{
  "abstract": "Let $P=(P_j)$ be a random mass partition and let $(X_j)$ be iid real marks, independent of $P$. We construct a fixed mark distribution for which the law of the real random variable $\\sum_j P_j X_j$ determines the law of $P$ on the Kingman simplex. The observation map is an affine topological embedding. One construction uses an infinite convolution of stable laws with rationally independent indices; for every prescribed $q\\in(0,\\infty)$, a determining mark can instead be chosen symmetric, centered, compound Poisson, and in $L^q$. At any known positive time, the law of the ranked block frequencies of a $Ξ$-coalescent started from singletons$\\unicode{x2014}$equivalently, the exchangeable partition probability functions of all its finite restrictions at that time$\\unicode{x2014}$determines the finite collision measure $Ξ$. Composing the two results therefore identifies $Ξ$ from the law of one real marked sum. Within the $Λ$ subclass, an asymmetric Bernoulli coloring determines the normalized collision measure, as does any centered nonconstant mark whose absolute moment of some order $a\\in(1,4)$ is finite while every higher absolute moment is infinite. We also prove that, for every $d\\ge 2$, a pair of distinct laws supported on strictly decreasing $(d+1)$-tuples of positive masses with a fixed total mass remains indistinguishable for every mark supported on at most $d$ points. For every fixed number $d\\ge 2$ of types, this obstruction yields distinct normalized collision measures with the same mutation-free neutral $d$-type $Ξ$-Fleming$\\unicode{x2013}$Viot transition semigroup.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12790",
  "author_total": 1,
  "authors": [
    {
      "name": "Jacopo Lenzi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8vdjxhtkgvz4n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12790",
  "fulltext_url": null,
  "id": "8vdjxhtkgvz4n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12790",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7212893080",
  "pdf_url": "https://arxiv.org/pdf/2609.12790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8vdjxhtkgvz4n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "One marked-sum law determines random mass partitions and $Ξ$-coalescents",
  "title_text": "One marked-sum law determines random mass partitions and Ξ-coalescents",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/8vdjxhtkgvz4n",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.5281/zenodo.22708550",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}