# Exact mathematical coordinates

Independent CC0 diagrams for [the polar-section and integral-equation proofs](../polar-sections-and-lipschitz-algebras.html). The pictures show exact examples of the general mechanisms; the full proofs remain alongside them.

Left: H={z3=z4=0} in C4 is reduced pure of complex dimension d=2 in N=4, meeting the original N>=d+2 requirement. Its conormal fibre F={xi1=xi2=0} is P1 in P3. In the chart xi3=1, the displayed real section has Im xi1=Im xi2=Im xi4=0; F is (0,0,xi4), and L2={xi4=0} is the displayed plane. Their entire complex intersection is [0:0:1:0]. The extra point of F at infinity is not displayed. For k=0, ell=N-1-d+k=1, p0(z)=(z1,z2,z3), rank(p0|TH)=2<3, so P0=H. For k=1, L1={xi3=xi4=0} lies outside this affine chart and is disjoint from F: combining its equations with F would make all four homogeneous coordinates zero. The projection p1(z)=(z1,z2) has full rank 2 on H, so P1 is empty. dim F=1 is respectively >=ell and <ell. This is the exact fibre threshold, not a Euclidean intersection estimate for arbitrary planes.

Right: I=(x^2,y^2), F=xy, with C=1 in |F|<=max(|x|^2,|y|^2). The actual projective graph [x^2:y^2:xy] has exceptional image the conic v3^2=v1 v2 in P2. Its tautological affine cone is w^2=u1u2. The plotted real section is parametrized by (u1,u2,w)=(r cos^2 theta,r sin^2 theta,r cos theta sin theta), theta in [0,pi], r in [-1,1]. The finite r interval is a display window, not a bound on the full cone. The vertical projective point [0:0:1] violates its equation and is excluded; its zero vector remains the cone vertex. The homogeneous equation has unit coefficient 1 on w^2. Substituting (u1,u2,w)=(x^2,y^2,xy) gives F^2-g1g2=0, with a1=0 in I and a2=-g1g2 in I^2. It is an actual ideal integral equation, not merely an equation with arbitrary coefficients.

General proof locators: PS1/PS2 actual compact-fibre generic flag and jump-image density; PS3 proper polar image; PS4 full fibre threshold. BI1 missing projective vertex; BI2 proper tautological-cone image; BI3 homogeneous Taylor extraction with the exact I^j coefficient powers. Human antecedents retained in those proof bodies: Teissier, Varietes polaires II, IV.4.1.1 and IV.6.1.1(a), pp.432–433 and 451; Demailly, Complex Analytic and Differential Geometry, II.5.4/II.5.9, II.6.2 and II.8.8, for actual closures, compact affine finiteness, one-equation purity and general proper analytic images. No protected source prose or figure is copied.