The Lorentz Group as an Automorphism of Ze Counting

Abstract

We demonstrate that the Lorentz group O(1,1) acts naturally and exactly on the dual-channel output of a Ze binary event counter. For any binary stream analyzed in a sliding window of width W, the stasis count Z_s (T-events) and transition count Z_t (S-events) define a pseudo-Euclidean quadratic form ds² = Z_s² − k²Z_t² on the Ze counting plane R^{1,1}. We prove analytically that the transformation T_v: (Z_s, Z_t) → (γ(Z_s − vZ_t), γ(Z_t − (v/k²)Z_s)) with γ = (1 − v²/k²)^{−1/2} preserves ds² exactly, constituting an element of the Lorentz group O(1,1). The parameter k plays the role of the speed of light: it is the Ze impedance limit separating timelike (v_Ze < k, ds² > 0) from spacelike (v_Ze > k, ds² < 0) intervals. Numerical simulations with up to 500,000-event streams and 2,495 sliding windows confirm: (i) ds² is preserved to within 5 × 10⁻¹³% under all tested Lorentz transforms; (ii) the γ-factor is recovered exactly as γ = Z_s'/(Z_s − vZ_t); (iii) the causal transition from timelike to spacelike occurs precisely at p = 0.5 (v_Ze = k = 1), which corresponds to the Minkowski lightcone. For non-stationary streams, the causal character changes dynamically, demonstrating that the Ze counting process generates Minkowski geometry locally. The Minkowski metric is therefore the unique O(1,1)-invariant quadratic form on the Ze counting plane.