A Structural Resolvent-Based Reduction of the General Sextic: Explicit Multi-Valued Algebraic Construction via Triple Sums

This paper presents a complete structural reduction of the general sextic equation to a quintic resolvent using triple sums of roots. Using representation theory of the symmetric group S6, the decomposition of the 20 triple sums into two irreducible components of dimension 5 is rigorously justified, yielding a quintic resolvent polynomial. Explicit degeneracy conditions are derived under which the resolvent collapses, leading to radical solutions. The main result is an explicit multi-valued algebraic construction for at least one root of the general sextic: x = (S + y0)/6 + t0, where y0 is a root of the resolvent quintic (understood as a multi-valued algebraic quantity) and t0 solves a cubic equation. The construction is multi-valued, consistent with Galois theory. An exact degenerate example is provided. This work complements the author's previous numerical framework for higher-degree equations and provides a theoretical foundation for the sextic case. The paper includes: - Reduction to Joubert canonical form - Representation-theoretic justification of the quintic resolvent - Rigorous derivation of the constant term c0 and the degeneracy condition 4α³ - β² - 4γ² = 0 - Main structural theorem and generic vs degenerate classification - Exact degenerate example with radical solutions - Complete references to classical works (Joubert, Klein) and the author's previous papers on cubic, quartic, quintic, sextic, octic, and decic equations.