A Unified Numerically Stable Framework for Cubic Equations: Classification by Injectivity (One-to-One) and Elimination of Catastrophic Cancellation

\begin{abstract}The general sextic equation is not solvable by radicals, a consequence of the Abel-Ruffini theorem and Galois theory. Classical solvable families, such as those obtained via Euler or Chebyshev substitutions, rely heavily on a reflective symmetry of the form \(z = x + a/x\), which imposes a palindromic structure on the polynomial coefficients. In this paper, we introduce a novel and comprehensive framework for generating solvable sextics that fundamentally break this classical symmetry. Our method is based on substituting a rational fraction with a cubic numerator into an auxiliary quadratic equation. Two principal families are derived: Family I, with a linear denominator, which automatically produces depressed sextics (lacking the \(x^5\) term); and Family II, with a quadratic denominator, which yields sextics with a vanishing linear term (\(x\)). For each family, we derive explicit parameterizations, exact coefficient extraction formulas, and rigorous membership conditions that any sextic must satisfy to belong to these families. Furthermore, we systematically classify seven distinct subfamilies within each framework by imposing specific constraints on the parameters, yielding highly specialized sextic forms with simplified coefficient relations. We prove the algebraic duality between the two families under the reciprocal transformation \(x \mapsto 1/x\), confirming that they are manifestations of a single underlying structure. A unified decision tree and solution algorithm are provided, reducing any qualifying sextic to a quadratic resolvent and two cubic equations, thus guaranteeing exact radical solutions. Numerical validation is provided through fourteen concrete examples, one for each subfamily, with full verification of membership conditions and detailed radical solutions for selected cases. A comparison with Traub's iterative method demonstrates the advantages of our closed-form approach in terms of accuracy, determinism, and the absence of convergence issues. This framework not only generalizes classical solvable cases but opens new avenues for capturing previously unknown solvable sextics, with potential applications in symbolic computation, computer algebra, and theoretical algebra.\end{abstract} \begin{keywords}Sextic equation, Solvable by radicals, Rational substitution, Depressed sextic, Non-classical symmetry, Traub method.\end{keywords}