Quantum Field Theory on Ramanujan Graphs: A Complete Formulation in GEUT

We present a rigorous construction of quantum field theory on d = 6 Ramanujan graphs within theGeometric Emergent Unity Theory (GEUT) framework. The theory yields finite quantum fields through: (1)discrete Laplacian propagators, (2) exact gauge invariance from graph holonomy, and (3) emergent continuum limits. All Standard Model interactions are derived from the graph spectrum λn ∈ [6 − 2√5, 6 + 2√5] with no external regularization required.我们在几何涌现统一理论（Geometric Emergent Unity Theory, GEUT）框架下，严格构建了维度$d=6$的拉马努金图（Ramanujan graphs）上的量子场论。该理论通过以下三种途径得到有限量子场：(1) 离散拉普拉斯传播子；(2) 源自图全环绕的精确规范不变性；(3) 涌现连续极限。所有标准模型（Standard Model）的相互作用均可从图的能谱$lambda_n in [6 - 2sqrt{5}, 6 + 2sqrt{5}]$推导得出，且无需引入外部正则化手段。