{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:U4C6GLARTSZSBFS6DBDU2P6DYG","short_pith_number":"pith:U4C6GLAR","canonical_record":{"source":{"id":"2604.11569","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T14:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"0cfbf483b6501d7856fa97b8b4d2dbbfd47a859ba3778594d881aa17a18d7f74","abstract_canon_sha256":"e358191404e921efd804a9213e01edfe5f18429123e770012efbe0f5ebcf715d"},"schema_version":"1.0"},"canonical_sha256":"a705e32c119cb320965e18474d3fc3c1a294abe75f65fd5bffc6969f958dfc0b","source":{"kind":"arxiv","id":"2604.11569","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.11569","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"arxiv_version","alias_value":"2604.11569v2","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11569","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_12","alias_value":"U4C6GLARTSZS","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_16","alias_value":"U4C6GLARTSZSBFS6","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_8","alias_value":"U4C6GLAR","created_at":"2026-06-02T02:04:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:U4C6GLARTSZSBFS6DBDU2P6DYG","target":"record","payload":{"canonical_record":{"source":{"id":"2604.11569","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-04-13T14:50:35Z","cross_cats_sorted":[],"title_canon_sha256":"0cfbf483b6501d7856fa97b8b4d2dbbfd47a859ba3778594d881aa17a18d7f74","abstract_canon_sha256":"e358191404e921efd804a9213e01edfe5f18429123e770012efbe0f5ebcf715d"},"schema_version":"1.0"},"canonical_sha256":"a705e32c119cb320965e18474d3fc3c1a294abe75f65fd5bffc6969f958dfc0b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:53.022766Z","signature_b64":"mFUv7ABD0evFwsrf0q9ROi4bR9huID8Augok0RApC0afatRlZhcjgmzt51G8mQIa8WLFKnRBc/DdcqTwsq2wCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a705e32c119cb320965e18474d3fc3c1a294abe75f65fd5bffc6969f958dfc0b","last_reissued_at":"2026-06-02T02:04:53.022346Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:53.022346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.11569","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rqWpNcPUT7+6Hdh7gugE/TRUUh/vvkxGRQIEJt+ds25MqI34/d6rnWEmy+pF56W4noxuRniMT2pUEK+1fQUZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:54:27.092993Z"},"content_sha256":"174ac33c7af366b05ba9b48771d27c474f73a3ace65323d7577305e41df6bb43","schema_version":"1.0","event_id":"sha256:174ac33c7af366b05ba9b48771d27c474f73a3ace65323d7577305e41df6bb43"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:U4C6GLARTSZSBFS6DBDU2P6DYG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Finite Generation in Polynomial Semirings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Mohammad El Asal, Wael Mahboub","submitted_at":"2026-04-13T14:50:35Z","abstract_excerpt":"We study the semiring $\\mathbb{N}_0[\\alpha]$ as an additive monoid where $\\alpha$ is a positive real algebraic number. 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our second main result shows that finite generation forces alpha to be a weak Perron number.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"97f436a5c16a47ef6b33c71b52ee2140cf4e449690a3e76cf0280abc8f498331"},"source":{"id":"2604.11569","kind":"arxiv","version":2},"verdict":{"id":"447d9663-ac86-4ae5-96ba-26cd0745c89b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:07:24.982845Z","strongest_claim":"Our second main result shows that finite generation forces alpha to be a weak Perron number.","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","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.","pith_extraction_headline":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11569/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"},"verdict_id":"447d9663-ac86-4ae5-96ba-26cd0745c89b"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WrHEZWiPZWQwWdjrvAR9jm1iwdUPQ88vTA7dVe1Zkl45RJu+DysL1nRm2FKIUrERCp2FIX8+e0rFOzby2HTXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T13:54:27.093473Z"},"content_sha256":"a5b1646865e8e9b508ad097c218a254c74058f3e3c23cca85abec46807aaea85","schema_version":"1.0","event_id":"sha256:a5b1646865e8e9b508ad097c218a254c74058f3e3c23cca85abec46807aaea85"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/bundle.json","state_url":"https://pith.science/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T13:54:27Z","links":{"resolver":"https://pith.science/pith/U4C6GLARTSZSBFS6DBDU2P6DYG","bundle":"https://pith.science/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/bundle.json","state":"https://pith.science/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U4C6GLARTSZSBFS6DBDU2P6DYG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:U4C6GLARTSZSBFS6DBDU2P6DYG","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":"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}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.11569","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"arxiv_version","alias_value":"2604.11569v2","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11569","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_12","alias_value":"U4C6GLARTSZS","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_16","alias_value":"U4C6GLARTSZSBFS6","created_at":"2026-06-02T02:04:53Z"},{"alias_kind":"pith_short_8","alias_value":"U4C6GLAR","created_at":"2026-06-02T02:04:53Z"}],"graph_snapshots":[{"event_id":"sha256:a5b1646865e8e9b508ad097c218a254c74058f3e3c23cca85abec46807aaea85","target":"graph","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":{"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":"Our second main result shows that finite generation forces alpha to be a weak Perron number."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Finite generation of the additive monoid N_0[alpha] forces the algebraic number alpha to be a weak Perron number."}],"snapshot_sha256":"97f436a5c16a47ef6b33c71b52ee2140cf4e449690a3e76cf0280abc8f498331"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.11569/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the semiring $\\mathbb{N}_0[\\alpha]$ as an additive monoid where $\\alpha$ is a positive real algebraic number. 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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rX4kFFvoZoiqhP77w51AZUEoUC+3NpM+i2Sjs9WjO9ss0wwdOzxoBtQhKUSR0i54XYTTFUVSTocoFr6OwcNrCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T13:54:27.095846Z","bundle_sha256":"59ae87ab9479affd91c6208bc94a261fbc8838c3e4c1590021e4ec3f69dcee1f"}}