{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:SDBNX4VMNCDRCF5BVIWN7UAEWH","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":"8f1f0aa45ba9151c2eeec9c066fba572bd5184148dba99648fbf92dee4eafecc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-04-28T13:37:24Z","title_canon_sha256":"607cdeb96860eda505894f457d541e7bd2b0f7fe7903dccdbb53173f08b691f6"},"schema_version":"1.0","source":{"id":"2604.25637","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.25637","created_at":"2026-06-24T01:15:03Z"},{"alias_kind":"arxiv_version","alias_value":"2604.25637v2","created_at":"2026-06-24T01:15:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.25637","created_at":"2026-06-24T01:15:03Z"},{"alias_kind":"pith_short_12","alias_value":"SDBNX4VMNCDR","created_at":"2026-06-24T01:15:03Z"},{"alias_kind":"pith_short_16","alias_value":"SDBNX4VMNCDRCF5B","created_at":"2026-06-24T01:15:03Z"},{"alias_kind":"pith_short_8","alias_value":"SDBNX4VM","created_at":"2026-06-24T01:15:03Z"}],"graph_snapshots":[{"event_id":"sha256:8fdab944f5581ca65d8b06326b42858361be036b4772d369f68484c3ba74ab05","target":"graph","created_at":"2026-06-24T01:15:03Z","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 consider a Ziegler pair of plane arrangements A:f=0 and A':f'=0 in P^3 such that L(A) ≅ L(A') but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the observed difference in Betti numbers is intrinsic and not an artifact of the choice of defining equations or of some undetected isomorphism between the algebras."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Ziegler pairs of plane arrangements in P^3 have isomorphic intersection lattices but different Betti numbers for Jacobian algebra resolutions and relate to cones over Ziegler pairs of line arrangements in P^2."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Ziegler pairs of plane arrangements in projective 3-space can have isomorphic intersection lattices yet different Betti numbers for the minimal resolutions of their Jacobian algebras."}],"snapshot_sha256":"41eedbd6653031e792af5aa53f60b0bb3e332ce98e96295ff610e1400e19218f"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T04:36:33.521694Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T20:53:37.353239Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.25637/integrity.json","findings":[],"snapshot_sha256":"61f45c258f1c3d4b324352d2147c863d03bc6c2bc688968fff726b5f3502395d","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\\mathcal{A}:f=0$ and $\\mathcal{A}':f'=0$ in the projective space $\\mathbb{P}^3$, such that the intersection lattices $L(\\mathcal{A})$ and $L(\\mathcal{A}')$ are isomorphic, but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same. We introduce several properties for such pairs and relate them to cones over Ziegler pairs of line arrangements in $\\mathbb{P}^2$.","authors_text":"Alexandru Dimca, Piotr Pokora","cross_cats":[],"headline":"Ziegler pairs of plane arrangements in projective 3-space can have isomorphic intersection lattices yet different Betti numbers for the minimal resolutions of their Jacobian algebras.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-04-28T13:37:24Z","title":"On the Jacobian algebras of Ziegler pairs of plane arrangements"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25637","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T15:29:33.196844Z","id":"1db782a3-b441-454a-932f-e80bbfca0e3e","model_set":{"reader":"grok-4.3"},"one_line_summary":"Ziegler pairs of plane arrangements in P^3 have isomorphic intersection lattices but different Betti numbers for Jacobian algebra resolutions and relate to cones over Ziegler pairs of line arrangements in P^2.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Ziegler pairs of plane arrangements in projective 3-space can have isomorphic intersection lattices yet different Betti numbers for the minimal resolutions of their Jacobian algebras.","strongest_claim":"We consider a Ziegler pair of plane arrangements A:f=0 and A':f'=0 in P^3 such that L(A) ≅ L(A') but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same.","weakest_assumption":"That the observed difference in Betti numbers is intrinsic and not an artifact of the choice of defining equations or of some undetected isomorphism between the algebras."}},"verdict_id":"1db782a3-b441-454a-932f-e80bbfca0e3e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:91058d2a6610858747d221cf9ce31983cb1141b012ed5e8cde20cbba80f8a4ee","target":"record","created_at":"2026-06-24T01:15:03Z","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":"8f1f0aa45ba9151c2eeec9c066fba572bd5184148dba99648fbf92dee4eafecc","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2026-04-28T13:37:24Z","title_canon_sha256":"607cdeb96860eda505894f457d541e7bd2b0f7fe7903dccdbb53173f08b691f6"},"schema_version":"1.0","source":{"id":"2604.25637","kind":"arxiv","version":2}},"canonical_sha256":"90c2dbf2ac68871117a1aa2cdfd004b1e0b74a905f85ed78db1320130e7af56f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90c2dbf2ac68871117a1aa2cdfd004b1e0b74a905f85ed78db1320130e7af56f","first_computed_at":"2026-06-24T01:15:03.088304Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-24T01:15:03.088304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AvLIDO9lN7GCKwSqQgGTOA3PKTq7oMioYNZUrh/Jtb4LvbnyiRVAkMTbVlT7y7Z5phMDqOf/PGpewqB2lbqDBQ==","signature_status":"signed_v1","signed_at":"2026-06-24T01:15:03.088760Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.25637","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91058d2a6610858747d221cf9ce31983cb1141b012ed5e8cde20cbba80f8a4ee","sha256:8fdab944f5581ca65d8b06326b42858361be036b4772d369f68484c3ba74ab05"],"state_sha256":"3926c90c88012ac0d9f7c2278785ad460551655de7ec8b58365b716761157c13"}