A Unified Transcendental Framework for Polynomial Equations of Degree n ≥ 5: The Դₙ Operators

We present a complete computational and theoretical framework for solving polynomial equations of arbitrary degree n ≥ 5 through the introduction of the Դₙ family of transcendental operations. While the Abel-Ruffini theorem establishes that no general algebraic solution exists for polynomials of degree ≥ 5 using radicals, we demonstrate that defining Դₙ as a new transcendental operation—computable via convergent numerical methods—provides a practical and theoretically sound resolution. We establish the mathematical properties of Դ₅ (pentadic root) and Դ₆ (sextic root), prove convergence of computational algorithms, provide verified numerical tables, and position these within a unified framework Դₙ applicable to all polynomial degrees. This approach parallels the historical development of logarithmic and trigonometric functions as transcendental operations that extend algebraic computation.