Formal Verification of the Dimensional Expansion Strategy in Yuanxian Physics: Lean 4 and Coq Implementation of Projection Operators, Self-Referential Field Equations, and Dimensional Reduction Mappings

Yuanxian Theory posits that observable physical phenomena are localized projections of a Self-Referential Mind Field residing on a 64-dimensional compact torus (T64) onto a standard four-dimensional spacetime cross-section. While the foundational framework outlined in "The Source of Projection: The Dimensional Up-Scaling Paradigm of Physics under Yuanxian Theory" systematically expounded this strategy from a conceptual and physical viewpoint—providing unified interpretations for gravity, quantum measurement, particle parameters, cosmological constant suppression, the arrow of time, and biophysical consciousness fields—its core mathematical structures have previously lacked rigorous, machine-checked validation. This paper directly bridges this critical gap by introducing a complete formal verification of these structures. Utilizing a coordinated dual-engine architecture consisting of the Lean 4 and Coq (Rocq) interactive theorem provers, we formally verify the entire mathematical chain of the Yuanxian dimensional expansion paradigm. Our primary contributions include: (1) Formally defining the topology of the 64-dimensional torus (T64) in Lean 4, proving its geometric compactness, discrete spectrum scaling, and the properties of its corresponding Laplacian operator; (2) Formalizing the high-dimensional projection functor and proving that it strictly preserves early homology ranks and the asymptotic distribution profiles of eigenvalues; (3) Proving the convergence dynamics and unique fixed-point existence of the discrete self-referential field equation via a fully modernized implementation of the Banach Fixed-Point Theorem, establishing an axiomatic foundation for the irreversible arrow of time; (4) Formalizing continuous dimensional reduction mappings in Coq, demonstrating via Fubini-based fiber integrals that the reduction from high-dimensional actions to a 4D effective field theory strictly preserves macroscopic Noether energy-momentum conservation laws; and (5) Formally deriving the explicit quantitative parameters of the alphaFSC⁶4 cosmological constant suppression factor, topological Casimir force boundaries, and discrete Kaluza-Klein (KK) mass spectra lines. All source codes have passed strict type-checking and are open-sourced in the YuanXian-FormalPhysics repository, establishing an unassailable, machine-checked mathematical foundation for high-dimensional physics under the Yuanxian paradigm. 元宪理论提出, 人类所能观测到的四维物理现象, 本质上是六十四维紧致环面 (T64) 上的自指心场在四维时空截面上的局部化投影。虽然奠基性文献《投影之源: 元宪理论下的物理学范式升维》从物理学和本体论视角系统阐述了这一升维方案——并对引力机制、量子测量问题、基本粒子参数、宇宙学常数压制、时间箭头以及生物物理意识场给出了统一的升维解释——但其核心数学结构此前一直缺乏严格的机器可验证性。本文直接填补了这一关键的实证与逻辑空白, 首次实现了对这些核心结构的完整形式化验证。 我们采用 Lean 4 和 Coq (Rocq) 交互式定理证明器构成的协同双引擎架构, 对元宪物理升维方案的完整数学链条进行了严密的机器检查。本文的核心贡献包括: (1) 在 Lean 4 中形式化定义了六十四维环面 (T64) 的拓扑结构, 证明了其几何紧致性、拉普拉斯算子的离散谱性质及特征值行为; (2) 形式化定义了高维投影函子, 并证明了该投影在低维截面上严格保持前四个同调群的秩以及拉普拉斯特征值的渐近分布; (3) 形式化实现了离散自指场方程的动力学演化, 利用标准 Banach 不动点定理严格证明了演化算子的压缩性、唯一宇宙基态不动点的存在性以及迭代收敛性, 从而为不可逆的时间箭头提供了公理化证明; (4) 在 Coq 中形式化了高维到低维的降维映射, 通过引入基于测度论的纤维积分, 证明了从六十四维高维作用量约化到四维有效场论的过程中, 严格保持了宏观的诺特能量-动量守恒律; (5) 在机器中成功形式化推导出了宇宙学常数精细结构常数64次方 (alphaFSC⁶4) 的拓扑压制因子、微观卡西米尔力的拓扑调制边界以及离散卡鲁扎-克莱因 (KK) 粒子质量谱公式。目前, 所有形式化证明代码均已完整通过定理证明器的类型检查, 并正式开源于 YuanXian-FormalPhysics 官方代码仓库, 为元宪理论的高维物理学范式奠定了无可辩驳的、机器验证的现代数学基石。