{"paper":{"title":"The sharp SAT/UNSAT phase transition in random ellipsoid fitting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","cs.DS","math.ST","stat.ML","stat.TH"],"primary_cat":"math.PR","authors_text":"Garrett G. Wen, Theodor Misiakiewicz","submitted_at":"2026-08-10T19:55:11Z","abstract_excerpt":"Let $x_1,\\ldots,x_n$ be independent standard Gaussian vectors in $\\mathbb{R}^d$. An \\emph{ellipsoid fit} is a matrix $S \\succeq 0$ such that $x_i^\\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\\{ x : x^\\top S x = d\\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \\to \\infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \\sim d^2/4$. We prove this conjecture. If $\\lim \\sup n/d^2 = \\alpha^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.10184","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/2608.10184/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"}