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  "slug": "cooper-spencer-temporal-unimodality",
  "title": "Counterexamples to the Cooper–Spencer temporal-unimodality conjecture",
  "shortTitle": "A random-walk probability falls, then rises",
  "url": "https://evidencepress.org/releases/cooper-spencer-temporal-unimodality/",
  "oneLine": "An exact dimension-26 counterexample and a written proof of failure in every dimension at least 64.",
  "abstract": "For simple symmetric lattice random walk, this unrefereed candidate gives an exact parity-restricted occupation-probability valley at times 65, 67 and 69 in dimension 26. A parity-sensitive Poissonisation lemma and rational Bessel bounds give counterexamples at (ceil(3d/5),0,...,0) for every integer d>=64. The threshold is sufficient, not optimal. Classical counting and transform machinery are explicitly distinguished from the counterexamples.",
  "datePublished": "2026-09-19",
  "dateModified": "2026-09-19",
  "version": "1.0.0-candidate",
  "doi": "10.5281/zenodo.22849717",
  "doiUrl": "https://doi.org/10.5281/zenodo.22849717",
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  "problem": {
    "name": "Cooper–Spencer Conjecture 3: parity-restricted temporal unimodality",
    "url": "https://arxiv.org/abs/math/0402323"
  },
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  "keywords": [
    "simple random walk",
    "temporal unimodality",
    "Poissonisation",
    "modified Bessel functions",
    "exact counterexample"
  ],
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    "Exact dimension-26 valley at supported times 65, 67 and 69.",
    "Counterexamples in every integer dimension d>=64 at m=ceil(3d/5).",
    "Parity-sensitive Poissonisation transfer with the essential a0=0 condition.",
    "Continuous-time dimension-40 certificate and discrete-time consequence."
  ],
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    "Determine the least dimension admitting a counterexample.",
    "Classify failing endpoints and the number of modes on the full time axis.",
    "Study valid parity-sensitive extensions to other walks and transforms."
  ],
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  "relatedWorks": [
    {
      "citation": "Cooper and Spencer (2006), Simulating a Random Walk with Constant Error: Conjecture 3 and parity convention.",
      "url": "https://doi.org/10.1017/S0963548306007565",
      "doi": "10.1017/S0963548306007565"
    },
    {
      "citation": "Karp, Vishnyakova and Zhang (2025), Unimodality Preservation by Ratios of Functional Series and Integral Transforms: classical sign-change input.",
      "url": "https://doi.org/10.1007/s00025-025-02431-4",
      "doi": "10.1007/s00025-025-02431-4"
    },
    {
      "citation": "NIST DLMF, modified Bessel series, derivatives and integral representations.",
      "url": "https://dlmf.nist.gov/10.25",
      "doi": null
    },
    {
      "citation": "OEIS A287318, classical lattice return-count array.",
      "url": "https://oeis.org/A287318",
      "doi": null
    }
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