{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:SOAKHILB5RXHHAKC634RIBWRHM","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":"6431819de6fad32465d3be66535f414e31cb737cd78339ccd4057739db33358d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-01-16T05:44:18Z","title_canon_sha256":"ce090d3088b8e91dfa0e9635fb560241c48539655d13498610d3b8f472f75f1a"},"schema_version":"1.0","source":{"id":"1301.3593","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.3593","created_at":"2026-05-18T03:17:46Z"},{"alias_kind":"arxiv_version","alias_value":"1301.3593v2","created_at":"2026-05-18T03:17:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.3593","created_at":"2026-05-18T03:17:46Z"},{"alias_kind":"pith_short_12","alias_value":"SOAKHILB5RXH","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"SOAKHILB5RXHHAKC","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"SOAKHILB","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:8334b8d4a81b64ec4f422d639274169864843065ecef2927f50dc3904af7961a","target":"graph","created_at":"2026-05-18T03:17:46Z","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"},"paper":{"abstract_excerpt":"We define what it means for a Cohen-Macaulay ring to to be super-stretched and show that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1,n), or (1,n,1). Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be ","authors_text":"Branden Stone","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-01-16T05:44:18Z","title":"Super-stretched and graded countable Cohen-Macaulay type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3593","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":"sha256:50aa676ed8e3fad0b2252582be5936cfdd12fb422fe1878c6b25fa1b6052bd1a","target":"record","created_at":"2026-05-18T03:17:46Z","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":"6431819de6fad32465d3be66535f414e31cb737cd78339ccd4057739db33358d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-01-16T05:44:18Z","title_canon_sha256":"ce090d3088b8e91dfa0e9635fb560241c48539655d13498610d3b8f472f75f1a"},"schema_version":"1.0","source":{"id":"1301.3593","kind":"arxiv","version":2}},"canonical_sha256":"9380a3a161ec6e738142f6f91406d13b3ecf39ae4ad03dcaffc38d56209e1a6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9380a3a161ec6e738142f6f91406d13b3ecf39ae4ad03dcaffc38d56209e1a6c","first_computed_at":"2026-05-18T03:17:46.704212Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:46.704212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"blfRluqkCrkFX0yIOVFwtvpzLwbI9Di4fmKHtEVdnmiUpf9UQRbo88Y5jwxpcQdxHuypF6fJN2gL0IGWDFncCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:46.704859Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.3593","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50aa676ed8e3fad0b2252582be5936cfdd12fb422fe1878c6b25fa1b6052bd1a","sha256:8334b8d4a81b64ec4f422d639274169864843065ecef2927f50dc3904af7961a"],"state_sha256":"7f11a5eca4e901b82be47408faa518de7dc3e6280efbdd45075ca62b7b638203"}