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  "slug": "unique-equilibrium-lsi-separation",
  "title": "A unique-equilibrium separation between particle and mean-field log–Sobolev constants",
  "shortTitle": "One equilibrium. Different sharp constants.",
  "url": "https://evidencepress.org/releases/unique-equilibrium-lsi-separation/",
  "oneLine": "A fixed smooth circle model has a unique free-energy minimizer, yet unrestricted particle and global nonlinear entropy-dissipation constants remain strictly separated.",
  "abstract": "For every fixed positive cosine coefficient a, an unrefereed written-proof candidate gives λN ≤ 1 < λMF ≤ min{1+a/2,4} for a smooth repulsive pair interaction on the circle of length 2π with a unique uniform free-energy minimizer. A tagged-coordinate test bounds the unrestricted particle constant; a global compactness/rigidity argument gives the nonlinear separation. The revision supplies an explicit conservative lower gap, an exact first-harmonic upper-bound formula, a finite-amplitude example with λMF(97/16) < 4 and a coordinate-block lifting/defect clarification. Neither sharp constant, a particle limit nor a symmetric-only particle result is determined.",
  "datePublished": "2026-10-14",
  "dateModified": "2026-10-14",
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  "doi": "10.5281/zenodo.23360417",
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  "problem": {
    "name": "Sharp equality of unrestricted particle and nonlinear mean-field constants in the identified arXiv formulation",
    "url": "https://arxiv.org/abs/2112.06304v1"
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  "metadataProvenance": "Revised manuscript and exact finite diagnostics, 14 October 2026; archive and site availability assessed separately.",
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  "keywords": [
    "log-Sobolev inequality",
    "mean-field limit",
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    "A fixed smooth model with unique uniform free-energy minimizer satisfies λN ≤ 1 < λMF for every N and every fixed a > 0.",
    "An explicit conservative δ(a) makes the global nonlinear lower gap quantitative without assuming an optimizer exists.",
    "An exact first-harmonic quotient yields finite-amplitude improvements, including λMF(97/16) < 4.",
    "Coordinate-block lifting explains the microscopic obstruction; the tagged test forces only an order-one additive defect, compatible with o(N)."
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  "evidencePackage": "Written global proof with explicit conservative gap and finite-amplitude upper family. Public original checker passed 386 exact comparisons and six rejected semantic controls in normal and optimized execution; supplied private floating audit replayed separately. Neither establishes the universal proof computationally.",
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  "relatedWorks": [
    {
      "citation": "Delgadino, Gvalani, Pavliotis and Smith (2021). Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions. arXiv:2112.06304v1, particle variational definition and Conjecture 1; comparison is to this version, not uninspected journal wording.",
      "url": "https://arxiv.org/abs/2112.06304v1",
      "doi": "10.48550/arXiv.2112.06304"
    },
    {
      "citation": "Émery and Yukich (1987). A simple proof of the logarithmic Sobolev inequality on the circle. Séminaire de probabilités 21, 173–175.",
      "url": "https://www.numdam.org/item/SPS_1987__21__173_0/",
      "doi": null
    },
    {
      "citation": "Monmarché (2026). Equivalence between N-particle log-Sobolev inequalities and non-linear Łojasiewicz inequalities for a class of mean field systems. arXiv:2609.06574v2; positivity and sharp equality remain distinct.",
      "url": "https://arxiv.org/abs/2609.06574v2",
      "doi": "10.48550/arXiv.2609.06574"
    },
    {
      "citation": "Monmarché (2024). arXiv:2409.17901v1, Section 1.4.5; the weaker sublinear-defect formulation is not refuted by the tagged-coordinate obstruction.",
      "url": "https://arxiv.org/abs/2409.17901v1",
      "doi": "10.48550/arXiv.2409.17901"
    }
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