{"paper":{"title":"An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\\mathbb Z[i]$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Junfeng Li, Yuchen Ding","submitted_at":"2026-06-11T00:39:01Z","abstract_excerpt":"This paper presents an AI proof that there is no algorithm deciding whether a polynomial equation over the Gaussian integers in $18$ unknowns has a solution. The proof improves the $20$-unknown theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, but saves two variables: the clearing-denominator variable is avoided by imposing two integer conditions, and the remaining nonzero condition is absorbed into the relation-combining lemma by the one-variable gadget $(2R+1)(3R+1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12776","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/2606.12776/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"}