Compositional U(1)-Residue in Complexified Synthetic Differential Geometry

We show that, in complexified synthetic differential geometry (SDG) with first-order nilpotent infinitesimals, a microlinear residue homomorphism from the based closed-traversal monoid of a microlinear pointed substrate (T, e_T) with tangent module a microlinear C-module V, satisfying compositionality, microlinear continuity, invertibility, nontriviality, compact closed-loop holonomy, second-order commutator-triviality, and infinitesimal generation of the residue Lie algebra, factors through a microlinear Lie group whose minimal nontrivial compact connected projected factor is U(1). The argument follows the standard classification of connected abelian Lie groups; the abelianness step is supplied by the second-order commutator-triviality hypothesis (vi) under the second-order Weil algebra W_C^[2] = C[ε_1, ε_2]/(ε_1^2, ε_2^2) (mixed term ε_1·ε_2 retained), together with the infinitesimal-generation hypothesis (vii) supplying the passage from infinitesimal traversal directions to the full residue Lie algebra. Both are stated as genuine additional hypotheses, not derived from substrate-tangent structure alone; this is the honest scope of the conditional theorem. This paper supplies a candidate theorem-grade route to the analogous minimality statement of Theorem U1minimal in §7.2 of Complementarity Theory: A Foundational Core for Finite Commit Geometry (CT 4.0), replacing the abelian-commutativity hypothesis on the residue role with a structurally narrower pair of second-order conditions under the stated substrate. The conclusion is correspondingly weakened from "image subgroup is U(1)" to "minimal nontrivial compact connected projected factor is U(1)." The construction sits in the neighborhood of the complex nilpotent residue calculus of the companion note (Zenodo 10.5281/zenodo.20208862), and admits an analogous post-geometric treatment of one-dimensional rapidity sectors developed in a sister paper deposited simultaneously. The paper is the v1.0 deposit-ready form after five iteration rounds: structural repair per peer review, mathematical hygiene per first scoped review (notational separation of Weil algebra / probing object / tangent module; based closed-traversal monoid; second-order infinitesimal object D_C^[2]), theorem-wording precision per second scoped review (condition (v) strengthened to require nontrivial identity component; projected-factor wording; monoid-with-reversal phrasing for the traversal space), and editorial cleanup with honest SDG-citation acknowledgment.