Unified Nilpotent Operational Framework: Foundations, Algebraic Exactness and Complexity

We develop the Unified Nilpotent Operational System S=(R,N,m,M_R), a framework that identifies nilpotent structure in truncated power series rings V_n=C[[ε]]/(εⁿ⁺¹) and exploits it to achieve optimal complexity. The framework unifies the computation of classical cumulants, free cumulants, Appell sequences, classical orthogonal polynomials, Stirling numbers of the second kind, Möbius inversion in incidence algebras, and Witt vectors. In all cases except Stirling numbers — where the SON provides algebraic exactness but not a speed improvement — strict complexity improvements are obtained over the classical algorithms explicitly analyzed. For classical cumulants, an equivalence with polynomial multiplication is established: L(C_n)=Θ(M(n)), where M(n) is the cost of multiplying two polynomials of degree n. For free cumulants, a complete equivalence TFC(n)=Θ(M(n)) is established through Theorem 2 bis, closing the structural classification in both regimes. The Diagnostic Theorem (conditions D1–D3) gives necessary and sufficient conditions for exact SON closure. The framework is grounded in the algebraic structure of the quotient ring C[ε]/(εⁿ⁺¹) and Newton's iteration with FFT-based polynomial arithmetic.