","target":"graph","created_at":"2026-06-09T01:05:17Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We prove that Γ_P(R) ≅ K_{|P|-1} ∨ K-bar_{|R|-|P|}, and a monomial x^α y^β belongs to the minimal generators of I(Γ_P(R))^n if and only if |α| + |β| = 2n, |β| ≤ n, and 0 ≤ α_i ≤ n for all i."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"R is a finite commutative ring with identity and P is a proper prime ideal; this finiteness and the prime property are used to obtain the explicit isomorphism and the counting formulas."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Prime ideal graphs are complete split graphs whose edge ideal powers have explicit minimal monomial generators satisfying exponent conditions, are polymatroidal with linear resolutions, and have computable analytic spread."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Prime ideal graphs of finite rings are complete split graphs whose edge ideal powers have explicit generators and are polymatroidal."}],"snapshot_sha256":"b23fee55ffd435a6cdfc28ef95459968db64efed4caef555970eede29f72c120"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"41f98122f9e7d5a9ad31ee148f446a9162a3cf5d54a54b0a4e2cbea066cb8c7a"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-20T03:01:35.866263Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.19408/integrity.json","findings":[],"snapshot_sha256":"729270eeb0145b918b5c11a20be73e2869561530f3e4a98db60f2ff991f19fa6","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $R$ be a finite commutative ring with identity and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\\Gamma_P(R)$ has vertex set $R\\setminus\\{0\\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\\in P$. We prove that prime ideal graphs form a ring-realizable subfamily of complete split graphs. More precisely, if $m=|P|$, $q=|R/P|$, then $q$ is a prime power and $\\Gamma_P(R)\\cong K_{m-1}\\vee \\overline{K}_{m(q-1)}$. We also prove a realization theorem showing that every complete split graph of this form arises from a prime ideal of a finite commutative ri","authors_text":"Tabinda Rasheed, Wang Yao","cross_cats":["math.CO"],"headline":"Prime ideal graphs of finite rings are complete split graphs whose edge ideal powers have explicit generators and are polymatroidal.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-04-21T12:35:40Z","title":"Edge Ideals of Prime Ideal Graphs over Finite Rings: Ordinary Powers, Fiber Cones, and Linear Powers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.19408","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-14T21:37:31.042953Z","id":"d1313ddd-4dff-418a-9836-333f16bdeb8f","model_set":{"reader":"grok-4.3"},"one_line_summary":"Prime ideal graphs are complete split graphs whose edge ideal powers have explicit minimal monomial generators satisfying exponent conditions, are polymatroidal with linear resolutions, and have computable analytic spread.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Prime ideal graphs of finite rings are complete split graphs whose edge ideal powers have explicit generators and are polymatroidal.","strongest_claim":"We prove that Γ_P(R) ≅ K_{|P|-1} ∨ K-bar_{|R|-|P|}, and a monomial x^α y^β belongs to the minimal generators of I(Γ_P(R))^n if and only if |α| + |β| = 2n, |β| ≤ n, and 0 ≤ α_i ≤ n for all i.","weakest_assumption":"R is a finite commutative ring with identity and P is a proper prime ideal; this finiteness and the prime property are used to obtain the explicit isomorphism and the counting formulas."}},"verdict_id":"d1313ddd-4dff-418a-9836-333f16bdeb8f"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0ebd28fb2e34423962e7c623df2d8ec31eaba49dcd0488154323e2e3a0fffa69","target":"record","created_at":"2026-06-09T01:05:17Z","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":"32bb321b4508c4102d50ddf81532f045bb4901c4ff2fe51d21591bad35e18baa","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2026-04-21T12:35:40Z","title_canon_sha256":"4ec064539a17e8148b399603ddfc945a1bf100085426c8000e288d39c575d70b"},"schema_version":"1.0","source":{"id":"2604.19408","kind":"arxiv","version":3}},"canonical_sha256":"4000fb89d768e7dc69ae1d96182e43fa4b5d0eb9106a45c9a647f4fe057571e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4000fb89d768e7dc69ae1d96182e43fa4b5d0eb9106a45c9a647f4fe057571e7","first_computed_at":"2026-06-09T01:05:17.545916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:17.545916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4xlQl3wcq0iQsgrvZvPlwupApIBmKsaAt/vNIWosLvIKGX3Y3fLqK1M9CxKq2/2mRykGBEi18UfQQEOKHpU7Bg==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:17.546376Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.19408","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ebd28fb2e34423962e7c623df2d8ec31eaba49dcd0488154323e2e3a0fffa69","sha256:652d9f65a2ee987c5b4f7b74e78b4b615a440f301930a3b0c332202d6685f432"],"state_sha256":"93e13a83dd5754a1a40171c1374554f96dc262f8fdb207e04c210913356e12f2"}