The Discrepancy: Excess Intersection 9
The 3x3 identity matrix we just saw has another explanation:

Every circle contains two `circular' (imaginary) points at infinity
Similarly, every sphere contains an imaginary `circular' conic at infinity,
and lines at infinity tangent to this conic are tangent to every sphere.

When k=1, there are 2 lines at infinity tangent to this conic that meet the last line, and the excess component has multiplicity two, which expains the discrepancy: 12 = 16 - 4.
When k=0, the case of lines tangent to 4 spheres, the solution of Macdonald, Pach, and Theobald used a choice of coordinates which excluded the lines at infinity and gave a Bézout number of 12.
These pictures complete the proof.
Their solution generalizes to higher dimensions:

Theorem. (S. & Theobald)
    There are 3 2n-1 lines tangent to 2n-2 general spheres in Rn