{"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 $K$. 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 "},"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; this is invoked when the explicit formulas are substituted and simplified.","pith_extraction_headline":"Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15452/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T15:51:55.932756Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:49:52.555130Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:37:53.680039Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T14:31:17.505243Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.108603Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.676884Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"e07290e4cc1d78792e7680b1ed57a833800f9c23bfa9106794726eda16636a46"},"references":{"count":4,"sample":[{"doi":"10.2140/ant","year":2025,"title":"Combing a hedgehog over a field","work_id":"e917cd6b-0e20-44c9-8529-8b698464506d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1145/3452143.3465545","year":2021,"title":"msolve: A Library for Solv- ing Polynomial Systems","work_id":"d193af2a-840e-426f-a875-295b7bd6314f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Müller.SageMath verification script.https://ypfmde.github.io/ verify_combing.html","work_id":"dedd0c48-c1eb-45d2-a6d5-eda9b0da3163","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"https://www.sagemath.org","work_id":"f4147711-4ec7-4c67-a9e4-1a37cee43f7b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":4,"snapshot_sha256":"34d0a737257eed923b83bf4e50323d312a77f328e1b51b504c2ddc811f142f04","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"}