{
  "abstract": "In this paper we present construction systems -- tuples $(X, BB, \\oplus, ν)$ comprising objects, building blocks, an assembly operation, and a joining multiplicity -- as a general algebraic framework for studying how complex objects are built from simpler parts. To each construction system we associate a toric ideal, a toric variety, and a matroid, obtaining analytical bounds on the growth function $N(a)$ (the number of objects of construction complexity $\\leq a$) purely from the design signature $(m, ν, n_0)$. For systems equipped with a type system and valence bounds, we define the composition polytope $P_{\\mathrm{val}} \\subset \\mathbb{R}^m$, whose integer points count the feasible compositions. Compositions outside $P_{\\mathrm{val}}$ -- termed zombies -- are combinatorially valid but physically unrealisable. We prove that the zombie classification is sound (zero false positives) and conservative: the true infeasibility rate is at least as high as the polytope predicts. Specialising to the molecular graph assembly system of Morales Parra et al. ($m = 19$ bond types, $5$ atom types with valences $1$--$4$), we identify a composition polytope whose lattice points capture the physically realisable compositions, and show that the resulting growth exponent tightens from $0.73$ to the exact value $\\log 2 \\approx 0.693$. While the difference of $0.037$ appears small, in the doubly-exponential regime it corresponds to a tightening of the bound by a factor exceeding $10^{198}$ at assembly index $10$. The framework also recovers the bioorthogonal click-chemistry system of the author's prior work as a second instance, with $m = 8$ and a partition matroid.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.21363",
  "author_total": 1,
  "authors": [
    {
      "name": "Vicent Ribas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a3vddz9q3nwst.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.21363",
  "fulltext_url": null,
  "id": "a3vddz9q3nwst",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.21363",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7165680264",
  "pdf_url": "https://arxiv.org/pdf/2606.21363",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a3vddz9q3nwst.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2606.21363",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.21363"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zombie Compositions in Assembly Algebras and an Upper Bound on the Size of Chemical Space",
  "title_text": "Zombie Compositions in Assembly Algebras and an Upper Bound on the Size of Chemical Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/a3vddz9q3nwst",
  "venue": "arXiv",
  "versions": []
}