Theory of Algebraic Spectra of Equilateral Polyhedra

This paper proposes and systematically expounds a complete algebraic generation theory for equilateral polyhedra. Its core framework is composed of the interwoven Forward World and Reverse World. The Forward World utilizes equilateral regular polygons and their derived polygons as materials, constructing polyhedra characterized by integer inter-edge angles through forward combinations. The Reverse World starts from the transcendental inter-edge angles generated by the reverse combination of faces in the Forward World, and reproduces a vast transcendental algebraic spectrum through generation rules. Both worlds construct polyhedra independently yet share the same self-consistent law. The algebraic foundation of this law is the golden ratio algebraic field Q(sqrt(5)). The screening condition is the polyhedral genetic code (36°, 60°, 72°, 90°, 108°, 120°, 144°), and the reproduction operator is a linear fractional transformation in cosine space: g(c1, c2) = (1 + 2c1 - c2) / (1 + c2). Starting from the algebraic structure of the generation rule, this paper rigorously proves a series of inevitable phenomena in the spectrum: the acute-obtuse pairing theorem, the complete classification of fixed points (absorbing and reflection types), the self-generating closed loop theorem (the unique acute angle solution 76.34542°, c = sqrt(5) - 2 = 1 / phi^3), and the self-dual generation theorem. The three fitting genes (golden rhombus, honeycomb rhombus, and double-Fibonacci rhombus) are all bound to the fixed point structure, and their self-dual products precisely cover three core node types: genetic code members, bridging nodes, and the self-generating closed loop. Furthermore, this paper reveals the continuous parameterization structure of the spectrum: the cosine values of all spectral angles can be uniformly expressed as c = 1 / phi^n, with corresponding angles theta = 2 * arccos(1 / phi^n). The precise n coordinates of the seven genetic code angles on the continuous curve are derived one by one. Among them, 72° and 144° are symmetric around 120°, and the characteristic point of 90°, n = ln(2) / (2 * ln(phi)), is the exact intersection of two-fold and five-fold symmetries. The constant ln(2) / ln(phi) repeatedly appears as the "core invariant" of the spectrum, unifying the discrete genetic code and continuous parameterization within the same mathematical framework.