Complex Nilpotent Residues in Synthetic Differential Geometry

We isolate a simple first-order construction in a complexified setting for synthetic differential geometry. Let R be a smooth real-lineobject with nilpotent first-order infinitesimals, let C = R[i], and let D_C = {ε ∈ C : ε² = 0} be the object of complex first-orderinfinitesimals. For a complex-valued map f and an infinitesimal displacement ε ∈ D_C, we define the nilpotent residue Res_ε(f; z) = f(z + ε) − f(z). For maps satisfying the complex first-order law f(z + ε) = f(z) + f'(z) ε, the residue is a phase-bearing first-order quantity: it mayhave a complex coefficient while remaining square-zero. We record the resulting linearity, product, chain, and conjugation rules. Thenote formalizes a residue operator and its holomorphic behavior inside established SDG-style assumptions. Contents: definition of the residue operator over D_C; the D_C-holomorphic first-order law; nilpotence of holomorphic residues;linearity, product, and chain rules with exact identities; worked examples (squares, powers, exponential, rank-one complex residue);holomorphic versus real-smooth first-order behavior; complex-linearity criterion; conjugation as anti-linear example; relation tocomplex dual numbers. MSC 2020: 58A03, 18F10, 30G30.