Spectral Geometric Prism Model: A Unified Field Framework via Finite-Action Constraints

Abstract

We propose the Spectral Geometric Prism Model (SGPM), a unified field framework built upon finite-action topological constraints to resolve infrared divergences arising from noncompact physical spacetime. Single compactification via finite Yang–Mills action yields a topological homeomorphism between finite-action gauge configuration spaces over ℝ⁴ and smooth gauge fields on S⁴. Under this isomorphism, standard spectral triples can be rigorously extended to compact four-manifolds. The heat-kernel asymptotic expansion of a single universal spectral trace functional simultaneously recovers gravitational Einstein–Hilbert dynamics, full Standard Model Yang–Mills gauge sectors, and Higgs mass generation terms, realizing a single master equation unifying all four fundamental interactions. All finite-dimensional trace identities, Dirac operator algebraic reductions and pointwise field decay estimates required for the unified spectral action are formally machine-verified within the Lean 4 proof assistant, eliminating manual algebraic derivation errors. We fully address low-energy renormalization decoupling, intrinsic type-I neutrino seesaw mass generation, and clearly outline all open mathematical and physical limitations of the framework. LaTeX source files and Lean 4 formal verification code will be released alongside the subsequent arXiv preprint. This deposit contains the revised English manuscript (v2) together with the first published full Chinese manuscript of the SGPM model. 我们提出了光谱几何棱镜模型(SGPM),这是一套建立在有限作用量拓扑约束之上的统一场框架,旨在解决非紧致物理时空带来的红外发散问题。借助有限杨–米尔斯作用量实现单点紧化,可以建立ℝ⁴有限作用规范构型空间与S⁴上光滑规范场之间的拓扑同胚。在此同胚映射下,标准谱三元组能够严格延拓至紧致四维流形。单一普适谱迹泛函的热核渐近展开同时复原引力爱因斯坦–希尔伯特动力学、完整标准模型杨–米尔斯规范扇区以及希格斯质量生成项,得到统一四类基本相互作用的主方程。统一谱作用所需全部有限维迹恒等式、狄拉克算符代数约化以及逐点场衰减估计均已在Lean4证明助手内完成形式化机器验证。本存档同时包含英文原版论文与完整中文译本。