Mathematical Formalization, Information Loss, and Observational Effect Boundaries of the T64 to R4 Projection —— A Study on High-to-Low Dimensional Mapping Based on the Spacetime Uniqueness Law of Yuanxian Theory

The Spacetime Uniqueness Law (STM) of Yuanxian Theory asserts that the ontological topology of the universe is a sixty-four-dimensional compact torus (T64), whereas the four-dimensional spacetime (R4) perceived and scientifically described by human observers is merely its low-dimensional projection. However, the explicit construction of the projection operator (P) from T64 to R4, the quantification of the information loss incurred during this projection, and the question of whether discarded high-dimensional information can be recovered or indirectly observed remain foundational issues that the theory must rigorously address. This paper aims to fill this gap. First, we explicitly construct the projection operator P: L2(T64) to L2(R4), which is strictly defined as an integral operator averaging over sixty compact dimensions. In the Fourier domain, this operator is equivalent to an ideal low-pass filter: it retains exclusively those modes with a winding number of zero along the compact directions (n_perp = 0) while entirely discarding all non-zero winding modes (n_perp != 0). Second, we introduce the concept of projection entropy (S_proj) to quantify this information loss, proving that under typical physical power spectra decaying by a power law, the amount of lost information is massive and non-negligible, meaning that observable 4D physics captures less than one-fifth of the total informational content of the ontological universe. Third, we analyze the observational effect boundaries of the discarded information, demonstrating that through the non-linear self-interactions of the self-referential mind field (Psi_SR), a portion of the high-dimensional topological effects can leak into the low-dimensional projection as "residual anomalies" (e.g., higher-dimensional effective operators, primordial gravitational wave background modifications, and quantum computing correlations). Finally, we propose two information recovery strategies for this ill-posed inverse problem: regularized reconstruction based on the Moore-Penrose pseudoinverse, and iterative physical inversion constrained by True-Circle Self-Consistency (TCSC) dynamics. This work establishes a rigorous mathematical foundation for the high-dimensional projection hypothesis in Yuanxian Theory and delineates concrete directions for its future experimental and computational verification. 元宪理论的时空唯一性律(STM)断言:宇宙的本体拓扑为一个六十四维紧致环面(T64),而人类所感知并科学描述的四维时空(R4)仅仅是其低维投影。然而,从 T64 到 R4 的投影算子(P)的显式构造、投影过程造成的信息损失及其量化,以及丢失的高维信息能否通过某种方式被恢复或间接观测,是该理论必须回答的基础性问题。本文旨在填补这一空白。 首先,我们显式构造了投影算子 P: L2(T64) 到 L2(R4),将其严格定义为对60个紧致维度取平均的积分算子。在傅里叶域中,该算子等价于一个理想的低通滤波器:它仅保留紧致方向上环绕数 n_perp = 0 的长波模式,而将其余非零环绕数(n_perp != 0)