{
 "artifacts": null,
 "category": "dynamics",
 "description": "Build a recurrent or state-space layer with fixed, spatially heterogeneous relaxation rates and measure its steady response to positive versus negative input offsets. Unlike a homogeneous linear state-space model, an individual heterogeneous realization can generate a controlled even response, allowing directional transport, asymmetric temporal context, or hysteresis-like sequence transformations without inserting an explicit quadratic input feature.",
 "formulas_latex": [
  "$$\\tau_i=\\tau_c\\exp(E_i/T),\\qquad \\phi(E)=T_g^{-1}\\exp(-E/T_g),\\qquad \\psi_\\nu(\\tau)=\\nu\\tau_c^\\nu\\tau^{-1-\\nu},\\quad \\tau\\geq\\tau_c,\\quad \\nu=T/T_g\u003e1.$$",
  "$$J(\\delta;\\rho)=a_1(\\rho)\\delta+a_2(\\rho)\\delta^2+a_3(\\rho)\\delta^3+O(\\delta^4),\\qquad R(\\delta)=\\frac{J(\\delta)+J(-\\delta)}{2}=a_2(\\rho)\\delta^2+O(\\delta^4).$$",
  "$$a_2=0\\quad\\text{if}\\quad p_i=p_{L+1-i}\\ \\text{for every position }i.$$",
  "$$h_{t+1}=Mh_t+Bu_t,\\qquad M=A+K,\\qquad A_{ii}=\\exp(-\\Delta t/\\tau_i),\\qquad \\rho(M)\u003c1.$$"
 ],
 "id": 3110,
 "implementation": "(1) Integration point: replace the diagonal transition matrix of a small recurrent or state-space model by a fixed quenched heterogeneous bank of time constants. For an input sequence u_t, use h_(t+1) = M h_t + B u_t and y_t = C h_t, initialize tau_i once, and keep these time constants fixed during training. Draw E_i from an exponential distribution with scale T_g, set tau_i = tau_c exp(E_i/T), and clip extreme values for numerical safety. Use a banded nearest-neighbor K so the hidden coordinates have a meaningful reflection operation. Rescale M after initialization to target spectral radius 0.8 or 0.9. (2) Pseudocode: sample E; compute tau; construct diagonal A; construct K; form M = A + K; estimate the largest eigenvalue magnitude; rescale K if needed; run the recurrence; train B, C, and optionally K by backpropagation. Add lambda times max(0, estimated_rho(M)-0.98)^2 if training pushes the system toward instability. For each held-out baseline state, evaluate converged outputs for drives +delta and -delta, fit a cubic polynomial, and record a2. (3) Computed from the paper: the Arrhenius time-scale construction and the symmetry prediction that a2 vanishes for a reflection-symmetric profile. Estimated empirically: spectral radius, equilibrium hidden profile p, profile asymmetry S = norm(p - R p), and response coefficients. Create a control by reflection-averaging two modules, one with tau_i and one with tau_(L+1-i). (4) First cheap experiment: compare 32-unit homogeneous, heterogeneous, and reflection-averaged models on a synthetic delayed directional-response task with target f(u) = u + 0.5 u^2, plus an adding problem and a sequence-classification task requiring asymmetric context. Match parameter counts and train with the same optimizer. Sweep delta from 0.01 to 0.2 and measure R(delta) = [J(delta)+J(-delta)]/2. The quantitative prediction is that R(delta)/delta^2 approaches a nonzero constant for an asymmetric realization, scales approximately linearly with S near S = 0, and falls to measurement noise in the reflection-averaged control. Independently sweep the spectral radius: hidden-state norms should remain bounded for rho(M) below one and grow rapidly after the predicted stability boundary rho(M) = 1.",
 "math_summary": "The paper uses quenched waiting times tau_i = tau_c exp(E_i/T), where E_i is a fixed trap depth, T is temperature, and tau_c is a microscopic time. Exponentially distributed trap depths induce psi_nu(tau) = nu tau_c^nu tau^(-1-nu) for tau \u003e= tau_c, with nu = T/T_g \u003e 1. For a fixed realization, expand the steady response around signed drive delta as J(delta; rho) = a1(rho) delta + a2(rho) delta^2 + a3(rho) delta^3 + ..., where rho is the baseline density or neural operating state. Spatial reflection maps position i to L+1-i. If the equilibrium profile obeys p_i = p_(L+1-i), then the paper predicts a2 = 0. The neural analogue uses h_(t+1) = M h_t + B u_t, with M = A + K, A_ii = exp(-Delta_t/tau_i), and K a local mixing matrix. The recurrence is contractive when the spectral radius rho(M) is less than one. Heterogeneity should be evaluated through the measured even-response coefficient a2, not assumed beneficial merely because weights are random.",
 "math_tags": [
  "statistical-mechanics",
  "dynamical-systems",
  "stochastic-processes",
  "linear-algebra"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.03560",
  "arxiv_url": "https://arxiv.org/abs/2609.03560",
  "summary_what_math_gives_to_ml": "The paper identifies a concrete transport mechanism: quenched, sample-specific spatial heterogeneity produces an even-in-drive current component, J(delta) = a1 delta + a2 delta^2 + ..., even though the disorder ensemble remains symmetric. The decisive condition is broken spatial-reflection symmetry of the realized equilibrium profile; reflection-symmetric samples have a2 = 0. A transferable neural analogue is a recurrent or state-space module with fixed heterogeneous local time scales, whose response to opposite signed inputs can be deliberately rectified. The paper therefore suggests a testable architecture for directional sequence processing, with a sharp symmetry prediction rather than only a benchmark hypothesis.",
  "title": "Sample-specific rectification-like response in a boundary-driven exclusion process",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "stability"
 ],
 "title": "Quenched-heterogeneity rectifying state-space layer",
 "url": "https://synthcore.org/idea/3110/quenched-heterogeneity-rectifying-state-space-layer",
 "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
   }
  }
 }
}