Spectral Convergence of Causal Matrices: From Nyström Approximation to SJ Vacuum

We prove that the eigenvalues of the rescaled causal matrix (V/N)·iA of a Poisson-sprinkled causal diamond converge to those of the continuum sign-kernel operator, at rate O(1/√N). The proof uses Atkinson's collectively compact perturbation theory. We extend this to eigenvectors, the regularised SJ two-point function, and entanglement entropy. The SJ mode fraction satisfies M/N → 1/2 (proven for D=2, numerically confirmed for D=3,4). The entropy density of the half-diamond is s ≈ (log 2)/2 ≈ 0.347, with restriction eigenvalues clustering at 1/2. Numerical verification covers D = 2–4 at N up to 1000.