Nilpotent Operational Framework for LTI Systems with Polynomial–Exponential Forcing: Exact Finite-Dimensional Closure and Applications to RLC Networks

We develop a unified algebraic framework for the exact solution of linear time-invariant differential systems with forcing of the form f(t) = P(t)eᵅᵗ, where P(t) is a vector of polynomials of degree at most n. The method rests on the nilpotency of the derivative operator D = d/dt on the space Vₙ = {p(t) ∈ ℝ[t] : deg p ≤ n}. Since Dⁿ⁺¹ = 0 exactly on Vₙ, the Neumann series associated with the inverse operator p(D)⁻¹ terminates after finitely many steps, producing the exact particular solution without recourse to the Laplace transform. The aim of the present work is not to replace classical methods, but to identify the finite algebraic mechanism that explains why such methods produce closed-form solutions in this sector of forcings. We establish the SON Closure Theorem for LTI systems and verify the diagnostic conditions D1 and D3. As canonical physical applications, we derive exact particular responses for damped oscillators, coupled resonators, and RLC networks. The algebraic cost is O(n²d² + d³) in a direct implementation, reducible to O(M(n)d² + d³) with fast polynomial multiplication. The algebraic core of the framework has been formally verified in Lean 4.29.1 with Mathlib (https://doi.org/10.5281/zenodo.20113599). MSC 2020: Primary 34A30, 68Q25. Secondary 47A60, 93C05, 94C05.