From the 1/e Coupling Corridor to the One-Third Wall Distributed Operating Space, Robustness Budgets, and Capture in Multi-Agent Systems

This technical note revises and retracts in part an earlier multi-agent simulation interpretation centered on a proposed \alpha = 1/e coupling corridor. Follow-up analysis showed that the original dynamical claims were not supported: variance and adversarial-resistance optima were parameter-dependent, and the identity R(1/e)=e-1 followed algebraically from the chosen ratio definition rather than from emergent system behavior. The surviving result is a corrected distributed-systems interpretation. Coherent multi-agent operation requires a protected operating space in which honest agents can retain bounded variation, local context, and adaptive flexibility. Variance is therefore not treated as failure. Failure begins when adversarial or capture-oriented influence exceeds the system’s robustness budget. In a fixed-trim W-MSR consensus model with consistent pinned adversaries, the relevant wall is an absolute-count condition, n_{\rm adv}\leq F, where F is the rejection budget. Normalizing this wall by population size produces F/N, which drifts as 1/N and is therefore not an invariant. A normalized population fraction becomes structurally meaningful only when the honest-to-adversary ratio is itself the object of study. In that framing, the relevant boundary is the one-third adversarial wall of distributed agreement, corresponding to an approximate two-thirds honest / one-third adversarial split under the assumptions of the classical Byzantine-agreement setting. The note’s contribution is methodological rather than theoretical: it retires an unsupported constant-based interpretation, preserves the underlying intuition of protected distributed operating space, and reframes the result in terms of robustness budgets, normalization discipline, and capture thresholds.