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

Let $\mathcal{N}(k,n)$ be the largest size of a family of affine $k$-planes in $\R^n$ 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 $\mathcal{N}(k,n)\leqslant 2^{\,2^{\,3(k+1)(n-k)}}$. For configurations whose direction sums have dimension at most $k+1$, we prove the exact reduction $\mathcal{N}_{\leqslant k+1}(k,n)=\mathcal{N}(1,n-k+1)$. Every codimension-two equidistant configuration splits off a common linear factor, giving $\mathcal{N}(n-2,n)=\mathcal{N}(1,3)$. 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 $S(c)=\mathcal{N}(c-1,2c-1)$ supermultiplicative. In particular, $S(c)\geqslant 2^{c\bmod 2}7^{\lfloor c/2\rfloor}$. The structural results and constructions do not depend on the exact value of $\mathcal{N}(1,3)$.