Height–plane contact along edges, corners, and cusps

We study pairs $\tilde{X}=(X,S)$ 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 $\mathrm{D}\mathrm{e}\mathrm{r}(-\log (g,b_s))$. 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 $\le 2$, with respect to the $\mathcal{R}(\tilde{X})$-equivalence relation. Second, for height functions $h_v(w)=⟨w,v⟩$ 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 $P$– ${\mathcal{R}}^{+}(\tilde{X})$–equivalence.