Resource-Driven Collective Attention Phase

1. Integration point: build a decentralized multi-agent RL environment with N agents moving in a two-dimensional periodic domain containing replenishable targets. Each agent receives only its own target-capture reward; target positions are masked, while local conspecific positions, velocities, and recent actions are visible. Implement a radius-limited attention encoder, with radius lambda as the experimental control. 2. Pseudocode: for each agent i, form N_i = {j: ||x_j-x_i|| <= lambda}; encode the self state and neighbor tokens; sample a_i from pi_theta(.|o_i); add reward 1 only when agent i captures a target. Compute lambda* from the Lambert-W equation using measured tau, r, ell, and B=0.64, then train identical policies at lambda/lambda* in {0.25, 0.5, 0.75, 1, 1.5, 2, 4}. 3. Computed from the paper are the logarithmic collective MFPT, the Lambert-W crossover, and the prediction that collective search becomes favorable above lambda*. Estimated empirically are ell from the displacement between an observing agent and a target it subsequently captures, tau from replenishment intervals, and T_ind and T_collective from episode first-passage times. 4. First cheap experiment: compare the proposed model against the same architecture with randomized neighbor features and against a model with an explicit proximity reward. Use 8-32 agents and small grids before scaling. Measure capture rate, duplicate-search rate, aggregation A, and spatial correlation length. 5. Pre-register the signature: A must show a reproducible kink or steep crossover near lambda*, while mean capture time must become weakly dependent on tau in the collective regime. Fit log T against log tau below and above the transition; above it, the collective model predicts logarithmic rather than power-law growth. Reject the transfer if the aggregation midpoint differs from lambda* by more than 20% across seeds, if removing conspecific observations leaves the same transition, or if the individual-only policy never outperforms the collective policy below the predicted boundary.