Dissipative Numeral System, Paper V: Structural Algebra — The Multidimensional Structural State Space

The Dissipative Numeral System (DNS) series has progressively established that numbers can carry structural properties beyond quantity: structural distinction between zeros (Paper 0), irreversible historical traces from decomposition (Paper II), cost-induced geometry (Paper III), and dynamic structural orbits (Paper IV). However, none of these works defined what a DNS number is algebraically. This paper fills that gap. We define the structural number as a finite linear combination over a structural basis: X = Σᵢ₌₀ⁿ xᵢ eᵢ where each eᵢ is a structural basis direction induced by the system's generation mechanism, and xᵢ ∈ ℝ is the coefficient along that direction. Classical arithmetic is recovered as the special case where only the e₀ direction is used. We introduce a family of four structural decomposition operators ∂_struct, ∂_full, ∂_unit, ∂_layer with a three-dimensional parameterization (handling × scope × cost reference), and identify ∂_FLR = ∂_(full, layer, rel) as the unique operator that preserves structural form under decomposition. We construct the DNS exponential function: E_DNS(λ) = Σₙ₌₀∞ λⁿ (Πₖ₌₁ⁿ 1/Cₖ) eₙ which satisfies ∂_FLR(E_DNS) = λ E_DNS. When Cₖ = k and λ = 1, this recovers the classical e series coefficients, demonstrating that the classical exponential is a special projection of the DNS structural exponential. This work establishes the algebraic foundations of DNS, showing that classical arithmetic is the degenerate limit (n = 0) of a richer multidimensional structural state space.