Pull-Back Discriminants and Transversality Indices: Regime Boundary Jumps on Policy Manifolds

Differential approaches to general equilibrium and related equilibrium systems represent equilibria as zeros of a residual equation F(p, x) = 0 and analyze the equilibrium correspondence through regularity and bifurcation properties on the primitive environment space X. In policy analysis, however, primitives are implemented by institutional instruments: a design parameter θ ranges over a policy manifold Θ and induces x = Φ(θ) through an implementation map Φ : Θ → X shaped by legal, administrative, or engineering constraints. This framework transfers the analysis of regime boundaries from the primitive space X to the policy-side geometry of Θ. We develop an ideal-theoretic formulation of policy-side regime boundaries by pulling back the discriminant locus D ⊂ X to the pull-back boundary BΘ = Φ−1(D) ⊂ Θ, thereby constructing a boundary atlas explicitly decomposed by mechanism via pull-back of discriminant ideals and their componentwise decompositions. To quantify whether boundary-jump classifications along policy paths are structurally robust transversal hits or perturbation-sensitive (grazing), we introduce a componentwise transversality index TIk ∈ [0, 1] defined via normal spaces to each boundary component, in a generator-independent manner. Our main results are: (i) a pull-back discriminant theorem transporting mechanism-specific boundary decompositions from X to Θ; (ii) a transversalhit stability theorem establishing robustness of boundary-jump classification under positive transversality; and (iii) a grazing instability theorem showing that near-tangential contacts yield intrinsically fragile jump classification. Two economic illustrations complement the theory: a motivating generalequilibrium demonstration where distinct mechanisms can share the same rank-loss signature, and a minimal two-cell economy with posted-price instruments in which Φ and two boundary mechanisms are explicit and TI heatmaps visualise where boundary separation is reliable or fragile. MSC (2020): 91B50; 58A15; 58K40; 13P10; 14Q20. JEL classification: C60; C61; D51; D78.