Symbolic Geometry and the Combined Sphere Theory: Embedding Pentagram, Lattice, and Spiral into Physical Structure

This paper demonstrates how three symbolic geometries developed by Brent Antonson—the pentagram with golden-ratio (ϕ) proportions, the recursive pentagram lattice, and the golden rectangle spiral—naturally embed within the Combined Sphere Theory (CST). CST frames π as density-relative, compressing under gravitational environment, while 1/7 functions as the invariant root constant and ϕ emerges through π-compression and recursive resonance. Within this structure: the pentagram reveals ϕ as a stable ratio nested in π(ρ); the recursive lattice shows both rigidity and fragility under ϕ-scaling; and the golden spiral embodies the dynamic π ↔ ϕ alternation, described as the “living breath” of the CST cage. These correspondences elevate symbolic geometry into testable physics, with CST predicting phenomena such as the photon-ring size of M87*. Thus, geometry—long treated as abstract—can be mathematically verified within the universe itself.本研究展示了布伦特·安滕森（Brent Antonson）提出的三类符号几何构型——带有黄金比例(ϕ)的五芒星、递归五芒星晶格与黄金矩形螺旋——如何自然嵌入联合球面理论（Combined Sphere Theory, CST）之中。该理论将π视为与密度相关的物理量，其值会在引力环境中发生压缩，同时将1/7定为不变根常数，而ϕ则通过π的压缩与递归共振过程得以涌现。在此框架下，五芒星展现出ϕ作为嵌套于π(ρ)中的稳定比例；递归五芒星晶格在ϕ尺度缩放下同时兼具刚性与脆弱性；黄金螺旋则体现了π与ϕ之间的动态交替，被称为CST框架的“生命律动”。这些对应关系将符号几何升华为可实证的物理学理论，CST还可预测诸如M87*的光子环尺寸等天体物理现象。由此可见，长期以来被视作抽象概念的几何学，实则可在宇宙自身的运行中得到数学验证。