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Let $S = \\bigoplus_{\\underline{n} \\in \\mathbb{N}^r} S_{\\underline{n}}$ be a Noetherian standard $\\mathbb{N}^r$-graded ring with $S_{\\underline{0}} = R$, and $\\mathcal{M} $ be a finitely generated $\\mathbb{Z}^r$-graded $S$-module. For $ \\underline{n} = (n_1,\\dots,n_r) \\in \\mathbb{N}^r$, set $G_{\\underline{n}} := \\mathcal{M}_{\\underline{n}}$ or $G_{\\underline{n}} := M/{\\bf I}^{\\underline{n}} N$, where ${\\bf I}^{\\underline{n}} = I_1^{n_1} \\cdots I_r^{n_r}$. Suppose $F$ is a coher"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.00529","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2025-05-31T12:25:23Z","cross_cats_sorted":[],"title_canon_sha256":"0a5f21ff89a72a4d8f7be83ea12d47fb690996100b69716b0cde2fcc92712631","abstract_canon_sha256":"7467469fbd527496ebbcffcb04266a640c0c098ef1d63c326e816e91ebd7a308"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:13:36.555057Z","signature_b64":"zaxhw9FVpTWha+LUofBkDbUb5M0Hq9Z+CdWkS/w2Y1q56JWWY9QQhXuA0L3k3MV37I9L/cRMW2/hNbj5ARaBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aee32ce2d0e70116c26043e57f2f33c72bc97e58d1de4a447f22c29b23383721","last_reissued_at":"2026-07-05T11:13:36.554586Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:13:36.554586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coherent functors, powers of ideals, and asymptotic stability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Dipankar Ghosh, Samarendra Sahoo, Siddhartha Pramanik, Souvik Dey, Tony J. Puthenpurakal","submitted_at":"2025-05-31T12:25:23Z","abstract_excerpt":"Let $R$ be a Noetherian ring, $I_1,\\ldots,I_r$ be ideals of $R$, and $N\\subseteq M$ be finitely generated $R$-modules. Let $S = \\bigoplus_{\\underline{n} \\in \\mathbb{N}^r} S_{\\underline{n}}$ be a Noetherian standard $\\mathbb{N}^r$-graded ring with $S_{\\underline{0}} = R$, and $\\mathcal{M} $ be a finitely generated $\\mathbb{Z}^r$-graded $S$-module. For $ \\underline{n} = (n_1,\\dots,n_r) \\in \\mathbb{N}^r$, set $G_{\\underline{n}} := \\mathcal{M}_{\\underline{n}}$ or $G_{\\underline{n}} := M/{\\bf I}^{\\underline{n}} N$, where ${\\bf I}^{\\underline{n}} = I_1^{n_1} \\cdots I_r^{n_r}$. Suppose $F$ is a coher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00529","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.00529/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.00529","created_at":"2026-07-05T11:13:36.554646+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.00529v1","created_at":"2026-07-05T11:13:36.554646+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.00529","created_at":"2026-07-05T11:13:36.554646+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3RSZYWQ44AR","created_at":"2026-07-05T11:13:36.554646+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3RSZYWQ44ARNQTA","created_at":"2026-07-05T11:13:36.554646+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3RSZYWQ","created_at":"2026-07-05T11:13:36.554646+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4","json":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4.json","graph_json":"https://pith.science/api/pith-number/V3RSZYWQ44ARNQTAIPSX6LZTY4/graph.json","events_json":"https://pith.science/api/pith-number/V3RSZYWQ44ARNQTAIPSX6LZTY4/events.json","paper":"https://pith.science/paper/V3RSZYWQ"},"agent_actions":{"view_html":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4","download_json":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4.json","view_paper":"https://pith.science/paper/V3RSZYWQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.00529&json=true","fetch_graph":"https://pith.science/api/pith-number/V3RSZYWQ44ARNQTAIPSX6LZTY4/graph.json","fetch_events":"https://pith.science/api/pith-number/V3RSZYWQ44ARNQTAIPSX6LZTY4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4/action/storage_attestation","attest_author":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4/action/author_attestation","sign_citation":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4/action/citation_signature","submit_replication":"https://pith.science/pith/V3RSZYWQ44ARNQTAIPSX6LZTY4/action/replication_record"}},"created_at":"2026-07-05T11:13:36.554646+00:00","updated_at":"2026-07-05T11:13:36.554646+00:00"}