Lorentzian Signature from Causal Dynamics on the Euclidean D4 Lattice

In the Selection–Stitch Model the D4 root lattice is taken as physical four-dimensional spacetime, with the face-centered cubic (FCC) lattice as each constant-time slice. The rank-four bond tensor of D4 is exactly the unique fully symmetric isotropic rank-four tensor (proved self-contained in an appendix), so the emergent long-wavelength dynamics is isotropic across all four directions at leading order. That symmetry is Euclidean SO(4), not Lorentzian SO(3,1), and we isolate by direct computation exactly what separates the two. We show that the D4 discrete Laplacian is second order in the temporal direction, entering at the same order and weight as the spatial part, so the wave operator is of d'Alembertian type rather than diffusive; that the spatial and temporal speeds coincide at a single coupling ratio r=1; and that the residual Euclidean form is converted to the Lorentzian one by a single ingredient, the reading of the cross-slice direction as a causal update rather than a static extremization. We further show that the second-order structure is forced, not assumed: the CSS stabilizer code on D4 has weight-24 checks whose cross-slice support is symmetric under e4 → −e4 (six forward, six backward, both check types, verified by computation), so the repair dynamics enforces a time-symmetric second difference and a hyperbolic, not diffusive, continuum limit, with measurement irreversibility supplying the causal arrow. Geometry and the code supply isotropy, a single speed, the second-order structure, and Lorentz violation suppressed to O((E/M_P)^4); the one ingredient neither supplies is which of the four symmetric axes is time. Given that designation, emergent leading-order Lorentz invariance follows; the framework does not at present derive it.