Why Spacetime Is Lorentzian: Forced Signature from a Discrete Cost Functional

We prove that a discrete lattice governed by the cost functional J(x) = 1/2 (x + x⁻¹) − 1 reproduces a Lorentzian manifold in the continuum limit N → ∞. The Lorentzian signature (−, +, +, +) is not assumed—it is forced by three structural asymmetries of the lattice: (i) temporal ticks are irreversible while spatial displacements are reversible, (ii) the cost symmetry Jlog(ε) = Jlog(−ε) imposes spatial isotropy, and (iii) the propagation bound of one voxel per tick fixes the speed of light and generates the light cone. We show that the second derivative J″(1) = 1 sets the canonical normalization of the spatial metric, that the lattice Laplacian converges to the continuum Laplacian ∇² with error O(a²) where a = L/N is the lattice spacing, and that the Arnowitt–Deser–Misner decomposition with unit lapse and zero shift recovers the Minkowski interval. When the lattice defect density is non-uniform, the effective metric acquires weak-field perturbations gμν = ημν + hμν governed by a Poisson equation with coupling κ = 8φ⁵, where φ = (1 + √5)/2 is the golden ratio—a constant derived within the framework, not fitted. We further prove that the lattice cost-action converges to the Einstein–Hilbert action at O(a²) in the weak-field regime: the linearized Euler–Lagrange equations reduce to the lattice Laplacian, the flat cubic lattice has zero deficit angle, and the linearization coefficient is unity. All results follow from a single functional equation with zero adjustable parameters.