# Finite covers and normal circles

The formal chapter gives the full relative mathematical proof, with exact linked preceding Morse, critical-value and homological Thom constructions. The learner chapter has four original examples and eight fully solved exercises totaling 100 points.

Run with Python 3, NumPy, mpmath and Matplotlib:

    python -B -X utf8 projective-cover-check-and-figures.py

The code regenerates the two PNG/SVG figures and exact geometry data. The first shows the base loop and the z-coordinate projection of both lifted arcs; their boundaries cancel. It is not a drawing of all four ambient real coordinates. The second shows the normal-first angular parameter square of an exact torus in C^2, with its identified edges, ordered tangent directions and pullback period. It is a parameter diagram rather than an ambient rendering.

The 563 finite identities supplement the full proof. They check branch equations, endpoint cancellation, period factors, coefficient invertibility, normal-disc/circle permutation signs and Euler smoothness on a repeated level. They do not prove a general theorem by testing samples.

Original material is CC0 1.0. The DejaVu font retains its own terms. Linked historical works are not reproduced.