Saturating Trail Memory for Asynchronous Multi-Agent Networks

Integrate this mechanism into a multi-agent RNN, recurrent world model, or graph network. Each agent has hidden state h_i, emits a soft visitation or write vector a_i in [0,1]^M over M shared memory cells, reads the bounded field u = g(m), and updates h_i using a GRU or small MLP. At each rollout step support two execution modes. In parallel mode, all agents read the same old m, produce h_i and a_i, and then all writes are summed. In sequential mode, choose a random permutation, read the current m, update one agent, write its a_i, and continue. Pseudocode is: m = rho*m; for i in order: u = g(m); h[i] = RNN(h[i], concat(obs[i], u)); a[i] = sigmoid(W*h[i]); m = m + a[i]; for parallel execution, compute all agent updates from the old m before adding writes. The paper supplies the saturation-based equivalence mechanism and scaling law; m_s, rho, latent walk dimension d_w, and exponent theta_k,N must be estimated empirically. First run a cheap synthetic experiment with N = 4 agents navigating a one-dimensional line toward a target. Compare shared saturating memory, additive nonsaturating memory, and no memory. Train with parallel execution, then evaluate with 100 random sequential permutations. Measure the Wasserstein distance between parallel and sequential distributions of K, the disagreement in final logits, and kth-arrival-time tails. The prediction is that disagreement falls sharply as m/m_s enters the saturation regime; saturated trials should have a near-zero distributional difference, while additive memory should retain order dependence. Rescale initial distances by z and fit the tail exponent; the measured slope of log F_k,N plus (1 + theta_k,N) log t versus log z should approach d_w theta_k,N.