Height–plane contact along edges, corners, and cusps

We study pairs $ \widetilde X = (X, S) $ which consist of regular surfaces $ X\subset{\mathbb{R}}^3 $ endowed with a distinguished boundary $ S $ given by $ y^2-x^s = 0 $ ($ s = 1, 2, 3 $: edge, corner, cusp). First, we give an explicit description of the logarithmic vector fields tangent to $ (g, b_s) $, $ g = z-f(x, y) $. In particular, the five fields $ E, L, G_g, G_x, G_y $ generate $ \mathrm{Der}(-\log(g, b_s)) $. This yields a concrete Kodaira–Spencer calculus and a relative $ 2 $-determinacy theorem (with a single parabolic exception). Then, we classify submersions and obtain normal forms and mini versal unfoldings for submersion germs in codimension $ \le2 $, with respect to the $ \mathcal R(\widetilde X) $-equivalence relation. Second, for height functions $ h_v(w) = \langle w, v\rangle $ we obtain sharp linear conditions on $ v\in S^2 $ that characterize the $ A_k $–contact along each boundary type. In particular, in the cuspidal case, the $ A_3/A_5 $ transition is governed by singular torsion. The resulting direction discriminant $ \mathcal{D}_s\subset S^2 $ corresponds, under the direction–parameter normalization, to a hyperplane arrangement of linear discriminant loci in the mini versal parameter spaces. Both the ambient discriminant $ \mathfrak{D}_X $ and the boundary contact discriminants are invariant under $ P $–$ \mathcal R^{+}(\widetilde X) $–equivalence.