Competition Between Ze Systems

Abstract

The Ze framework models any binary observation stream as a causal counter whose proper time is the Minkowski interval τ = √(T² − X²), where T = N_T + N_S counts total events and X = N_S counts state-change events. This paper addresses the multi-Ze scenario: what happens when two or more Ze systems co-exist, compete for causal dominance, and interact through adversarial mechanisms? We formalize τ-dominance (Ze_k dominates Ze_j iff τ_k > τ_j), identify two primary strategies—T-amplification (inserting redundant T-events) and S-injection (inserting random S-events into a rival)—and derive their quantitative effects. T-amplification by factor m+1 boosts τ proportionally, with ×10 amplification raising τ by a factor of 11.53. S-injection at rate r = 40% degrades the rival's τ by 10.9%. Game-theoretic analysis shows that symmetric mutual attack converges to a Nash equilibrium at v = 0.5 (maximum entropy), destroying τ for both parties—a causal Prisoner's Dilemma. For three or more Ze systems, a τ-dominance hierarchy emerges whose stability depends on pairwise velocity differences. We present falsifiable predictions and discuss the ontological interpretation: competing Ze systems generate distinguishable experiential realities, with the dominant system experiencing slower, richer time.