# Exact figure mathematics

Independent CC0 diagram. Example only; the general proof retains arbitrary h and singular/nonisolated loci.

h(z,y)=z² on C²; Y={z=0}; π=y; N={y=0}; dim_C Y=1. For the perverse P=K_{C²}[2], the correctly normalized restriction is P|N[-1]=K_N[1]. In T*_{C}N, its microsupport is the zero section ξ=0. The original graph ξ=2z meets it at (0,0); the perturbation z²+(1/8)z meets it at z=-1/16. Panel one is precisely the real section Im z=Im ξ=0, not the whole complex cotangent surface.

On N write z=x+it. The perturbed real height is u_a=x²-t²+x/8=(x+1/16)²-t²-1/256. Its unique critical point has one negative direction t and one positive direction x. Choose |z|<=1/2, ε=1/64, η=1/16, with fixed sides |Im(z²)|=|2xt|<=η. The displayed blue/green contours are sampled exact levels u_a=+ε and u_a=-ε in that fixed cap; grey shading is the intervening band. The critical value -1/256 lies strictly inside it. At a radial-face point in this band, |Re(z²)|<=ε+a/2=5/64 and |Im(z²)|<=1/16, so |z²|<=sqrt(41)/64<1/4; that contradicts |z|=1/2. Thus this example has no radial-face point in the band, and introduces no hidden boundary jump.

The actual finite jump of K_N[1] is its point-normal object in ordinary degree zero: M_{z²}(K_N[1])=K. The dimension is the integer 1 over every field. Unshifted Φ_{z²}K_N[1]=K[1]. Ambient unshifted Φ_h P=K_Y[2], so the transverse normalization is essential. The ambient critical locus is the entire complex line Y, not an isolated point.

Proof locators: [visible normal slicing](../general-critical-support.html#GS2), [the finite visible Morse filtration](../general-critical-support.html#IV1), and [the critical-support argument](../general-critical-support.html#HC2). The complete lesson links its scholarly prerequisites.