{
 "artifacts": null,
 "category": "dynamics",
 "description": "Add calibrated multiplicative noise to a binary energy-based model, Hopfield network, or discrete recurrent attractor system so that its effective coupling follows a chosen distance from the collective critical point. Unlike ordinary temperature annealing, the controller distinguishes microscopic channel noise from Gibbs or sampling noise and predicts when ordered attractors should disappear through a measurable susceptibility peak.",
 "formulas_latex": [
  "$$P(\\{s_i\\})=Z^{-1}\\exp[-\\beta\\mathcal{H}(s)],\\qquad \\mathcal{H}(s)=-J_{\\mathrm{eff}}\\sum_{\\langle i,j\\rangle}s_i s_j-H_{\\mathrm{eff}}\\sum_i s_i.$$",
  "$$J_{\\mathrm{eff}}^{\\mathrm{dep}}(p)=(1-p)^2J_0,\\qquad H_{\\mathrm{eff}}=0\\ \\text{on the neutrality branch}.$$",
  "$$K(p)=\\beta(1-p)^2J_0,\\qquad K_c=\\frac12\\log(1+\\sqrt{2}),\\qquad p_*=1-\\sqrt{\\frac{K_c}{\\beta J_0}}.$$",
  "$$p_t=1-\\sqrt{\\frac{K_{\\mathrm{target}}}{\\beta_t\\widehat{J}_t}},\\qquad \\widehat{J}_t\\approx\\lambda_{\\max}\\left(\\frac{W_t+W_t^T}{2}\\right).$$"
 ],
 "id": 3116,
 "implementation": "1. Exact integration point: use this controller in a binary energy-based model, Hopfield-style recurrent layer, or discrete latent-variable module. Let the pre-noise update be $h_i=\\sum_jW_{ij}s_j+b_i$, and attenuate pair interactions with $W_t^{\\mathrm{eff}}=(1-p_t)^2W_t$; keep Gibbs sampling temperature $\\beta_t$ separate from channel noise $p_t$. 2. Pseudocode: initialize $W$ and choose $K_{target}$; at each epoch estimate $\\widehat J_t$ from $\\lambda_{max}((W_t+W_t^T)/2)$, or from the mean coupling on a fixed interaction graph; compute $p_t=\\operatorname{clip}(1-\\sqrt{K_{target}/(\\beta_t\\widehat J_t+\\epsilon)},0,1)$; use $h_i=\\sum_j(1-p_t)^2W_{ij}s_j+b_i$ in Gibbs or recurrent updates; record magnetization, susceptibility, and state autocorrelation. 3. Computed from the paper: the $(1-p)^2$ renormalization law, the neutrality condition, and the square-lattice critical value $K_c$. Estimated empirically: the neural interaction scale $\\widehat J_t$, finite-size susceptibility, and whether the learned graph has a comparable effective coordination number. 4. First cheap experiment: train a $32\\times32$ binary Hopfield or energy model on binarized MNIST patches, comparing fixed noise, ordinary temperature annealing, and this controller at $K_{target}=0.35,0.44,0.55$. Sweep $p$ after training. The falsifiable prediction is a magnetization loss and susceptibility peak when $\\beta(1-p)^2\\widehat J$ crosses $K_c$; on a square-grid model, the measured transition should be within approximately 20% of $p_*=1-\\sqrt{K_c/(\\beta\\widehat J)}$. Above the boundary, ordered attractors should disappear and autocorrelation times should peak near the transition, indicating critical slowing down rather than a smooth monotonic noise effect.",
 "math_summary": "Represent binary states by $s_i\\in\\{-1,+1\\}$ and the effective population energy by $\\mathcal{H}(s)=-J_{\\mathrm{eff}}\\sum_{\\langle i,j\\rangle}s_i s_j-H_{\\mathrm{eff}}\\sum_i s_i$, where $\\langle i,j\\rangle$ denotes neighboring or selected interacting units, $J_{\\mathrm{eff}}$ is the collective coupling, and $H_{\\mathrm{eff}}$ is a field or bias. The Gibbs distribution is $P(s)\\propto\\exp[-\\beta\\mathcal{H}(s)]$, with inverse sampling temperature $\\beta$. For depolarization, the paper's channel-to-Ising mapping gives $J_{\\mathrm{eff}}(p)=(1-p)^2J_0$ while the neutrality branch keeps $H_{\\mathrm{eff}}=0$. The dimensionless coupling is $K(p)=\\beta J_{\\mathrm{eff}}(p)$, and the square-lattice critical value is $K_c=\\frac12\\log(1+\\sqrt{2})\\approx0.4406868$. Thus the predicted critical channel strength is $p_*=1-\\sqrt{K_c/(\\beta J_0)}$, provided $\\beta J_0\u003eK_c$. In a neural model, replace $J_0$ by an empirically estimated dominant collective coupling $J_{\\mathrm{NN}}$, such as the largest eigenvalue or average coupling after normalization.",
 "math_tags": [
  "statistical-mechanics",
  "dynamical-systems",
  "bifurcations",
  "stochastic-processes"
 ],
 "ml_areas": [
  "training-dynamics",
  "regularization",
  "sampling",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2609.02957",
  "arxiv_url": "https://arxiv.org/abs/2609.02957",
  "summary_what_math_gives_to_ml": "The paper provides an exact mechanism by which microscopic decoherence renormalizes the effective interaction and field of a lattice population, while leaving strategic Gibbs fluctuations as a separate source of randomness. In particular, depolarization preserves the microscopic neutrality condition but suppresses the collective Ising coupling as $(1-p)^2$, producing a sharp transition from ordered coexistence to disordered crossover at a calculable noise level. The most transferable neural-network construction is a coupling-aware noise controller for binary energy-based models or recurrent attractor networks: estimate the effective interaction strength and tune injected noise to remain below, at, or above the collective critical boundary rather than treating noise as an arbitrary hyperparameter.",
  "title": "Decoherence-controlled collective criticality in a two-dimensional quantum Stag Hunt",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "generalization",
  "sampling"
 ],
 "title": "Critical-Coupling Noise Controller",
 "url": "https://synthcore.org/idea/3116/critical-coupling-noise-controller",
 "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
   }
  }
 }
}