This is the eighth paper in the series on the purely geometric unified theory. Based on the encoding-sphere
C geometric architecture and the fermion geometric dynamics equation established in previous work, we demonstrate the ability of this framework to be "downward compatible" with standard quantum mechanics. First, starting from photon-sphere geometry and the holographic principle, we rigorously derive the effective Coulomb potential between two charges,
V(r)=−α/r, where the fine-structure constant
α is uniquely determined by the information capacity of the triple projections of the encoding sphere
C, without experimental input. On this basis, we explicitly write down the full dynamics equation for the electron in the hydrogen atom and prove that, after separating the rooting degree of freedom, the ground-state solution automatically reduces to the standard Schrödinger equation of quantum mechanics, thereby incorporating all the successes of quantum mechanics in atomic physics into the encoding-sphere
C geometric framework. Furthermore, in a manner completely independent of standard quantum mechanics, starting from the rooting geometry of the encoding sphere
C, we derive the hydrogen atom ground-state binding energy via a purely geometric self-consistent iteration:
Ebind=13.596 eV, with a deviation from the experimental value
13.598 eV of only **0.018%**, with zero free parameters. The Pauli exclusion principle also acquires an ontological explanation within this framework—it is a necessary consequence of the minimisation of information-encoding complexity in the rooting geometry of the encoding sphere
C. A Python verification code is provided; all numerical values are independently reproducible. 本文是纯几何统一理论系列的第八篇。基于前序工作确立的编码球
C 几何架构和费米子几何动力学方程,我们展示了该框架向标准量子力学的"向下兼容"能力。首先,从光子球面几何与全息原理出发,严格导出两个电荷之间的有效库仑势
V(r)=−α/r,其中精细结构常数
α 由编码球
C 三重投影的信息容量唯一确定,无需实验输入。在此基础上,显式写出氢原子中电子的完整动力学方程,并证明在分离扎根自由度后,基态解自动退化为标准量子力学的薛定谔方程,从而将量子力学在原子物理中的全部成功纳入编码球
C 几何框架。 进一步,以完全独立于标准量子力学的方式,从编码球
C 扎根几何出发,通过纯几何自洽迭代导出氢原子基态束缚能
Ebind=13.596 eV,与实验值
13.598 eV 偏差仅 **0.018%**,零自由参数。泡利不相容原理也在本框架中获得了本体论层面的解释——它是信息编码复杂度极小化在编码球
C 扎根几何中的必然推论。附Python验算代码,所有数值可独立复现。
Complex numbers are the mathematical foundation of quantum mechanics, yet a fundamental question has remained unasked: where does the imaginary unit i come from? The standard formulation treats it as a primitive mathematical object without tracing its physical or geometric origin. This paper proves that i is not a mysterious attribute peculiar to quantum mechanics, but an algebraic necessity of spatial geometry. Starting from the Oh point group of a simple cubic lattice, the rigorous construction of the generators of the antisymmetric T1g subspace demonstrates that the commutation relation [J^a, J^b] = iε^{abc}J^c necessarily contains i, which is the algebraic signature of the su(2) structure constants arising from the non-commutativity of rotations about the three spatial axes. Furthermore, within the Clifford algebra Cl(3,0), the square of a bivector equals -1, providing a direct geometric counterpart for i. Recent experimental tests have confirmed the physical reality of complex numbers in quantum theory with extremely high confidence—this paper establishes the geometric necessity of i: it is not an introduced mathematical tool, but an intrinsic feature of the algebraic structure of any vector degree of freedom in three-dimensional space.