{
 "artifacts": null,
 "category": "dynamics",
 "description": "Apply an inverse-square Calogero barrier to the eigenvalues of a recurrent or state-space transition Jacobian, discouraging unstable eigenvalues and pathological eigenvalue collisions without forcing the matrix to be Hermitian. The paper's non-Hermitian scattering picture motivates treating the spectrum as correlated rather than assuming an ordinary pairwise Coulomb gas; the inverse-square term is used as a local, computable surrogate for that mechanism.",
 "formulas_latex": [
  "$$\\mathcal{H}_{C}=-\\sum_{i=1}^{N}\\frac{\\partial^2}{\\partial x_i^2}+g\\sum_{i\u003cj}\\frac{1}{(x_i-x_j)^2},$$",
  "$$R_C(J_t)=\\sum_{i\u003cj}\\frac{g}{|z_i(J_t)-z_j(J_t)|^2+\\epsilon^2},\\qquad R_\\rho(J_t)=\\left[\\max\\left(0,\\rho(J_t)-(1-\\delta)\\right)\\right]^2,$$",
  "$$h_{t+k}-\\tilde h_{t+k}\\approx J_{t+k-1}\\cdots J_t(h_t-\\tilde h_t),\\qquad \\|h_{t+k}-\\tilde h_{t+k}\\|\\lesssim (1-\\delta)^k\\|h_t-\\tilde h_t\\|\\ \\text{if}\\ \\rho(J_s)\\leq 1-\\delta.$$"
 ],
 "id": 2966,
 "implementation": "Integrate the method into a small recurrent network, linear state-space model, or neural ODE discretization. At every K training steps, collect representative hidden states from a minibatch and compute the transition Jacobian J with automatic differentiation; for a linear state-space model this is simply the learned transition matrix A. Compute its eigenvalues z_i exactly when the hidden dimension is below roughly 256. For larger systems, estimate leading eigenvalues with Arnoldi iteration and use a random subset of eigenvalue pairs. Add $\\lambda_C R_C+\\lambda_\\rho R_\\rho$ to the task loss, detach the spectrum between updates if eigenvalue gradients are unstable, and clip the total gradient. Pseudocode is: sample states; form J; eigvals = eig(J); collision = sum over i\u003cj of g/(abs(eigvals[i]-eigvals[j])^2+eps^2); radius = max(abs(eigvals)); loss = task_loss + lambda_C*collision + lambda_rho*relu(radius-(1-delta))^2; backpropagate. The Calogero operator and inverse-square mechanism come from the paper; the Jacobian penalty, softening, and contraction term are engineering adaptations. First test on copy/addition and sequential MNIST with equal parameter, wall-clock, and optimizer budgets against the same model with only spectral-radius control, plus an ablation removing $R_C$. Pre-register that the measured divergence boundary occurs at spectral radius 1 within 20%, hidden perturbation norms decay with slope at least $\\log(1-\\delta)$ over long rollouts, and the number of eigenvalue pairs with spacing below a fixed threshold $s$ is reduced by at least 30% at fixed radius. Reject the transfer if the Calogero penalty does not reduce near-collision frequency, or if it improves neither rollout decay nor validation loss while costing more than 10% training time.",
 "math_summary": "For a non-Hermitian matrix with eigenvalues $z_i$, the paper identifies the spectral density, after a Vandermonde factor, with a Calogero scattering state. The underlying Calogero operator is $\\mathcal{H}_{C}=-\\sum_{i=1}^{N}\\partial_{x_i}^{2}+g\\sum_{i\u003cj}(x_i-x_j)^{-2}$, where $x_i$ are particle coordinates and $g$ is the symmetry-dependent inverse-square coupling. For a neural transition map $h_{t+1}=F_\\theta(h_t,u_t)$, let $J_t=\\partial F_\\theta/\\partial h_t$ and let $z_i(J_t)$ be its eigenvalues. Use the adapted regularizer $R_C(J_t)=\\sum_{i\u003cj}g/(|z_i-z_j|^2+\\epsilon^2)$, with softening $\\epsilon\u003e0$, together with a contraction penalty $R_\\rho=[\\max(0,\\rho(J_t)-(1-\\delta))]^2$, where $\\rho(J_t)=\\max_i|z_i|$ and $\\delta\u003e0$ is the desired stability margin. The discrete-time stability prediction is that perturbations decay when $\\rho(J_t)\\leq 1-\\delta$ and grow exponentially when the effective spectral radius exceeds one; the Calogero term predicts suppression of near-collision events.",
 "math_tags": [
  "statistical-mechanics",
  "random-matrix",
  "spectral-theory",
  "dynamical-systems"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "optimizer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.00164",
  "arxiv_url": "https://arxiv.org/abs/2609.00164",
  "summary_what_math_gives_to_ml": "The paper provides an exact spectral mechanism for non-Hermitian matrices with transposition symmetry: after factoring out a Vandermonde term, the joint eigenvalue density is a Calogero-model scattering wavefunction with inverse-square interactions. This differs from a conventional Coulomb-gas model and predicts non-Gaussian, power-law spectral correlations and spacing tails. A transferable neural-network construction is to regularize the spectrum of recurrent, state-space, or neural-ODE Jacobians with a Calogero-inspired inverse-square spectral barrier while separately enforcing a contraction boundary. The experiment should test eigenvalue-collision statistics and long-horizon stability at a predicted spectral-radius boundary, rather than merely reporting a benchmark improvement.",
  "title": "Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Calogero Spectral Barrier for Recurrent Dynamics",
 "url": "https://synthcore.org/idea/2966/calogero-spectral-barrier-for-recurrent-dynamics",
 "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
   }
  }
 }
}