Algebraic Symmetries and Modular Properties of Ramanujan Polynomials in the Cube Decomposition Problem

This paper investigates hidden algebraic symmetries in Srinivasa Ramanujan's parametric system of Diophantine equations for sums of three cubes. We previously showed that this system possesses a telescoping property, allowing for the construction of deterministic decompositions of an integer cube into an arbitrarily large number of positive cubes. In this work, we prove the fundamental antisymmetry identity Z(a, b - a^3) = -Y(a, b), which connects the two main generating polynomials of the system. This identity explains the fractal structure of the generated sequences. In addition, we analyze the modular properties of the resulting elements (their digital roots) and rigorously prove the empirically observed proportions of the distribution of residues modulo 9, which lead to an "algebraic sieve" effect and an anomalously high density of prime numbers. Finally, we investigate the system for algebraic rigidity and demonstrate the uniqueness of the parameterization found by Ramanujan in the context of the search for generalized "meta-generators".