Metaphysical Termination of the Darkness of Night: Ontological Dissolution of Olbers' Paradox under the Compact Torus T⁶⁴

Abstract

Olbers’ paradox remains the sharpest logical touchstone of classical cosmology. Its modern mainstream resolutions still harbour, at the infinite-distance limit, a metaphysical residue of “infinitely many stars beyond the horizon.” On the basis of the $\mathcal{T}^{64}$ topological law system of Yuanxian Theory, this paper proves that the darkness of night is not an observational truncation arising from “light that has not yet arrived,” but a necessary corollary of the discreteness of closed-chain homotopy classes on $\mathcal{T}^{64}$. We rigorously establish: (1) the ADM Hamiltonian constraint on a compact manifold forces the total energy to vanish, thereby cutting off any divergent source of total mass; (2) the topological rigidity of the fundamental group $\pi_1(\mathcal{T}^{64})\cong\mathbb{Z}^{64}$ confines photon modes to a finite set of homotopy classes, generating an intrinsic upper bound on the density of states without any artificial cut-off; (3) the pull-back of the Jacobian matrix of the higher-dimensional projection map produces a non-expansion geometric redshift, so that dark energy appears naturally as the four-dimensional projection residual of fibre-direction curvature. The scheme completely severs the mathematical premise of infinite accumulation and agrees with Planck satellite CMB data to within $1\sigma$. --- 奥伯斯佯谬作为经典宇宙学最尖锐的逻辑试金石,其现代主流消解方案在无穷远极限下仍隐含“视界外无限多恒星”的形而上学残余。本文基于元宪理论 $\mathcal{T}^{64}$ 拓扑规律体系,证明夜黑现象并非源于“光尚未到达”的观测截断,而是 $\mathcal{T}^{64}$ 上闭链同伦类离散性的必然推论。我们严格证明:(1)紧致流形上的ADM哈密顿约束迫使总能量归零,切断发散的总质量来源;(2)$\mathcal{T}^{64}$ 基本群 $\pi_1(\mathcal{T}^{64})\cong\mathbb{Z}^{64}$ 的拓扑刚性导致光子模式被限制在有限的同伦类中,形成内禀的态密度上限,无需引入人为截断;(3)高维投影映射的雅可比矩阵拉回导致非膨胀的几何红移,暗能量自然呈现为纤维方向曲率的4维投影残差。该方案彻底切断了无限累积的数学前提,与Planck卫星CMB数据在 $1\sigma$ 内精确吻合。