{
 "artifacts": [
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1137/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1137/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1137/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Train decentralized agents using only individual rewards for discovering replenishable targets, while their observations contain conspecifics but not target locations. Give the policy a tunable visual or attention radius and test whether aggregation and improved search emerge above the predicted crossover, without adding alignment, proximity, or group rewards. This creates a controllable collective phase that can reduce redundant exploration and improve multi-agent resource discovery.",
 "download_zip": "https://synthcore.org/download/1137",
 "formulas_latex": [
  "$$T_c(\\lambda) \\simeq \\frac{b^2}{2D}\\ln\\!\\left[\\frac{\\tau}{\\lambda e^{-\\ell/\\lambda}}\\right]$$",
  "$$\\lambda^* = \\frac{\\ell}{W\\!\\left(\\frac{\\ell e^K}{\\tau}\\right)}, \\qquad K=\\frac{\\pi}{4r}\\left(1-B\\frac{r}{\\tau}\\right), \\quad B\\approx0.64$$",
  "$$\\mathcal{A}=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{1}{|\\mathcal{N}_i|}\\sum_{j\\in\\mathcal{N}_i}\\mathbf{1}\\{\\|x_i-x_j\\|\u003cr_a\\}$$",
  "$$\\Delta T(\\lambda)=T_{\\mathrm{ind}}(\\lambda)-T_{\\mathrm{collective}}(\\lambda),\\qquad \\Delta T(\\lambda^*)=0$$"
 ],
 "id": 2928,
 "implementation": "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|| \u003c= 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.",
 "math_summary": "Let ell be the reactive length associated with observing another agent and successfully exploiting the target it found, tau be the target depletion or replenishment time, b be the characteristic search-domain length, D be the diffusive coefficient, and lambda be the agents' detection radius. The collective strategy has mean first-passage time T_c approximately equal to (b^2/(2D)) log[tau/(lambda exp(-ell/lambda))], which grows logarithmically with tau. The crossover is predicted by equating collective and single-agent search times, yielding lambda* = ell/W(ell exp(K)/tau), where W is the Lambert W function, K = (pi/(4r))(1-Br/tau), r is the target or encounter length scale, and B is approximately 0.64. The transfer hypothesis is that the aggregation order parameter and collective-search advantage exhibit a crossover near lambda* as the policy's conspecific attention radius is varied.",
 "math_tags": [
  "statistical-mechanics",
  "stochastic-processes",
  "bifurcations",
  "probability"
 ],
 "ml_areas": [
  "rl",
  "attention",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.28046",
  "arxiv_url": "https://arxiv.org/abs/2608.28046",
  "summary_what_math_gives_to_ml": "The paper provides a transferable mechanism for emergent coordination: agents optimizing only an individual replenishable-resource reward can spontaneously aggregate when their sensory range exceeds a calculable crossover. The analytical signature is an equality between a single-agent search mean first-passage time and a collective search time, with the collective time growing only logarithmically with depletion time and the crossover radius given by a Lambert-W expression. In neural multi-agent systems, this can be implemented as decentralized agents with conspecific-only attention and individual resource rewards, then tested for a sharp aggregation transition without any social bonus. A second use is to turn the crossover formula into an adaptive attention-range controller whose predicted transition can be falsified quantitatively.",
  "title": "Emergent aggregation from collective foraging",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "sample-efficiency",
  "scalability",
  "accuracy"
 ],
 "title": "Resource-Driven Collective Attention Phase",
 "url": "https://synthcore.org/idea/2928/resource-driven-collective-attention-phase",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 8,
   "verdict": "Built a decentralized hidden-target multi-agent toy simulator with radius-limited conspecific attention, individual capture rewards, replenishing targets, an independent baseline, and Lambert-W prediction checks. The analytic collective MFPT showed the claimed logarithmic dependence on tau to machine precision, and lambda* satisfied its defining equation, but the simulation showed no reproducible aggregation kink or capture advantage near lambda*: capture rate peaked weakly at 0.75–1.0 lambda* and matched the independent baseline overall.",
   "metrics": {
    "baseline": "capture rate 16.67 per 1000 steps, aggregation A 0.1208, duplicate fraction 0.00542",
    "idea": "Across 0.25, 0.5, 0.75, 1, 1.5, 2, 4 lambda*: capture rates 15.00, 16.67, 18.33, 18.33, 16.67, 16.67, 16.67; aggregation A 0.1304, 0.1454, 0.1542, 0.1163, 0.1042, 0.1435, 0.1165"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "This is a small 8-agent, 300-step toy simulation with two seeds, not trained RL; it uses transient motion cues as a simplified proxy for learned attention and does not estimate empirical ell, tau, or MFPT. The short horizon and sparse captures make the radius metrics noisy, so larger trained-policy experiments are needed to test transfer beyond this MVP."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-02T12:10:26",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": false,
    "tested": true
   }
  }
 }
}