{
  "abstract": "Törnquist (arXiv:1310.3250) proved that if there exists a $\\boldsymbolΣ^1_2$ maximal almost disjoint (MAD) family, then there exists a $\\boldsymbolΠ^1_1$ MAD family. Since then, several other \"complexity reduction\" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on $\\boldsymbolΠ^1_1$-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call \"Functional Projection\", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in $L[U]$, the canonical inner model for a measurable cardinal, Functional Projection holds for $\\boldsymbolΠ^1_2$ relations. As an immediate corollary, we obtain that in $L[U]$ there are $\\boldsymbolΠ^1_2$ witnesses for many combinatorial families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17455",
  "author_total": 4,
  "authors": [
    {
      "name": "Vera Fischer"
    },
    {
      "name": "Yurii Khomskii"
    },
    {
      "name": "Yipu Li"
    },
    {
      "name": "Julia Millhouse"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45n9hpzfscsc7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17455",
  "fulltext_url": null,
  "id": "45n9hpzfscsc7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17455",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7221805411",
  "pdf_url": "https://arxiv.org/pdf/2610.17455",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45n9hpzfscsc7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.17455",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17455"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Functional Projection and Complexity Reduction",
  "title_text": "Functional Projection and Complexity Reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/45n9hpzfscsc7",
  "venue": "arXiv",
  "versions": []
}