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  "title": "How fast can a driven Markov network oscillate? A four-state counterexample to the affine Karpelevich conjecture, and the exact dimension at which it fails",
  "shortTitle": "A four-state network breaks the affine Karpelevich conjecture",
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  "oneLine": "The rescaled eigenvalues of Markov rate matrices do not always fill the affinity-squashed Karpelevich region conjectured by Xu, Kolchinsky, Delvenne and Ito (2025): an integer four-state network escapes it, three states cannot, and on fully connected networks every size from four up fails; the single-cycle case remains open.",
  "abstract": "The eigenvalues of a continuous-time Markov rate matrix decide how fast its probability distribution relaxes and how strongly it can oscillate. For discrete-time transition matrices the exact set of possible eigenvalues is a classical object, the Karpelevich region Ω_n. Xu, Kolchinsky, Delvenne and Ito (Phys. Rev. Lett. 135, 257102, 2025; Supplement Eq. (S3)) conjectured that the eigenvalues of rate matrices fill a squashed copy of this region: stretched by the fastest escape rate α and flattened vertically by β′ = tanh(max_c F_c/(2n_c)), set by the largest cycle affinity per step. This candidate shows the conjecture is false. An explicit four-state network with integer rates and about 8.4 k_BT of driving around its strongest cycle has the eigenvalue −4745 + 450√3 i, whose rescaled image ζ ≈ −0.7062 + 0.3161 i lies 0.0087 above the left boundary arc of Ω₄; one integer inequality proves it. For at most three states the containment holds, so four is the smallest failing size. The mechanism is weak resetting: it moves eigenvalues by O(ε) but relaxes the affinity factor by order ε^{2/3}. This gives counterexamples near a whole boundary arc in four states, an explicit interval 0 < ε ≤ 1/100, exactly certified five- and six-state instances whose data are printed in an appendix and re-checked by a stand-alone checker, and an asymptotic construction in every dimension n ≥ 4. All counterexamples use networks in which every pair of states is connected; whether the conjecture holds for single-cycle networks, the setting in which it was tested, remains open. The violations are thin, and the proved affinity ellipse and Farey-polygon bounds are unaffected. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-10-27",
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  "keywords": [
    "Markov rate matrices",
    "Karpelevich region",
    "cycle affinity",
    "stochastic thermodynamics",
    "eigenvalue bounds",
    "counterexample",
    "resetting",
    "exact certificate"
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    "Four-state counterexample: the integer rate matrix Q_w has eigenvalue −4745 + 450√3 i with α = 2781 and dominant triangle ratio 177853829/38529; its rescaled image ζ ≈ −0.7062 + 0.3161 i lies 0.0087 above the left boundary arc of Ω₄, so Supplement Eq. (S3) of Xu et al. (2025) fails (Proposition 2.1, Theorem 3.1, proof with an exact integer certificate).",
    "Smallest size: for at most three states the rescaled spectrum lies in Ω_n, so four is the smallest failing dimension (Theorem 4.1, proof, using the sector bound of Ohga, Ito and Kolchinsky).",
    "Mechanism: weak resetting moves eigenvalues by O(ε) but relaxes the affinity factor by order ε^{2/3}, giving counterexamples near every point of the open left arc of Ω₄, an explicit interval 0 < ε ≤ 1/100, and an open set of positive-rate counterexamples (Theorems 5.1, 6.1, Corollary 6.2, proof).",
    "Every dimension n ≥ 4 fails on fully connected networks: fix n, then a small t, then let ε → 0; the thresholds are not explicit (Theorem 7.4, proof).",
    "Explicit five- and six-state instances are certified exactly, with the certificate data printed in Appendix A and re-checked by a separate checker (Theorem 7.6, computer-assisted exact certificate).",
    "Open: whether the conjecture holds for single-cycle networks (Open problem 8.1); a small numerical ring search found no violation, which is evidence, not proof."
  ],
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    {
      "citation": "Xu, Kolchinsky, Delvenne and Ito (2025), Thermodynamic geometric constraint on the spectrum of Markov rate matrices. Physical Review Letters 135, 257102 (source of the refuted Supplement Eq. (S3); supplement read on arXiv:2507.08938v2).",
      "url": "https://doi.org/10.1103/z4t2-18cx",
      "doi": "10.1103/z4t2-18cx"
    },
    {
      "citation": "Kolchinsky, Ohga and Ito (2026), Cycle affinity and winding localize eigenvalues of Markov generators. arXiv:2605.15884v3 (proves the affinity ellipse and the Farey-polygon bound, both unaffected).",
      "url": "https://doi.org/10.48550/arXiv.2605.15884",
      "doi": "10.48550/arXiv.2605.15884"
    },
    {
      "citation": "Uhl and Seifert (2019), Affinity-dependent bound on the spectrum of stochastic matrices. Journal of Physics A 52, 405002 (the ellipse conjecture).",
      "url": "https://doi.org/10.1088/1751-8121/ab3a7a",
      "doi": "10.1088/1751-8121/ab3a7a"
    },
    {
      "citation": "Ohga, Ito and Kolchinsky (2023), Thermodynamic bound on the asymmetry of cross-correlations. Physical Review Letters 131, 077101 (the sector bound used for three states).",
      "url": "https://doi.org/10.1103/PhysRevLett.131.077101",
      "doi": "10.1103/PhysRevLett.131.077101"
    },
    {
      "citation": "Barato and Seifert (2017), Coherence of biochemical oscillations is bounded by driving force and network topology. Physical Review E 95, 062409.",
      "url": "https://doi.org/10.1103/PhysRevE.95.062409",
      "doi": "10.1103/PhysRevE.95.062409"
    },
    {
      "citation": "Karpelevich (1951; English translation 1988), On the characteristic roots of matrices with nonnegative elements. American Mathematical Society Translations, Series 2, 140, 79–100.",
      "url": "https://doi.org/10.1090/trans2/140/05",
      "doi": "10.1090/trans2/140/05"
    },
    {
      "citation": "Ito (1997), A new statement about the theorem determining the region of eigenvalues of stochastic matrices. Linear Algebra and its Applications 267, 241–246.",
      "url": "https://doi.org/10.1016/S0024-3795(97)00018-9",
      "doi": "10.1016/S0024-3795(97)00018-9"
    },
    {
      "citation": "Johnson and Paparella (2017), A matricial view of the Karpelevič theorem. Linear Algebra and its Applications 520, 1–15 (boundary-realising matrices).",
      "url": "https://doi.org/10.1016/j.laa.2017.01.009",
      "doi": "10.1016/j.laa.2017.01.009"
    },
    {
      "citation": "Kirkland and Šmigoc (2022), Stochastic matrices realising the boundary of the Karpelevič region. Linear Algebra and its Applications 635, 116–138.",
      "url": "https://doi.org/10.1016/j.laa.2021.11.016",
      "doi": "10.1016/j.laa.2021.11.016"
    },
    {
      "citation": "Rose, Touchette, Lesanovsky and Garrahan (2018), Spectral properties of simple classical and quantum reset processes. Physical Review E 98, 022129 (the resetting spectral shift).",
      "url": "https://doi.org/10.1103/PhysRevE.98.022129",
      "doi": "10.1103/PhysRevE.98.022129"
    },
    {
      "citation": "Chruściński, Denisov, Tarnowski and Życzkowski (2025), Spectral delineation of Markov generators: classical vs quantum. arXiv:2504.07903.",
      "url": "https://doi.org/10.48550/arXiv.2504.07903",
      "doi": "10.48550/arXiv.2504.07903"
    },
    {
      "citation": "Verbeken and Ginis (2026), On the spectral region of n-cycle stochastic matrices. arXiv:2606.10143.",
      "url": "https://doi.org/10.48550/arXiv.2606.10143",
      "doi": "10.48550/arXiv.2606.10143"
    },
    {
      "citation": "Xu and Meng (2026), Kinetic heterogeneity tightens thermodynamic bounds on spectra of Markov cycles. arXiv:2608.22934.",
      "url": "https://doi.org/10.48550/arXiv.2608.22934",
      "doi": "10.48550/arXiv.2608.22934"
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