Zero-Divisor-Free Coordinate Cells in the Cayley-Dickson Tower

A classification, uniform in n, of the coordinate cells of the Cayley–Dickson algebra A_n that contain no nonzero zero divisor. The free cells are exactly the subspaces lying inside the octonion block or inside a vertical quaternion block; they number 2^(n+1) − n + 2 and form an order ideal generated by 2^n − n − 4 maximal blocks. Freeness is absolute under the tower inclusions, the basis-permuting symmetry group is GL(3,2) at every level, the free cells fall into 2^(n−1) − n + 2 orbits, and the canonical section of projective support closure admits no equivariant refinement, with exactly two obstructed orbits supported on the octonion block. All results are machine-verified for n = 4, 5, 6 by an included script.