Zero Closure Was Four-Dimensional: An Interpretive Note on the Geometry and Algebra of Sum xₙ² = 0

An interpretive note. No new theorems are claimed. Known results are rearranged under the single closure condition Sum xₙ² = 0 in order to fix what is determined by it and what is not. Claim 0 shows that central projection reduces the many-body zero-closure problem to a single quadratic equality constraint, and that this reduction survives complexification: writing z = q + ip, zero closure is equivalent to |q|² = |p|² together with q. p = 0, so at fixed C the solution set is the Stiefel manifold of orthonormal 2-frames. Claim 2B states the title of the paper: the basic representation is the four-dimensional (r, t, R, Q), on which zero closure imposes the light cone r² - t² - R² - Q² = 0. Claims 4 to 9 collect the geometric statements (parallelotopes satisfy the signed closure, the inertia ellipsoid saturates at multipole l <= 2, complex numbers are not mandatory). Claims 10 to 18 report measurements on the numerical model of the series at resolution N = 16 = 2⁴ with T = 40000 steps: the transition at tau about 9000, the 15 principal axes, the conserved signed trace, and the diffusion of principal-axis orientation together with the quadratic forms that survive it. Undischarged assumptions are stated explicitly throughout.