[
  {
    "id": "ARR-2026-31ED4MTKC18JPSDE-v2",
    "type": "article",
    "title": "Clustering and the Transfer-Operator Gap: A Machine-Checked Dense-Family Criterion",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-31ED4MTKC18JPSDE",
    "version": "v2",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-31ED4MTKC18JPSDE/versions/v2/",
    "note": "Research preprint; ARR-2026-31ED4MTKC18JPSDE v2. Not peer reviewed by AIRR.",
    "abstract": "Inside a Lean 4 formalization programme for four-dimensional SU(Nc) lattice Yang–Mills, this paper studies the operator-theoretic criterion between exponential decay of a connected correlator and a spectral gap. Let T be a bounded self-adjoint operator on a Hilbert space, let Ω be a fixed unit vector, and set S = T − |Ω⟩⟨Ω|. The dense-family criterion states that if the vectors whose S-orbits decay at one common exponential rate r form a dense set, then ‖S‖ ≤ r, even when each vector has its own arbitrary finite prefactor. The decay domain is a linear subspace, so a densely spanning observable family suffices. The corrected manuscript supplies an elementary dyadic proof alongside the historical Lean functional-calculus formalization. The underlying criterion is classical; the contribution is its mechanization and packaging for observable families. The paper constructs no gauge-theory Hilbert space, proves no reflection positivity of the Wilson measure, and identifies no physical correlator with an operator matrix element. It makes no continuum-limit or Clay-problem claim. The release preserves historical formal evidence and distinguishes it from the paper-level additions; no fresh complete Lean build is certified."
  }
]