Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products

Let N(k,n) be the largest size of a family of affine k-planes in ⁿ at pairwise separation distance one. We develop a unified treatment of upper bounds, dimension reduction, and product constructions. Orthogonal-projection coordinates give a rank-stratified distance identity; a Veronese lifting converts its polynomial consequences into simultaneous total isotropy. Connected Nash decomposition and an analytic diagonal argument replace the linear-independence step in Höfer's three-dimensional method. An explicit cylindrical decomposition estimate yields N(k,n) 2^\,2\,3(k+1)(n-k). For configurations whose direction sums have dimension at most $k+1$, we prove the exact reduction Nₖ₊₁(k,n)=N(1,n-k+1). Every codimension-two equidistant configuration splits off a common linear factor, giving N(n-2,n)=N(1,3). We also determine the sharp bound $2(n-k)$ when all directions lie in a fixed $(k+1)$-dimensional subspace. A transverse product construction gives multiplicative inequalities and makes S(c)=N(c-1,2c-1) supermultiplicative. In particular, S(c) 2c 27c/2. The structural results and constructions do not depend on the exact value of N(1,3).