{
 "artifacts": null,
 "category": "architecture",
 "description": "Add a fixed, spatially correlated perturbation field to every layer of a CNN or 2D state-space model, with the perturbation decomposed into transverse and longitudinal Fourier components. Unlike ordinary injected noise, the same field is reused for all training examples and all forward passes, allowing it to act as a structured architectural flow that can promote global feature alignment. Sweep the transverse fraction at fixed total perturbation variance and test for the predicted ordering transition and continuously varying depth-relaxation exponent.",
 "formulas_latex": [
  "$$P^\\perp_{ij}(k)=\\delta_{ij}-\\frac{k_i k_j}{|k|^2},\\qquad P^\\parallel_{ij}(k)=\\frac{k_i k_j}{|k|^2},\\qquad \\hat\\xi^\\perp_i(k)=P^\\perp_{ij}(k)\\hat\\xi_j(k).$$",
  "$$\\mathbb{E}[\\xi_i(x)\\xi_j(0)]\\sim |x|^{-a},\\qquad S(k)=\\mathbb{E}|\\hat\\xi(k)|^2\\sim |k|^{a-2}\\quad (d=2).$$",
  "$$h_{\\ell+1}(x)=\\phi\\!\\left(F_\\ell[h_\\ell](x)+\\gamma\\,\\xi_i(x)\\,\\nabla_i h_\\ell(x)+\\sigma_n\\varepsilon_\\ell(x)\\right),\\qquad \\mu=\\frac{\\sum_k\\|\\hat\\xi^\\perp(k)\\|^2}{\\sum_k\\|\\hat\\xi(k)\\|^2}.$$$$",
  "$$m_L=\\frac{1}{L^2}\\left\\|\\sum_x \\frac{h(x)}{\\|h(x)\\|+\\epsilon}\\right\\|,\\qquad \\tau_L\\sim L^z.$$"
 ],
 "id": 2926,
 "implementation": "1. Integration point: use a CNN with periodic padding and vector-valued feature maps, or a convolutional state-space block whose hidden state has two spatial dimensions. Before training, sample one smooth random vector field $\\xi$ per model seed in Fourier space with amplitude proportional to $|k|^{(a-2)/2}$, project it to obtain a prescribed transverse fraction $\\mu$, inverse FFT it, and normalize its RMS to one. Insert the fixed advection term $\\gamma(\\xi_xD_xh+\\xi_yD_yh)$ before the activation in every residual block; use fresh dropout or Gaussian noise only for the separate $\\sigma_n\\varepsilon$ term. 2. Pseudocode: sample $\\hat\\xi(k)$; apply transverse and longitudinal projectors; mix them to target $\\mu$; normalize; at each block compute $q=F(h)+\\gamma(\\xi_xD_xh+\\xi_yD_yh)+\\sigma_n\\epsilon$ and set $h=\\phi(q)$. Stop gradients through $\\xi$. 3. Computed from the paper's mechanism: Fourier projectors, quenched reuse, long-range spectrum, and the transverse-fraction control. Estimated empirically: the ordering parameter $m_L$, susceptibility $\\chi_L=L^2(\\langle m_L^2\\rangle-\\langle m_L\\rangle^2)$, and relaxation time from perturbing a trained feature map and fitting $\\tau_L\\propto L^z$. 4. First experiment: CIFAR-10 or a synthetic oriented-texture task with identical-width ResNet baselines, comparing clean, iid-noise, longitudinal-quenched, and transverse-quenched models at equal RMS perturbation and equal training compute. Evaluate multiple grid resolutions $L$ and sweep $\\mu\\in\\{0,.25,.5,.75,1\\}$ while holding $\\gamma^2/\\sigma_n^2$ fixed. 5. Preregistered signature: transverse quenched disorder must yield larger $m_L$ and a finite-size susceptibility peak that shifts with $\\mu$; fitted $z(\\mu)$ must vary systematically with $\\mu$ while the iid-noise control does not show the same variation. Reject the transfer if transverse and longitudinal fields have indistinguishable $m_L$, $\\chi_L$, and $z$ within 20%, or if no reproducible transition-like feature appears across at least three grid sizes.",
 "math_summary": "Let $\\xi_i(x)$ be a zero-mean quenched disorder field on a periodic $L\\times L$ feature grid, sampled once and held fixed. Its long-range covariance is $\\mathbb{E}[\\xi_i(x)\\xi_j(0)]\\sim |x|^{-a}$, or equivalently its low-frequency spectrum scales as $S(k)\\sim |k|^{a-2}$ in two dimensions. In Fourier space, decompose the field with the transverse projector $P^\\perp_{ij}(k)=\\delta_{ij}-k_i k_j/|k|^2$ and longitudinal projector $P^\\parallel_{ij}(k)=k_i k_j/|k|^2$. Define the transverse fraction $\\mu=\\sum_k\\|P^\\perp\\hat\\xi(k)\\|^2/\\sum_k\\|\\hat\\xi(k)\\|^2$. The adapted feature-flow update is $h_{\\ell+1}=\\phi(F_\\ell(h_\\ell)+\\gamma\\,\\xi\\cdot\\nabla h_\\ell+\\sigma_n\\varepsilon_\\ell)$, where $F_\\ell$ is the trainable convolution or SSM operator, $\\xi\\cdot\\nabla$ is implemented by finite differences, $\\varepsilon_\\ell$ is fresh zero-mean noise, $\\gamma$ is the quenched-flow strength, and $\\sigma_n$ is ordinary noise strength. The paper's mechanism predicts that changing $\\mu$ at fixed $\\gamma^2/\\sigma_n^2$ can cross an ordering transition and that the measured relaxation exponent $z$ is nonuniversal and varies with $\\mu$, rather than remaining at the clean diffusive value $z\\approx2$.",
 "math_tags": [
  "statistical-mechanics",
  "dynamical-systems",
  "pde",
  "harmonic-analysis",
  "bifurcations"
 ],
 "ml_areas": [
  "cnn",
  "ssm",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.28012",
  "arxiv_url": "https://arxiv.org/abs/2608.28012",
  "summary_what_math_gives_to_ml": "The paper identifies a transferable mechanism in which long-range correlated quenched disorder, unlike temporally refreshed noise, can induce global ordering and qualitatively change relaxation exponents. The effect is controlled by the disorder's transversality, with continuous nonuniversal exponents and transitions between ordered/disordered or strong-coupling/free regimes at fixed disorder and noise variance. A concrete neural analogue is a CNN or spatial state-space model whose feature maps receive a fixed, long-range-correlated transverse perturbation at every layer, rather than independent dropout-like noise. The transfer is valuable if it produces measurable finite-size ordering, a tunable transition as the transverse fraction changes, and altered depth-relaxation exponents.",
  "title": "Correlated disorder versus correlated noise: Ordering in active systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Transverse Quenched Feature Flow",
 "url": "https://synthcore.org/idea/2926/transverse-quenched-feature-flow",
 "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
   }
  }
 }
}