{
 "artifacts": null,
 "category": "dynamics",
 "description": "Regularize the state-transition or input-output Jacobian of a recurrent, state-space, or implicit neural network so that its complex eigenvalue cloud belongs to a selected non-Hermitian symmetry class and has the corresponding unfolded pair statistics. Combine this statistical-shape constraint with an explicit spectral-abscissa or spectral-radius margin, preventing the network from obtaining good average singular values while remaining highly non-normal and transiently unstable.",
 "formulas_latex": [
  "\\mathcal T_+H^*\\mathcal T_+^{-1}=H,\\qquad \\mathcal T_+\\mathcal T_+^*=\\pm1",
  "\\mathcal C_+H^{\\mathrm T}\\mathcal C_+^{-1}=H,\\qquad \\mathcal C_+\\mathcal C_+^*=\\pm1",
  "L(\\theta)=L_{task}(\\theta)+\\lambda_g\\int_{r_{min}}^{r_{max}}w(r)[g_\\theta(r)-g_c(r)]^2\\,dr+\\lambda_s\\left[\\max_i\\max(0,|z_i|-\\rho)\\right]^2",
  "x_{t+1}=f_\\theta(x_t,u_t),\\quad J_t=\\frac{\\partial f_\\theta}{\\partial x_t},\\quad \\text{linearized discrete-time stability if }\\rho(J_t)=\\max_i|z_i|\u003c1"
 ],
 "id": 2965,
 "implementation": "1. Integration point: apply the method to the recurrent transition matrix of an RNN or SSM, or to minibatch Jacobians of an implicit layer. Start with state widths 64 to 256, where dense eigendecomposition is affordable. 2. Pseudocode: for each batch, compute \\(J_t\\) by automatic differentiation; for small states, calculate all eigenvalues \\(z_i\\); estimate the two-dimensional spectral density with a Gaussian kernel; locally unfold eigenvalue coordinates using that density; bin pair distances to obtain \\(g_\\theta(r)\\); evaluate the selected class curve \\(g_c(r)\\); add the displayed correlation and stability penalties to the task loss; then backpropagate. For larger states, estimate leading eigenvalues with Arnoldi and use randomized Jacobian-vector products. 3. The paper supplies the symmetry classes and universal target statistics; the finite-width density, eigenvalues, and pair histogram are estimated empirically. 4. First experiment: use a small long-horizon sequence-prediction task and compare a vanilla RNN, spectral-radius regularization alone, and this method at equal parameter count and optimizer budget. Ablate the pair-statistics term while retaining the radius margin. Measure rollout loss, transient amplification, and distance to the target correlation curve. 5. Pre-register the signature: increasing \\(\\lambda_g\\) must reduce correlation error monotonically, and the result must remain similar for widths 64, 128, and 256. At matched measured \\(\\rho(J)\\), transient amplification should be lower than with radius-only control. Reject the transfer if pair-correlation error fails to decrease by 30%, long-horizon error fails to improve by 10% over the radius-only baseline, or the inferred class statistics vary substantially with width.",
 "math_summary": "The paper distinguishes Hermitian matrices \\(\\mathcal H=\\mathcal H^\\dagger\\) from non-Hermitian matrices \\(H\\), and defines non-Hermitian time-reversal constraints by \\(\\mathcal T_+H^*\\mathcal T_+^{-1}=H\\), with unitary \\(\\mathcal T_+\\) and \\(\\mathcal T_+\\mathcal T_+^*=\\pm1\\), or by \\(\\mathcal C_+H^{\\mathrm T}\\mathcal C_+^{-1}=H\\), with unitary \\(\\mathcal C_+\\) and \\(\\mathcal C_+\\mathcal C_+^*=\\pm1\\). These constraints generate distinct universal complex-eigenvalue pair statistics through analytic continuation of the corresponding Hermitian replica sigma model. For a neural state map \\(x_{t+1}=f_\\theta(x_t,u_t)\\), let \\(J_t=\\partial f_\\theta/\\partial x_t\\), let \\(z_i\\) be its eigenvalues, and let \\(\\tilde z_i\\) be locally unfolded eigenvalues obtained by dividing distances by the estimated local spectral density. Define \\(g_\\theta(r)\\) as the empirical radial pair-correlation function of \\(\\tilde z_i\\), and \\(g_c(r)\\) as the class-specific universal curve read from the paper. Optimize the task loss plus a correlation penalty and a spectral-radius margin.",
 "math_tags": [
  "random-matrix",
  "statistics",
  "spectral-theory",
  "linear-algebra",
  "dynamical-systems"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "optimizer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.00162",
  "arxiv_url": "https://arxiv.org/abs/2609.00162",
  "summary_what_math_gives_to_ml": "The paper provides an exact large-N duality between Hermitian and non-Hermitian random-matrix level statistics, organized by transposition symmetry rather than only conventional time-reversal symmetry. Its transferable asset is a principled, symmetry-class-dependent target distribution for the complex eigenvalues of non-normal operators, including bulk pair correlations and hard-edge statistics near the origin. A neural-network use is to treat the recurrent or implicit-network Jacobian as a non-Hermitian random matrix and regularize its unfolded spectrum toward the appropriate universal class while separately enforcing a stability margin. The strongest first test is whether this produces longer stable rollouts at equal parameter count and whether the predicted eigenvalue pair-correlation transition survives width changes.",
  "title": "Duality between the level statistics of Hermitian and non-Hermitian random matrices",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Duality-Calibrated Jacobian Spectrum",
 "url": "https://synthcore.org/idea/2965/duality-calibrated-jacobian-spectrum",
 "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
   }
  }
 }
}