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In the atomic case, the atoms of $\\mathbb{N}_0[\\alpha]$ are precisely the powers $\\alpha^n$ up to a certain nonnegative integer $n$, and finite generation is governed by divisibility of the minimal polynomial by a negative-tail polynomial. Our first main result gives a complete characterization when the minimal polynomial has the form $\\mathfrak{m}_\\alpha(X)=p_\\alpha(X)-c$ with $c\\in\\mathbb{N}$. Our second main result shows that finite generation forces $\\alpha$ to be a weak P","authors_text":"Mohammad El Asal, Wael Mahboub","cross_cats":[],"headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T14:50:35Z","title":"Finite Generation in Polynomial Semirings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.11569","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T16:07:24.982845Z","id":"447d9663-ac86-4ae5-96ba-26cd0745c89b","model_set":{"reader":"grok-4.3"},"one_line_summary":"Finite generation of the additive monoid N_0[alpha] is fully characterized for minimal polynomials of the form p(X) - c and implies that alpha must be a weak Perron number, with applications to cubic cases and rank-3 monoids.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","strongest_claim":"Our second main result shows that finite generation forces alpha to be a weak Perron number.","weakest_assumption":"The analysis assumes the atomic case where atoms are precisely the powers alpha^n up to a certain n, and relies on divisibility conditions involving negative-tail polynomials without independent verification of atomicity in all cases."}},"verdict_id":"447d9663-ac86-4ae5-96ba-26cd0745c89b"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:174ac33c7af366b05ba9b48771d27c474f73a3ace65323d7577305e41df6bb43","target":"record","created_at":"2026-06-02T02:04:53Z","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":"e358191404e921efd804a9213e01edfe5f18429123e770012efbe0f5ebcf715d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T14:50:35Z","title_canon_sha256":"0cfbf483b6501d7856fa97b8b4d2dbbfd47a859ba3778594d881aa17a18d7f74"},"schema_version":"1.0","source":{"id":"2604.11569","kind":"arxiv","version":2}},"canonical_sha256":"a705e32c119cb320965e18474d3fc3c1a294abe75f65fd5bffc6969f958dfc0b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a705e32c119cb320965e18474d3fc3c1a294abe75f65fd5bffc6969f958dfc0b","first_computed_at":"2026-06-02T02:04:53.022346Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:53.022346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mFUv7ABD0evFwsrf0q9ROi4bR9huID8Augok0RApC0afatRlZhcjgmzt51G8mQIa8WLFKnRBc/DdcqTwsq2wCg==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:53.022766Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.11569","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:174ac33c7af366b05ba9b48771d27c474f73a3ace65323d7577305e41df6bb43","sha256:a5b1646865e8e9b508ad097c218a254c74058f3e3c23cca85abec46807aaea85"],"state_sha256":"f23600d2b46187b2883ac9935d0c3c6beb0746d6fb9135d89ba4ca06619cf0a4"}