The Spectral Geometric Prism Model: Convergence Mechanism of Gauge Interactions under Finite Action Constraints and Prediction of Novel States of Matter

Within the framework of the Spectral Geometric Prism Model (SGPM)(https://doi.org/10.5281/zenodo.21810152), this papersystematically elucidates how fundamental interactions emerge naturally from thespectral action of noncommutative geometry. Unlike the Standard Model, whichintroduces the gauge group SU(3)_c × SU(2)_L × U(1)_Y ad hoc, the SGPM is basedon topological constraints imposed by finite action. We demonstrate that gauge fieldsare inevitable convergence terms arising from the spectral properties of thegeneralized Dirac operator D_A under the renormalization group flow. Through aterm-by-term convergence analysis of higher-order corrections in the heat kernelexpansion, we rigorously derive an effective Lagrangian containing higher-ordercurvature terms and provide the complete running equations for gauge couplingconstants under the renormalization group flow. Based on the variational principle ofthis framework, we predict a novel class of scalar bound states induced by gluoncondensation, termed “Prism Glueballs,” and present a formula for estimatingtheir mass. Furthermore, in conjunction with experimental directions at the BESIII(Beijing Spectrometer III), LHC (Large Hadron Collider), CEPC (Circular ElectronPositron Collider), and ILC (International Linear Collider), we propose specificdetection windows and signal characteristics. All key algebraic identities have beenformally verified using Lean 4. 本文在光谱几何棱镜模型(Spectral Geometric Prism Model, SGPM)框架下(https://doi.org/10.5281/zenodo.21810152),系统阐述了基本相互作用如何从非交换几何的谱作用量中自然涌现。不同于传统标准模型人为引入规范群SU(3)_c × SU(2)_L × U(1)_Y,SGPM 基于有限作用量拓扑约束,证明了规范场是广义狄拉克算子 D_A 谱性质在重整化群流下的必然收敛项。通过对热核展开的高阶修正进行逐阶收敛分析,我们严格导出了包含高阶曲率项的有效拉格朗日量,并给出了规范耦合常数在重整化群流下的完整跑动方程。基于该框架的变分原理,我们预言了一类由胶子凝聚诱导的新型标量束缚态——“棱镜胶球”(Prism Glueball),并给出了其质量估算公式。本文进一步结合北京正负电子对撞机 BESIII、欧洲核子研究中心 LHC、未来环形正负电子对撞机 CEPC 及国际直线对撞机 ILC 的实验方向,提出了具体的探测窗口与信号特征。所有关键代数恒等式均已通过Lean 4 形式化验证。