{"paper":{"title":"The Structure of Extremal Bad Science Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Shridhar Sinha","submitted_at":"2025-09-11T20:47:43Z","abstract_excerpt":"We study the 'bad science matrix problem': among all matrices $A\\in\\mathbb{R}^{n\\times n}$ whose rows have unit $\\ell_2$-norm, determine the maximum of $\\beta(A)=\\frac{1}{2^n}\\sum_{x\\in\\{\\pm1\\}^n}\\|Ax\\|_\\infty$. Steinerberger [1] (arXiv:2402.03205) showed that the optimal asymptotic rate is $(1+o(1))\\sqrt{2\\log n}$, and that this rate is attained with high probability by matrices with i.i.d. $\\pm1$ entries after normalization. More recent explicit constructions [2] (arXiv:2408.00933) achieve $\\beta(A)\\ge\\sqrt{\\log_2(n)+1}$, which lies within a constant factor of the asymptotic optimum. In this"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.10580","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/2509.10580/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"}