Non-Crystallographic Orders of Integral Lorentzian Lattices and Where They Are Realized (Geometry of the vacuum, floor 0)

Finite-order integral isometries of a lattice of signature (3,1) have order in {1,2,3,4,6}, so orders 5, 8 and 12 are excluded there. This paper asks where those orders are realized instead, and answers on two levels. First, one rank up: for each n in {5,8,12} the cyclotomic carrier M_n -- the rank-4 lattice Z^4 with a chosen companion(Phi_n)-invariant positive definite form -- gives M_n + <-1>, an integral lattice of signature (4,1) carrying an isometry of order n, so the forbidden order sits exactly one rank above in the same Lorentzian tower rather than in some unrelated object. Second, inside rank 4: the invariant real 2-planes of companion(Phi_n) contain no nonzero lattice vector, and this is forced rather than arranged, since Phi_n is irreducible of degree equal to the dimension, which makes Q^4 a simple Q[g]-module; the irreducibility itself is proved here rather than cited. Without a window the projection of M_n onto an invariant plane is dense, so a discrete point set costs one external input; with a bounded window the resulting point set is proved aperiodic. The rotational symmetry of the shadow is measured at orders 10, 8 and 12 with the search scope named in each row, and the doubling 5 -> 10 is explained exactly: it happens if and only if -I is not a power of g. The same criterion separates three distinct groups that a single word had conflated -- the rotation group , the Galois dihedral group arising from complex conjugation of the cyclotomic field, and their join -- which at n = 5 are C_10 and D_5, of equal order and neither contained in the other. The space of invariant forms is two-dimensional, so the carrier is pinned by an explicit Gram matrix and each lattice identification (A_4*, Z^4, A_2+A_2) is closed by an explicit change of basis in GL_4(Z) rather than by matching invariants. A Vinberg cell for I_(3,1) with diagram [3,4,4] is included, where three-dimensional point groups sit at the finite vertices and two-dimensional wallpaper crystallography at the cusp. No physical reading of any object is made anywhere in the work.