Geometry of Equilibrium Regimes I: Pull-Back Policy Varieties from Multi-Channel Systems: Boundary Ideals, Overlap Thickness and Identification Wedges

We develop an applied-algebraic-geometry framework for policy boundaries generated by multi-channel Karush–Kuhn–Tucker (KKT) systems and use it to formalize a central applied phenomenon: identification wedges—parameter regions where distinct institutional mechanisms produce observationally similar outcomes until a regime boundary is approached. Given a parametric family of constrained optimization problems, the KKT equations define an algebraic set in the joint space of policy parameters, primal variables, and multipliers. Projecting (eliminating) the non-policy variables yields a pull-back policy variety in parameter space. The central objects of this paper are (i) the boundary ideal encoding regime changes (active-set switching), (ii) the Jacobian ideal encoding criticality of the projection, (iii) transversality indices that quantify how policy paths cross boundary components, and (iv) overlap thickness invariants (local algebra lengths) that quantify non-transversal multi-component intersections (boundary crowding). In dimension two, boundary germs admit an ADE-type refinement; we further show that, under mild hypotheses, overlap multiplicities determine the Milnor number and hence certify ADE type, making the identification-wedge geometry computably testable. We conclude by discussing practical implications for policy design and economics.