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.