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University of Galway, ´Aras De Br´un, Gaillimh/Galway, H91 H3CY, Ireland Email address:i","work_id":"5f4cd730-2875-4d40-8813-a6c9530ad7bf","year":1992}],"snapshot_sha256":"978c8ab66ede05d90d7f348cb229fca95b25decdce4678dc7d56602dfd0ce0f5"},"source":{"id":"2605.16974","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T19:00:51.445382Z","id":"b4aecb82-c6f5-4084-a9e9-efa2a6461e6b","model_set":{"reader":"grok-4.3"},"one_line_summary":"Introduces n-ary elliptic groups and rings in which Dirichlet's theorem on arithmetic progressions reduces to Euclid's theorem for an+1 progressions, while defining an n-ary class group that captures unique factorization and proving a Dedekind-type theorem for nEl(Z).","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Dirichlet's theorem on arithmetic progressions reduces to Euclid's theorem inside n-ary elliptic rings for sequences an + 1.","strongest_claim":"Dirichlet's famous theorem on arithmetic progressions becomes simply Euclid's theorem in these n-ary rings, at least for progressions of the form an + 1.","weakest_assumption":"The n-ary operation distributes over the monoidal structure in an n-ary sense, allowing the arithmetic properties (including reduction of Dirichlet's theorem to Euclid's) to hold in the defined n-ary elliptic rings."}},"verdict_id":"b4aecb82-c6f5-4084-a9e9-efa2a6461e6b"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2ccb85e1862951234315f0356e8bdd450cd3440db714ffb0ac347c9cd7225c47","target":"record","created_at":"2026-05-20T00:03:33Z","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":"3c49512d71650bd79b397e8ac71c7e998128b5503fefd2bb29fa8c2874cf45f5","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-05-16T12:51:51Z","title_canon_sha256":"b9d95c8ed12b21f02baaa35ede73e46948cbbc2e5c5034a7db5bf222c62d828b"},"schema_version":"1.0","source":{"id":"2605.16974","kind":"arxiv","version":1}},"canonical_sha256":"2f10e0882da3d2f9e3638faf31108ed4099fda8a285d15d0117da3c95472eddb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f10e0882da3d2f9e3638faf31108ed4099fda8a285d15d0117da3c95472eddb","first_computed_at":"2026-05-20T00:03:33.818827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:33.818827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8JHn06NrSl87ymwEvWZV/nvxLpHsjylxYNFgljmS9X04bMEjk3GeMQZ/N4Lle236d3NIjEJudgt+5Y/X2Sz/AA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:33.819452Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.16974","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ccb85e1862951234315f0356e8bdd450cd3440db714ffb0ac347c9cd7225c47","sha256:03e79cd684dffe799d2775d1154a50102735aa4a682cbd7722d0ca23978556c8"],"state_sha256":"a59058fedd833e3ee470a5642d70462ebf2c2a6d6e917a4ab00c00f76f8bab8e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"maqTLHw4w9mwBUZ4hqUYLuuAHp3kEorEdkQ5exOZo+ervNfU7ft+aLRQbduF7+qmxay7Cezwbo6zR/G75PW9Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T11:27:09.039306Z","bundle_sha256":"97fdd8cda9b079d09dd717e69c5a06a293cc819a9cf6ff8a68f1d8644169cefb"}}