Einstein–Maxwell from One Ordered Response: An Exact Local SL(3,\( \mathbb C \)) Construction

Can geometry, electromagnetic propagation, and electromagnetic backreaction be derivatives of one function before a metric or a matter action is introduced? We exhibit a local affirmative construction. The only dynamical field in the parent action is an \( SL(3,\mathbb C) \) connection, and a degree-one homogeneous adjoint-invariant phase \( \Phi(X) \) of curvature wedge products supplies its constitutive response \( B_I=2\Phi_{,IJ}F^J \). Curvature-derived spectral projectors define an analytic parent phase near an ordered 3+4+1 response. Its exact, consistently embedded neutral sector admits an algebraic stationary representation that enforces the simplicity constraints on the full gravitational response at finite electromagnetic amplitude within the regular local neutral branch. The resulting triplet reconstructs the geometry; the central derivative gives Maxwell propagation in that geometry; and the gravitational derivative gives its Maxwell source, controlled by the same parent coefficient. On the specified Lorentzian real branch, the neutral equations are exactly the Einstein–Maxwell–\( \Lambda \) electrovacuum equations, not an amplitude truncation or an inference from quadratic matching. We prove the local invariant completion, compute its ordered connection two-jet, and give explicit curvature and coupling conventions. The construction realizes one classical ordered branch of a single phase; its domain is the regular analytic neighborhood of that branch.