A Generalized Fundamental Theorem of Algebra and Vieta's Formulas Over Operator Algebras: A Unified Framework for Higher-Degree Forms

This paper establishes a comprehensive and rigorous framework extending polynomial theory from Clifford algebras to operator algebras, specifically finite C^*-algebras with a faithful normal tracial state, and more generally to n-ary operator algebras associated with fully symmetric n-linear forms. By introducing ellipticity conditions and the concept of regular solutions, we prove that the regular solution set of a monic polynomial with real coefficients decomposes into discrete scalar roots and a finite number of smooth manifolds, each of which is a homogeneous space carrying a unique invariant probability state. Using the path ordering operator to handle noncommutativity, we derive a generalized Vieta formula. We provide complete, self-contained proofs of all theorems, including: a rigorous characterization of regular solutions via minimal polynomials (Theorem thm: minimal-polynomial) ; a complete proof of the Faddeev-LeVerrier algorithm with termination criterion (Theorem thm: faddeev-leverrier) ; a proof that the solution manifolds are homogeneous spaces U (d) /U (d/2) U (d/2) (Theorem thm: homogeneous) ; a proof of the well-definedness and spectral decomposition of the path ordering operator (Theorems thm: path-ordering-well-defined, thm: path-ordering-spectral) ; a proof of its uniqueness (Theorem thm: path-ordering-unique) ; and a complete derivation of the generalized Vieta formula (Theorem thm: vieta-final). We extend the theory to non-compact cases using L² Euler characteristics (Theorem thm: noncompact-vieta), establish connections with Shirokov's noncommutative Vieta theorem (Theorem thm: shirokov-unified), and generalize to n-ary operator algebras (Theorems thm: nary-structure, thm: nary-fundamental, thm: nary-vieta-final). We provide a complete classification of n-ary operator algebras via discriminants, Hasse invariants, and n-ary signatures (Theorem thm: classification-nary-clifford). We construct the quantization of solution manifolds and prove the quantum generalized Vieta formula (Theorem thm: quantum-vieta). We establish the arithmetic-geometric interpretation as conjectures connecting to Frobenius traces and Hasse-Weil L-functions (Conjectures conj: arithmetic-geometric-weak, conj: arithmetic-geometric-strong). We prove a conditional version of the connection with the Langlands program (Theorem thm: langlands). Finally, we construct the Absolute Unifier, a universal object unifying all n-ary operator algebras and their polynomial theories, and prove its existence and uniqueness (Theorem thm: absolute-unifier), with further structures presented as conjectures (Conjectures conj: unifier-ring--conj: unifier-mtheory). The results unify polynomial theory over quaternions, octonions, Clifford algebras, and operator algebras, providing new tools for noncommutative algebra, differential geometry, topology, number theory, and mathematical physics.