Geometric Structures from Bézout Decompositions of Semiprimes

For a semiprime N=sp, the normalized CRT–Bézout map assigns each unit modulo N a unique pair (m,n) in the rectangle {1,…,p−1}×{1,…,s−1}. The coordinates mn, Δ=ms+np, and δ=ms−np satisfy Δ2−δ2=4Nmn and give the geometric diagrams studied here. We describe the bounded affine slices Am+Bn=λ, including their point counts, complement symmetry, vertices, and spacings. We also write modular squaring in these coordinates: the two components evolve independently, while the nonempty fibers are orbits of a four-element Klein group. The complement symmetry of the slices is compatible with these dynamics. Finally, we determine the two admissible points with |δ|=1 and prove an exact count for the points in a central strip |δ|<T. These results describe the geometry attached to a known factorization; they do not give a new factorization algorithm.