{
 "artifacts": null,
 "category": "training",
 "description": "Use the two-species phase diagram as a training controller: gradually increase one-way mutation \\(\\lambda\\) and mutant suppression \\(\\mu\\) while monitoring whether feature activity approaches the absorbing boundary. This creates a measurable curriculum from independent learning to controlled competition instead of turning on strong destructive interactions at initialization.",
 "formulas_latex": [
  "$$\\partial_t\\rho_A=\\cdots-\\mu\\rho_A\\rho_B,\\qquad \\partial_t\\rho_B=\\cdots+\\lambda\\rho_A,$$",
  "$$M_s(t)=\\frac{1}{N}\\sum_{i=1}^{N}\\rho_s(i,t),\\qquad \\widehat g_s(t)=\\frac{\\log(M_s(t+W)+\\varepsilon)-\\log(M_s(t)+\\varepsilon)}{W},\\quad s\\in\\{A,B\\},$$",
  "$$\\lambda_{q+1}=\\operatorname{clip}\\left(\\lambda_q-\\eta_\\lambda\\widehat g_B(q),0,\\lambda_{\\max}\\right),$$",
  "$$\\mu_{q+1}=\\operatorname{clip}\\left(\\mu_q+\\eta_\\mu\\widehat g_A(q),0,\\mu_{\\max}\\right).$$"
 ],
 "id": 3129,
 "implementation": "Integration point: apply this controller to the coupling coefficients of the nonreciprocal feature block during optimization. Keep a1, b1, a2, and b2 fixed initially, and update lambda and mu only every Q optimizer steps using batch-averaged activities. Pseudocode is: `for optimizer_step: y,A,B=net(x,lambda,mu); loss=task_loss(y,target); loss.backward(); optimizer.step(); if step%Q==0: MA=mean(A.detach()); MB=mean(B.detach()); gA=(log(MA+eps)-log(MA_prev+eps))/window; gB=(log(MB+eps)-log(MB_prev+eps))/window; lambda=clip(lambda-eta_lambda*gB,0,lmax); mu=clip(mu+eta_mu*gA,0,mmax); MA_prev=MA; MB_prev=MB`. The paper supplies the coupled and decoupled limits and the multicritical interpretation; the growth-rate feedback law is an empirical control adaptation. Compute activities from actual hidden tensors, smooth them with an exponential moving average, and freeze the controller during validation and testing. First cheap experiment: use a two-layer recurrent classifier on sequential MNIST, comparing zero coupling, fixed strong coupling, random coupling, and the adaptive schedule. Sweep initial a1 and b1 around zero. The predicted signature is a stable nonzero-activity plateau with both estimated growth rates near zero under feedback, while fixed coupling should cross into extinction or runaway activity as mu and lambda increase. Identify the empirical absorbing boundary by plotting extinction probability against coupling strength; the adaptive schedule should remain within one activity standard deviation of this boundary. Near the boundary, A traces should show a statistically detectable power-times-log correction relative to pure-power fits if the paper's critical mechanism survives neural discretization.",
 "math_summary": "The decoupled limit is obtained with \\(\\lambda=\\mu=0\\), while coupling adds B production \\(\\lambda\\rho_A\\) and A suppression \\(-\\mu\\rho_A\\rho_B\\). Define empirical activities \\(M_A\\) and \\(M_B\\) as spatial and batch averages of the nonnegative hidden fields. Estimate their logarithmic growth rates over a window W. The controller increases mutation when B is decaying too quickly and increases suppression when A is growing too quickly, targeting near-zero growth without allowing either activity to vanish. The paper's multicritical point is near \\(a_1=b_1=0\\), and its distinct A/B scaling means the controller must monitor both species separately rather than use one global feature norm.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "bifurcations",
  "statistical-mechanics"
 ],
 "ml_areas": [
  "training-dynamics",
  "scheduler",
  "regularization",
  "rnn"
 ],
 "paper": {
  "arxiv_id": "2609.03123",
  "arxiv_url": "https://arxiv.org/abs/2609.03123",
  "summary_what_math_gives_to_ml": "The paper provides a transferable mechanism for designing nonreciprocally coupled neural populations: an active species A generates a mutant species B, while B suppresses A, producing a multicritical active-to-absorbing transition. Its useful asset is the explicit coupled Langevin system with asymmetric cross-couplings, positivity-preserving demographic noise, and a predicted separation of critical behavior between A and B, including logarithmic corrections for A below its upper critical dimension. A neural implementation should use two nonnegative feature fields or residual streams with learnable reaction rates, then map the phase diagram and critical decay laws rather than treating the coupling as an unconstrained extra layer.",
  "title": "Death by mutants: unusual multicritical dynamics in a two-species model for absorbing state transitions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 8,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "generalization",
  "sample-efficiency"
 ],
 "title": "Criticality-controlled mutation schedule",
 "url": "https://synthcore.org/idea/3129/criticality-controlled-mutation-schedule",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "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": null,
    "tested": false
   }
  }
 }
}