{
 "artifacts": [
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1142/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1142/report.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1142/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Treat a recurrent or state-space network as a locally linear dynamical system and select a small set of hidden-state or module coordinates that have unusually high leverage on a target output through a dominant unstable or weakly damped eigenmode. Use the ranking both for red-team targeted perturbations and for defense: penalize, prune, or damp selected coordinates so that target amplification is reduced without uniformly shrinking all recurrent dynamics.",
 "download_zip": "https://synthcore.org/download/1142",
 "formulas_latex": [
  "$$x_{t+1}=f_\\theta(x_t,u_t),\\qquad A_t=\\frac{\\partial f_\\theta}{\\partial x}(x_t,u_t),\\qquad y_t=Cx_t.$$",
  "$$Av=\\lambda v,\\qquad w^\\top A=\\lambda w^\\top,\\qquad w^\\top v=1;\\qquad |\\lambda|\u003e1\\ \\text{is unstable and}\\ |\\lambda|\\approx1\\ \\text{is weakly damped}.$$",
  "$$\\max_{z\\in\\{0,1\\}^n,\\,\\mathbf{1}^\\top z\\le k}\\frac{\\|Cv\\|_2^2}{\\varepsilon+\\sum_{i=1}^n z_i\\|b_i v_i\\|_2^2},\\qquad\\text{or defensively}\\qquad\\min_z\\ \\|Cv\\|_2^2+\\rho\\sum_i z_i\\|b_i v_i\\|_2^2.$$",
  "$$\\|\\Delta y\\|_2\\approx\\left\\|C(I-\\lambda A)^{-1}B_S\\Delta a\\right\\|_2,\\qquad\\text{so near }|\\lambda|=1,\\quad\\|\\Delta y\\|_2\\propto(1-|\\lambda|)^{-1}.$$"
 ],
 "id": 2944,
 "implementation": "Integrate this into a GRU, vanilla RNN, S4-like state-space layer, or recurrent transformer memory, treating hidden coordinates or groups of recurrent parameters as intervention channels. After training warmup, collect representative hidden trajectories and estimate the local Jacobian operator $A$ at several checkpoints. For small hidden states, form $A$ explicitly with automatic differentiation; for large states, use Arnoldi iteration and Jacobian-vector products to estimate the eigenvalue of largest modulus and its right eigenvector $v$. Define each intervention direction $b_i$ as a unit basis vector for a hidden coordinate, channel, head, or parameter block. Compute the score $s_i=\\|Cv\\|_2^2/(\\varepsilon+\\|b_i v_i\\|_2^2)$ and select the top $k$ coordinates, optionally followed by one-swap local search over the binary subset. For a red-team test, inject $\\Delta x_t=B_Sa_t$ with $a_t$ aligned to the left-eigenvector projection and measure target-output growth. For a defense, add a mode-gain penalty or apply coordinate damping $x_{t+1}\\leftarrow x_{t+1}-\\gamma D_Sx_t$, where $D_S$ is diagonal on selected coordinates. Compare with random selection, gradient saliency, magnitude pruning, and uniform spectral normalization under equal compute and parameter budgets. The paper-derived quantities are eigenmode extraction and subset ranking; Jacobian estimates, nonlinear rollout gain, and the target readout $C$ are measured empirically. Pre-register that amplification scales approximately as $(1-|\\lambda|)^{-1}$ and that the selected top-$k$ set yields at least 2x the target amplification of a random set at identical $k$. Reject the transfer if the fitted log-amplification slope versus $-\\log(1-|\\lambda|)$ is below 0.5, or if eigenmode selection fails to beat random selection by 20% across three seeds.",
 "math_summary": "Let the neural state update be $x_{t+1}=f_\\theta(x_t,u_t)$, with local Jacobian $A=\\partial f_\\theta/\\partial x$ and target readout $y=Cx$. Let $\\lambda$ be an eigenvalue of $A$ with right eigenvector $v$ and left eigenvector $w$, normalized by $w^\\top v=1$. A mode is unstable in discrete time when $|\\lambda|\u003e1$, or weakly damped when $|\\lambda|$ is close to one. Let $B=[b_1,\\ldots,b_n]$ contain intervention directions for hidden units, channel groups, recurrent modules, or parameter blocks, and let binary variables $z_i\\in\\{0,1\\}$ select at most $k$ directions. The target exposure is $\\|Cv\\|_2$, while the selected intervention footprint is $\\|B\\operatorname{diag}(z)v\\|_2^2=\\sum_i z_i\\|b_i v_i\\|_2^2$. The eigenvector ranking identifies a small subset with high target-mode leverage. The forced linear response contains the resolvent $(I-\\lambda A)^{-1}$, producing amplification that grows sharply as $|\\lambda|$ approaches one.",
 "math_tags": [
  "dynamical-systems",
  "spectral-theory",
  "linear-algebra",
  "control-theory",
  "optimization"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.28533",
  "arxiv_url": "https://arxiv.org/abs/2608.28533",
  "summary_what_math_gives_to_ml": "The paper provides a transferable actuator-selection mechanism: select a small subset of controllable coordinates by exploiting an unstable eigenmode whose amplitude is large at designated targets and small at the compromised coordinates. Its concrete asset is an eigenvector-based ranking combined with a binary quadratic subset-selection problem, rather than generic gradient saliency. For recurrent or state-space neural networks, the same construction can identify hidden-state or module interventions that selectively amplify target outputs, or conversely identify and damp the most dangerous mode-exposure channels. The transfer is falsifiable through a predicted sharp dependence of target amplification on the real part or modulus of the dominant Jacobian eigenvalue and on the selected subset size.",
  "title": "Targeted Power System Frequency Attack via the Selection of Maliciously Controlled Inverters",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Eigenmode-Targeted Hidden-State Actuator Selection",
 "url": "https://synthcore.org/idea/2944/eigenmode-targeted-hidden-state-actuator-selection",
 "verification": {
  "peer_reviewed": false,
  "stage1_mechanism_check": {
   "worked": false,
   "confidence": 9,
   "verdict": "The toy system verified accurate eigenpairs and strong resolvent growth near a unit eigenvalue. Modal coordinate selection improved target gain over random by 1.54x to 1.78x across three seeds, but missed the stated 2x target; the literal inverse-footprint ranking was worse than random, so the overall idea is not a demonstrated win.",
   "metrics": {
    "baseline": "Random k=2 mean gains: 2.915, 16.904, 15.036.",
    "idea": "Modal gains: 4.475, 29.718, 26.810; mean ratio 1.692x. Literal score mean ratio 0.420x."
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "Only an 8D linear system was tested. No trained nonlinear RNN, defense, pruning, optimization, FLOP match, or larger statistical evaluation was performed."
  },
  "status": "mechanism_failed",
  "status_label": "Mechanism failed",
  "updated_at": "2026-09-02T12:14:40",
  "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": false,
    "tested": true
   }
  }
 }
}