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Umberto Zannier showed that there exists a matrix in $\\operatorname{SL}_3(K[x,y,z])$ with first row $(x,y,z)$ for $K=\\mathbb Q_p$, the field of $p$-adic numbers for an odd prime $p$, or more generally, if $-1$ is a sum of two squares in $K$. The case $K=\\mathbb Q_2$ remained open and was subsequently posed and discussed by Zannier with numerous researchers, thereby bringing the problem to broader attention.\n  In 2025, Alexey Ananyevskiy and Marc Levine showed that such a matrix exists "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.15452","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T22:29:02Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"2b452565bd7fd92a06a11bb264ad7fa58892fc738d408e3d31bdfc1731dfe24f","abstract_canon_sha256":"f7c504951f1de88b8024a0e201ce3d5e0396317ab354865831294a83aa233a15"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:59.347652Z","signature_b64":"YCfqjYFSOnkkPKBOr3mabdsJC1mgvt0oWy7zZmv14moUrE9G9ub/Jw1iWCDo+BAdE8snRDdtnnf0aELYp37MBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a2f6cac1a103e3e9d929b0409ab1b6390eab98b7fd2a277e555d0606473fa63","last_reissued_at":"2026-05-20T00:00:59.347040Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:59.347040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicitly combing hedgehogs over fields of Stufe 4","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one.","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Peter M\\\"uller","submitted_at":"2026-05-14T22:29:02Z","abstract_excerpt":"Let $K[x,y,z]=K[X,Y,Z]/(X^2+Y^2+Z^2-1)$ be the coordinate ring of the algebraic unit sphere over a field $K$. 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The case $K=\\mathbb Q_2$ remained open and was subsequently posed and discussed by Zannier with numerous researchers, thereby bringing the problem to broader attention.\n  In 2025, Alexey Ananyevskiy and Marc Levine showed that such a matrix exists "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct an explicit example in terms of a,b,c,d of a matrix in SL_3(K[x,y,z]) with first row (x,y,z) whenever a² + b² + c² + d² = -1 in K.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algebraic identities that define the matrix entries from a,b,c,d remain valid inside the quotient ring K[x,y,z]/(x²+y²+z²-1) and produce determinant 1; this is invoked when the explicit formulas are substituted and simplified.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit matrix in SL_3(K[x,y,z]) with first row (x,y,z) for any field K of Stufe at most 4, expressed in terms of a,b,c,d satisfying a²+b²+c²+d²=-1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"02e501e714f10bc8ebf4adc9dfc310a95674f48754b1ae0eb10bc863e461548c"},"source":{"id":"2605.15452","kind":"arxiv","version":1},"verdict":{"id":"3eeaf89d-52d2-4b53-953c-d3279eb6e072","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T14:24:23.929754Z","strongest_claim":"We construct an explicit example in terms of a,b,c,d of a matrix in SL_3(K[x,y,z]) with first row (x,y,z) whenever a² + b² + c² + d² = -1 in K.","one_line_summary":"Explicit matrix in SL_3(K[x,y,z]) with first row (x,y,z) for any field K of Stufe at most 4, expressed in terms of a,b,c,d satisfying a²+b²+c²+d²=-1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algebraic identities that define the matrix entries from a,b,c,d remain valid inside the quotient ring K[x,y,z]/(x²+y²+z²-1) and produce determinant 1; 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