Dissipative Numeral System, Paper VII: Structural Differential Operators: A Cost-Aware Extension of Differential Calculus via Dissipative Numerical Structures

Classical differential calculus rests on an implicit assumption: variation occurs within a frictionless, cost-uniform continuum where the derivative operator acts uniformly on the monomial basis. This paper introduces Structural Differential Equations (SDEs) as a formal topological extension of standard calculus via the Dissipative Numeral System (DNS) framework. By replacing the classical derivative with the DNS deconstruction operator ∂(en)=Cnen−1∂(en)=Cnen−1, we prove that classical calculus is strictly the non-dissipative, uniform-cost limit (Cn=n,0T→0Cn=n,0T→0) of a broader deconstructive dynamics (Theorem 1). Furthermore, we demonstrate that continuously differentiable strict Lyapunov functions V(x)V(x) admit a constructive discrete realization as DNS cost landscapes (Theorem 2), and that a class of inaccessible dynamical boundaries can be canonically represented as structured zeros (0T0T) — the historical artifacts of saturated structural construction (Theorem 3). The governing law ∂S(t)=F(S(t),C)+RT(S)∂S(t)=F(S(t),C)+RT(S) unifies smooth ordinary evolution, distributed spatial fields, and discrete reset dynamics within a single algebraic framework. This work establishes that differentiation is a special case of structural deconstruction, shifting the mathematical ontology from passive variation to active, cost-bounded structural reduction.