Continuum Yang–Mills Limit of SU(2) Holonomies on the Diamond Lattice in Granular Entropic Physics

We perform the continuum expansion of SU(2) Wilson loops on the diamond lattice as prescribed by Granular Entropic Physics (GEP). For a slowly varying gauge field, the holonomy around the minimal hexagonal cycle expands as W = I + ia^2 Sigmamu nu Fmu nu + O(a^3), where Sigma is the oriented area tensor and F is the Yang-Mills field strength. We prove an isotropy theorem: the four tetrahedrally related hexagon classes satisfy Sum_I n_I^rho n_I^sigma = (4/3) deltarho sigma, producing an exactly isotropic leading-order effective action despite the discrete tetrahedral geometry. The resulting theory is 3D SU(2) Yang-Mills with coupling g^2 = 3/(2|Sigma|^2 a). The area tensor Sigma is dimensionless (constructed from unit vectors), and g^2 diverges as a goes to 0, consistent with the super-renormalisable nature of 3D Yang-Mills. We note that the GEP vacuum has Tr(W_0) = -10/27, so a physically motivated expansion should be performed around the frustrated background rather than trivial holonomy. This is the first explicit leading-order lattice-to-continuum bridge in GEP, connecting the discrete diamond lattice structure to standard gauge theory.