{
  "abstract": "We define a class of classes of complete first-order theories, denoted by $\\mathfrak{D}_{n\\text{-var}}$, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP$_1$, NTP$_2$, NATP, NCTP, NBTP, NWP, NGP, and NPM$^{(k)}$ theories. For any dividing line $D\\in\\mathfrak{D}_{n\\text{-var}}$ and any complete first-order theory $T\\notin D$, we can always find a formula $φ(x,y)$ witnessing $T\\notin D$ with an indexed set of parameters such that the set of its instances required to be consistent has a realization whose algebraic dimension over the whole set of parameters is $|x|$. We call statements of this form $n$-variable theorems, as they may be regarded as weak versions of one-variable theorems. Under the appropriate hypotheses on $T$, we prove that $T\\in D$ if and only if the corresponding theory belongs to $D$ for (i) $T^{gt}$, (ii) $T_P$, (iii) $T^{ind}$, (iv) $T^G_K$, (v) ACF$_T$, and (vi) any completion of $T^{\\barδ,nc}_g$. These are, respectively, the theories of generic trivializations, lovely pair expansions, $H$-structure expansions, vector spaces with a dense-codense generic $K$-subspace, algebraically closed fields with a distinguished subfield, and generic non-commutative derivations of algebraically bounded fields. The $n$-variable theorem is essential in proving (i)--(v). We also prove preservation for a special case of Mordell-Lang theories. We extend the framework to a larger class $\\mathfrak{D}^h_{n\\text{-var}}\\supseteq\\mathfrak{D}_{n\\text{-var}}$, capturing higher-arity dividing lines such as NOP$_k$, NFOP$_k$, and NIP$_k$ for $1<k<ω$. The $n$-variable theorem and preservation results (ii), (iii), (iv), and (vi) also hold for this larger class, while (i) holds assuming ${\\rm acl=dcl}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36394",
  "author_total": 1,
  "authors": [
    {
      "name": "Joonhee Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fnp0j1bwzrdbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36394",
  "fulltext_url": null,
  "id": "fnp0j1bwzrdbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36394",
  "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": "W7214887107",
  "pdf_url": "https://arxiv.org/pdf/2609.36394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fnp0j1bwzrdbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.36394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "$n$-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems",
  "title_text": "n-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fnp0j1bwzrdbf",
  "venue": "arXiv",
  "versions": []
}