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  "slug": "symmetric-unimodality-free-convolution",
  "title": "Symmetric unimodality under free additive convolution",
  "shortTitle": "One peak under free addition",
  "url": "https://evidencepress.org/releases/symmetric-unimodality-free-convolution/",
  "oneLine": "An analytic proof candidate that free addition preserves every symmetric unimodal probability law.",
  "abstract": "The candidate proves symmetric unimodality under free additive convolution for arbitrary Borel probabilities, without support, moment or smoothness assumptions. Centered-uniform mixtures, a two-real-moment endpoint-contact reduction and an escaping-radius limit yield a reciprocal-derivative inequality. Established subordination and weak continuity transfer it to free addition. A separate regularity corollary assumes bounded locally Hölder-continuous input densities and gives a bounded globally continuous output, analytic and strictly decreasing away from its unique central maximum on its positivity interval. The result remains unrefereed; nine symbolic identities check selected algebra only.",
  "datePublished": "2026-09-07",
  "dateModified": "2026-09-07",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22649756",
  "doiUrl": "https://doi.org/10.5281/zenodo.22649756",
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  "pdfUrl": "https://github.com/ipitchford/symmetric-unimodality-free-convolution/releases/download/v0.1.0-candidate/symmetric-unimodality-free-convolution-0.1.0-candidate.pdf",
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    "name": "AIM-PROBABILITY-0118: Symmetric unimodality under free additive convolution",
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  "keywords": [
    "free probability",
    "free additive convolution",
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    "General weak symmetric-unimodality closure under free addition.",
    "A reciprocal-derivative transform inequality for finite centered uniform mixtures.",
    "Endpoint-contact reduction and explicit escaping-mass limit.",
    "Bounded locally Hölder input densities give continuous output with a unique mode; no global edge smoothness."
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    {
      "citation": "AIM Free Analysis problem list (2006), section 0.5.",
      "url": "https://aimath.org/WWN/freeanalysis/freeanalysis.pdf",
      "doi": null
    },
    {
      "citation": "Hasebe and Ueda (2018), Conjecture 3.5 and semicircular-input Theorem 3.6.",
      "url": "https://alea.math.cnrs.fr/articles/v15/15-15.pdf",
      "doi": null
    },
    {
      "citation": "Belinschi, Mai and Speicher, arXiv:1303.3196v3, Theorem 2.2.",
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    {
      "citation": "Bercovici and Voiculescu (1993), Proposition 4.13: weak continuity.",
      "url": "https://doi.org/10.1512/iumj.1993.42.42033",
      "doi": "10.1512/iumj.1993.42.42033"
    },
    {
      "citation": "Belinschi, arXiv:math/0603104v2, Theorems 2.16 and 3.3(1).",
      "url": "https://arxiv.org/abs/math/0603104v2",
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