[
  {
    "id": "ARR-2026-5PXDYNR2ER89HTQG-v1",
    "type": "article",
    "title": "Recurrence-Amos Proof of the Unit-Step Order-Monotonicity of (log I_nu)', with a Feynman-Hellmann Application to Two-Dimensional Lattice Gauge Theory",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          7,
          9
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5PXDYNR2ER89HTQG",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-5PXDYNR2ER89HTQG/versions/v1/",
    "note": "Archived manuscript; ARR-2026-5PXDYNR2ER89HTQG v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "Let I_nu denote the modified Bessel function of the first kind and, forx>0, let rho_nu(x) = I_{nu+1}(x)/I_nu(x). We give a four-step, fullyelementary proof of the sharp difference inequality0 < rho_nu(x) - rho_{nu+1}(x) < 1/x (x>0, nu>=0), whose right-handinequality is exactly the strict increase of the logarithmic derivative(log I_nu)'(x) under the unit shift nu -> nu+1; consequentlynu -> (log I_nu)'(x) is strictly increasing along every unit-spaced gridnu_0 + N, in particular on the integer and half-integer orders arising inthe application. The stronger continuous-order statement is known(Freitas-Laugesen, arXiv:1810.07461, Lemma 10, via Bessel zeros); we makeno elementary claim about fractional steps. The proof given here uses noinformation about Bessel zeros: it combines the three-term recurrence withthe classical Amos-type upper bound rho_nu < x/(a+sqrt(a^2+x^2)),a = nu+1/2, and rests on the observation that this bound is exactlycalibrated for the problem: 1/U - U = 2a/x is an algebraic identity, and(2nu+1)/x is precisely the threshold the unit step requires; in fact theunit-step monotonicity and the Amos bound are equivalent. As anapplication we record the following consequence in two-dimensional latticegauge theory: for the Wilson action, every mass gap between charactersectors of the 2D transfer operator - for U(1) and SU(2) alike - is astrictly decreasing function of the bare coupling beta, by theFeynman-Hellmann identity. The algebraic core of the proof ismachine-checked in Lean 4/Mathlib (no sorry; axiom oracle: Lean's threestandard axioms), and an independent high-precision numerical audit ofevery inequality used is reported."
  }
]