The Symmetry Centre of an Integral Lorentzian Lattice: Cell, Breaking, and Two Roots (Geometry of the vacuum, floor 0)

We describe the symmetry centre of an integral Lorentzian lattice of rank 4 and show that three previously unrelated features of it reduce to two roots, not to one. The fundamental domain of the group generated by the mirrors of I_(3,1) is a finite-volume hyperbolic simplex with Coxeter diagram [3,4,4]; three of its four vertices carry three-dimensional crystallographic point groups and the fourth, a cusp, carries two-dimensional crystallography, so both floors of crystallography sit inside one cell. The axes of orders 5, 8 and 12, forbidden in this arithmetic, live one level up on classical root lattices and their duals (A_4*, Z^4, A_2+A_2, with automorphism groups of orders 240, 384, 288) and descend only as a non-translational shadow of orders 10, 8, 12; the irrationality of the directions is an irreducibility theorem rather than a choice of parameter, the order of the shadow is exact by Kronecker's theorem with no search bound, and discreteness costs exactly one external input, a window, without which the projection is dense and with which the point set is proved aperiodic. The change of arena is itself forced: the invariant forms of companion(Phi_n) realise only the signatures (4,0), (2,2), (0,4), so a forbidden axis cannot live on any lattice of signature (3,1). Breakings of the cell's mirrors on sublattices are free -- the realisable sets of surviving mirrors form the full boolean lattice, so no breaking arithmetically forces its neighbours -- while the only descent preserving the finite covolume of the fixed four-mirror system is classified completely as m*Lambda_0 with odd scalar m and Lambda_0 of index a power of two, whence all nontrivial unbreakability is 2-adic; the two computational steps behind that classification are printed as a checkable certificate rather than asserted. Finally the mirror machinery embeds one floor up: a nilpotent wedge in so(5,3) has an 18-dimensional centralizer inside a 28-dimensional algebra, that centralizer is (so(3,1) + sl(2,R)) semidirect h_9, and the whole O(3,1) of the lattice lifts precisely into its six-dimensional simple factor. The limit of the reduction is stated rather than hidden: it covers the mirror line and not the shadow line, which has its own ancestor, so the work carries two roots and says so. No physical reading of any object is made anywhere.