Unified Nilpotent Operational Framework: Foundations, Algebraic Exactness, and Complexity

A unified algebraic framework is developed to study nilpotency as a structural mechanism for exactness in operational, combinatorial, and computational problems.

The central object is the Nilpotent Operational System (SON), formalized as a tuple (R, N, m, M_R), where R is a C-algebra, N satisfies N^{m+1}=0, and M_R(m) is the arithmetic cost of a product in R.

The basic result is the Exact Termination Lemma: every formal series evaluated at N collapses to an exact finite sum of m+1 terms.

Three complexity regimes are obtained: truncated series (quasi-linear bound via Newton iteration), nilpotent operators (linear bound via Horner evaluation), and incidence algebras (quasi-quadratic bound).

Applications include classical and free cumulants, Appell sequences, orthogonal polynomials, Stirling numbers, Möbius inversion, Witt vectors, and local holonomic functions.

In all cases except Stirling numbers, strict complexity improvements over classical algorithms are obtained.

For classical cumulants, equivalence with polynomial multiplication is achieved: L(C_n)=Theta(M(n)).

For free cumulants, the complete equivalence T_{FC}(n)=Theta(M(n)) is established via the Voiculescu functional equation and compositional reversal.