D4 Symmetry in Square Lattice QCD and Pedagogical Resources — E8 Intelligence Research

FINDING: The search results are primarily pedagogical videos and one lattice QCD paper; no direct new result on Dedekind eta modular symmetry order 4 or C4 partition is presented. The mathematical core is the crystallographic point group D4 (order 8) and its action on the square lattice, plus a lattice QCD mass calculation. MATH: - Square lattice point group: D4 (dihedral of order 8), generated by 90° rotation \( r \) and reflection \( s \), with relations \( r^4 = s^2 = (rs)^2 = e \). - Crystallographic restriction: only rotations of order 1,2,3,4,6 are allowed in 2D lattices — order 4 is maximal for square symmetry. - Dedekind eta function: \( \eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n) \), \( q = e^{2\pi i \tau} \). Modular transformations of order 4 correspond to \( \tau \mapsto \frac{a\tau+b}{c\tau+d} \) with \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2,\mathbb{Z}) \) and specific congruence conditions (e.g., \( \Gamma(4) \) or \( \Gamma_0(4) \)). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com