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We say that a positive integer ℓ₀ is the strong persistence index of I if ℓ₀ is the smallest integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀. The first aim of this paper is to study this notion for monomial ideals."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The definitions assume that the strong persistence index exists (i.e., there is a finite smallest ℓ₀ satisfying the eventual equality) and that fluctuations can be meaningfully detected by checking finitely many exponents a < b < c; this is not justified in the abstract and may require the Noetherian hypothesis or specific properties of monomial ideals."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The paper defines the strong persistence index and fluctuation phenomena for colon powers of ideals, then investigates both concepts specifically for monomial ideals."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ."}],"snapshot_sha256":"3637aadd13fbab1390af835b2570d8ce22bb8b82c8fcaf993de390e71a3b5c88"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.11475/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\\ell_0$ is the strong persistence index of $I$ if $\\ell_0$ is the smallest integer such that $(I^{\\ell+1} :_R I) = I^{\\ell}$ for all $\\ell \\geq \\ell_0$. The first aim of this paper is to study this notion for monomial ideals.\n  We also introduce the notion of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs:\n  (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \\neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$.\n  (ii) $(I^{a} : I) \\neq I^{a-1}$, ","authors_text":"Jonathan Toledo, Mehrdad Nasernejad","cross_cats":[],"headline":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T13:45:05Z","title":"Strong persistence index and fluctuations in colon powers of monomial ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.11475","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T15:39:55.417868Z","id":"a5988952-016a-4e2f-a07d-a5c125493593","model_set":{"reader":"grok-4.3"},"one_line_summary":"The paper defines the strong persistence index and fluctuation phenomena for colon powers of ideals, then investigates both concepts specifically for monomial ideals.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Monomial ideals possess a finite strong persistence index after which (I^{ℓ+1} : I) equals I^ℓ for all larger ℓ.","strongest_claim":"Let I be an ideal in a commutative Noetherian ring R. We say that a positive integer ℓ₀ is the strong persistence index of I if ℓ₀ is the smallest integer such that (I^{ℓ+1} :_R I) = I^ℓ for all ℓ ≥ ℓ₀. The first aim of this paper is to study this notion for monomial ideals.","weakest_assumption":"The definitions assume that the strong persistence index exists (i.e., there is a finite smallest ℓ₀ satisfying the eventual equality) and that fluctuations can be meaningfully detected by checking finitely many exponents a < b < c; this is not justified in the abstract and may require the Noetherian hypothesis or specific properties of monomial ideals."}},"verdict_id":"a5988952-016a-4e2f-a07d-a5c125493593"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:28a6f0046460a152f6bc18a6ca7868b0297871f830ed3bfef5438e07682834d3","target":"record","created_at":"2026-05-25T02:01:19Z","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":"66721a8c9ddafaf937dbfd5e52e4ff3fb1740107573fd756140484e95e3c65ae","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T13:45:05Z","title_canon_sha256":"f5a9aa4af8a3bc95c2adbb3a0a0d216f8d8ec4f7a5532c1d4eb610b233bbebfc"},"schema_version":"1.0","source":{"id":"2604.11475","kind":"arxiv","version":2}},"canonical_sha256":"217b3fc248bc665af997586532bfb9cc9af169a80459721be186ee9357786eaf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"217b3fc248bc665af997586532bfb9cc9af169a80459721be186ee9357786eaf","first_computed_at":"2026-05-25T02:01:19.412041Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:01:19.412041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uRMTtRC06BfLvgVVeuOT//F0mvXnuo7Ttl1xtmwz3uccXGdZx5F3FCTWq81kATGy4XH/xjiA1z8GZhUkkxbGBQ==","signature_status":"signed_v1","signed_at":"2026-05-25T02:01:19.412816Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.11475","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:28a6f0046460a152f6bc18a6ca7868b0297871f830ed3bfef5438e07682834d3","sha256:5a557bd9229d60681ebc743020727e3ccadff7b6ed3bd3ff0e8af9342901d6b9"],"state_sha256":"fc32849911cef341c25e08db9a2a992235cbc7d8a2877d3022d326c8a38b1fe5"}