Character Positivity of Wilson Kernels for Compact Simple Lie Groups

We prove that for any compact simple Lie group G and any coupling β > 0, allcharacter expansion coefficients of the Wilson lattice gauge theory kernel are strictlypositive: aλ(β) > 0 for every irreducible representation λ. The proof uses Peter–Weyl decomposition, the non-negativity of Clebsch–Gordan multiplicities, and thestandard fact that every irreducible of a compact simple Lie group occurs in sometensor power of any faithful representation. This replaces the Li–Yau heat kernelbound, which requires a geometric comparison argument, with a purely algebraicproof that applies uniformly to all compact simple groups including the exceptionaltypes. As a corollary, the character ratio r(β) is strictly increasing on (0, ∞).As a second result, we prove that n = 4 is the unique dimension in which thef-vector of the n-cube simultaneously matches the root system decomposition of theCoxeter group Bn.We conclude with a discussion of open problems in the lattice approach to theYang–Mills mass gap, identifying the specific analytical steps that remain unresolved.