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  "title": "When a network looks exactly like a tree",
  "shortTitle": "K2P tree–network coincidence",
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  "oneLine": "Two reticulations can leave exactly the same complete pattern probabilities as a tree—even with continuous-time, transition-biased mutation.",
  "abstract": "An exact K2,3 construction in the uniform-root displayed-tree model matches all 64 three-leaf pattern probabilities of a star tree, with strictly interior edge-specific continuous-time transition-biased K2P parameters in both models. On the larger stochastic region, anchoring classifies reduced trinets through level three: 30 of 83 are distinguishable and 53 admit coincidences. Written proofs propagate examples to arbitrary leaf counts and levels. A separate convexity obstruction excludes the same construction when both models share one prescribed rate ratio greater than one.",
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      "citation": "Brits et al. (2026), arXiv version 3. Higher-level separation under JC; K2P extension left open.",
      "url": "https://arxiv.org/abs/2607.12919v3",
      "doi": "10.48550/arXiv.2607.12919"
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    {
      "citation": "Allman, Rhodes and Sullivant (2012). Mixture mimicry and topology-qualified non-mimicking results.",
      "url": "https://doi.org/10.1093/sysbio/sys064",
      "doi": "10.1093/sysbio/sys064"
    },
    {
      "citation": "Sullivant (2026). Graphical-model local modification and suppression framework.",
      "url": "https://doi.org/10.1007/s11538-026-01695-3",
      "doi": "10.1007/s11538-026-01695-3"
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    },
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      "citation": "Holtgrefe et al. (2025), version 2. Quartet information under specified level-two structural conditions.",
      "url": "https://arxiv.org/abs/2507.17308v2",
      "doi": "10.48550/arXiv.2507.17308"
    },
    {
      "citation": "Evidence Press: When a network cannot look like a tree. Different single-cycle and shared-generator hypotheses; not contradicted by this two-reticulation example.",
      "url": "https://evidencepress.org/releases/tree-network-separation/",
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