A Parametric Structural Representation of the Sextic Equation via Radical Maps and Discriminant Factors

This paper presents a parametric structural representation of the reduced sextic equation x^6 + x + t = 0 using the symmetric ansatz x = u+v, p = uv, q = u+v. A discriminant factor Φ(q,t) of degree 7 arises from the elimination of p, governing the branching behavior of the radical map. The construction yields a triple-layer decomposition: (1) a polynomial Φ(q,t), (2) a multi-valued radical map x(q), and (3) an admissibility condition F(q)=0 selecting parameters consistent with the original equation. The framework does not claim to solve the sextic in radicals; rather, it reorganizes the solution space into a multi-sheeted algebraic correspondence compatible with Galois theory. Explicit numerical examples and degeneracy conditions are provided. The work includes a comparison with the quintic case and discusses parity phenomena in the Φ-hierarchy.