We introduce a spectral anisotropy functional on periodic two-dimensional lattices and derive exact analytic expressions comparing the six-direction and four-direction nearest-neighbor coordination classes. For equal shell distance and equal directional weights, the six-direction coordination class (realized by the hexagonal and triangular lattices) satisfies Ahex = Atri = w²/3 0. Because the shell weight w² factors out of the ordering, the comparison reduces to a purely geometric statement carried by the coordination-normalized coefficients 1/3 and 1/2. The 1/zg normalization is derived from a duplication-invariance condition and independently confirmed through a Brillouin-zone trace identity. This establishes an exact analytic pairwise comparison between the square lattice and the six-direction nearest-neighbor coordination class and provides the first verified case of a conjectured coordination-normalized anisotropy relation Cg = 2/zg. Possible connections between geometric constants associated with the hexagonal primitive cell and physical coupling scales lie outside the scope of the present analysis and are addressed separately in companion technical works. Companion works Universal Grid Mechanics (UGM): An Axiomatic, Admissibility-First Framework for Physical Reality. Zenodo (February 2026). A Pre-Phenomenological Update Constraint for Admissible Physical States. Zenodo (January 2026). These works develop a broader theoretical framework motivating the lattice-based operator structures analyzed here, but are not required for the mathematical results derived in the present paper.
H. et al. (Wed,) studied this question.