{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1035/bench_report.json"
  },
  {
   "name": "energy_gradient_bench.py",
   "url": "https://synthcore.org/code/1035/energy_gradient_bench.py"
  },
  {
   "name": "experiment.py",
   "url": "https://synthcore.org/code/1035/experiment.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1035/report.md"
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  {
   "name": "report_bench_2026-09-01T191432.md",
   "url": "https://synthcore.org/code/1035/report_bench_2026-09-01T191432.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1035/results.json"
  }
 ],
 "category": "dynamics",
 "description": "Replace an unconstrained neural ODE or recurrent update field with the negative gradient of a learned scalar energy \\(E_\\theta(z,t)\\). The resulting hidden-state dynamics have an exact Lyapunov certificate: energy decreases continuously, bounded trajectories cannot exhibit nonstationary recurrence, and the Łojasiewicz mechanism predicts convergence to a single equilibrium rather than persistent oscillation or chaos.",
 "download_zip": "https://synthcore.org/download/1035",
 "formulas_latex": [
  "$$DF(x)=DF(x)^T\\quad\\Longleftrightarrow\\quad F(x)=\\nabla V(x)$$",
  "$$\\dot z=-\\nabla_zE_\\theta(z),\\qquad \\frac{d}{dt}E_\\theta(z(t))=-\\|\\nabla_zE_\\theta(z(t))\\|_2^2\\leq0$$",
  "$$|E(z)-E(z_*)|^{1-\\alpha}\\leq C\\|\\nabla E(z)\\|,\\qquad \\alpha\\in(0,1/2],\\quad\\Longrightarrow\\quad z(t)\\to z_*$$",
  "$$E(z_{k+1})\\leq E(z_k)-\\eta\\left(1-\\frac{\\eta L}{2}\\right)\\|\\nabla E(z_k)\\|_2^2,\\qquad 0\u003c\\eta\u003c\\frac{2}{L}$$"
 ],
 "id": 2678,
 "implementation": "Use this as a replacement for the vector field in a neural ODE, continuous-depth residual block, recurrent state-space model, or iterative refinement module. Let an MLP receive the current state \\(z\\) and optional fixed conditioning vector \\(c\\), but output a scalar energy \\(E_\\theta(z,c)\\); compute the vector field by automatic differentiation with respect to \\(z\\), rather than predicting its coordinates independently. Add a coercive term \\(\\mu\\|z\\|^2/2\\), with \\(\\mu\\geq0\\), or apply bounded-state normalization to encourage compact sublevel sets. The update is: `E = energy(z,c); g = grad(E,z); Lhat = power_iteration_hessian_norm(E,z); eta = gamma*2/(Lhat+eps); z_next = z - eta*g`; use \\(\\gamma\\in[0.2,0.8]\\). During training, log \\(E_k\\), \\(\\|g_k\\|\\), state norm, and cumulative path length; optionally add `relu(E_next-E)^2` as a numerical-descent penalty. The paper supplies the gradient structure, exact continuous-time dissipation, and convergence implication; estimate the local Lipschitz constant \\(L\\), boundedness, and the Łojasiewicz exponent empirically. First test on MNIST denoising or CIFAR-10 fixed-point refinement against an unconstrained residual MLP with matched parameters and depth. Sweep \\(\\eta\\). The quantitative prediction is a practical instability or monotonicity boundary near \\(\\eta_c=2/L\\): below it energy decreases, while above it energy increases or trajectories diverge. Stable trajectories should have finite cumulative path length, vanishing gradient norm, and no sustained energy oscillation.",
 "math_summary": "Let \\(z(t)\\in\\mathbb{R}^d\\) be the hidden state and let \\(E_\\theta:\\mathbb{R}^d\\to\\mathbb{R}\\) be a continuously differentiable scalar neural energy. Define \\(\\dot z=-\\nabla_zE_\\theta(z)\\). Its Jacobian is \\(Df(z)=-\\nabla_z^2E_\\theta(z)\\), hence symmetric, transferring the paper's symmetric-Jacobian-to-gradient mechanism. Along every solution, \\(dE_\\theta(z(t))/dt=-\\|\\nabla E_\\theta(z(t))\\|_2^2\\leq0\\). If the trajectory remains in a compact sublevel set and the energy satisfies the Łojasiewicz inequality near an equilibrium \\(z_*\\), \\(|E(z)-E(z_*)|^{1-\\alpha}\\leq C\\|\\nabla E(z)\\|\\), with \\(C\u003e0\\) and \\(\\alpha\\in(0,1/2]\\), then the trajectory has finite length and converges to one equilibrium. For explicit Euler, if \\(\\nabla E\\) is locally \\(L\\)-Lipschitz, then \\(E(z_{k+1})\\leq E(z_k)-\\eta(1-\\eta L/2)\\|\\nabla E(z_k)\\|^2\\), giving the sufficient descent boundary \\(0\u003c\\eta\u003c2/L\\).",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "optimization",
  "differential-geometry"
 ],
 "ml_areas": [
  "mlp",
  "ssm",
  "training-dynamics",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.21681",
  "arxiv_url": "https://arxiv.org/abs/2608.21681",
  "summary_what_math_gives_to_ml": "The paper identifies a strong structural mechanism: a vector field on simply connected Euclidean space has a symmetric Jacobian if and only if it is a global gradient field, so the entire quadratic family becomes a cubic gradient flow. Along the dissipative convention \\(\\dot z=-\\nabla E(z)\\), the scalar energy decreases as \\(dE/dt=-\\|\\nabla E\\|^2\\), and the Łojasiewicz gradient inequality gives convergence of every bounded trajectory to one equilibrium even when the equilibrium set is non-isolated. This can be transferred into continuous-depth networks or recurrent inference modules by parameterizing the vector field as the gradient of a learned scalar energy, yielding an explicit energy monitor and a falsifiable no-recurrence and convergence prediction.",
  "title": "An Elegant Analytical Resolution of the Sprott-Zeraoulia Conjecture for Three-Dimensional Quadratic Differential Systems with Symmetric Jacobian Matrices",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Energy-Gradient Neural Flow",
 "url": "https://synthcore.org/idea/2678/energy-gradient-neural-flow",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.000164712047080684,
    "idea_mean": 0.000707463565049693,
    "delta_mean": 0.000542751517969009,
    "per_seed_diffs": [
     0.00044560455717146397,
     0.000601749838097021,
     0.0004958258505212143,
     0.0005931293126195669,
     0.0005573933885898441,
     0.0005051088373875245,
     0.0005889362946618348,
     0.0005542640647036023
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.0081,
    "mde": 0.00004639948729729596,
    "mde_rel_pct": 28.17006303404585,
    "verdict": "idea worse (significant)",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "lr": 0.002
    },
    "sweep": [
     {
      "cfg": {
       "lr": 0.001
      },
      "mean": 0.00028059784381184727
     },
     {
      "cfg": {
       "lr": 0.002
      },
      "mean": 0.0001786892862583045
     },
     {
      "cfg": {
       "lr": 0.003
      },
      "mean": 0.00018342551447858568
     }
    ],
    "full": {
     "mean": 0.000164712047080684,
     "std": 0.00004489169459326852,
     "per_seed": [
      0.00020323635544627905,
      0.00021873260266147554,
      0.00013271479110699147,
      0.0001600733958184719,
      0.0001428304531145841,
      0.00023414976021740586,
      0.00011625964543782175,
      0.00010969937284244224
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.000707463565049693,
    "std": 0.00005862115619486818,
    "per_seed": [
     0.000648840912617743,
     0.0008204824407584965,
     0.0006285406416282058,
     0.0007532027084380388,
     0.0007002238417044282,
     0.0007392585976049304,
     0.0007051959400996566,
     0.0006639634375460446
    ],
    "n": 8
   },
   "mechanism_signature": {
    "confirmed": true,
    "observed_energy_first_last": [
     -2.293245553970337,
     -3.0686049461364746
    ],
    "observed_energy_nonincreasing_fraction": 1,
    "observed_grad_norm_first_last": [
     1.0507688522338867,
     0.8547213077545166
    ],
    "observed_local_energy_boundary": "not estimated (learned Hessian unavailable in budget)",
    "predicted": "energy should not increase under sufficiently small Euler steps",
    "predicted_boundary_eta_L": 2
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "harms",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 9,
   "verdict": "Built an energy-gradient flow and verified the mechanism on a quadratic energy. The observed Euler transition occurred at gamma=1.00 (eta*L=2): energy was monotone for gamma\u003c=1, increased above 1, and strongly diverged for gamma\u003e=1.2; measured contraction factors exactly matched |1-eta*lambda|, and the normalized descent coefficient was invariant across lambda_max. In the small denoising experiment, the idea preserved monotonic energy and reduced gradient norm, but its 12-step MSE (0.4641) was worse than the residual baseline (0.4081), so the stability mechanism worked without an accuracy win.",
   "metrics": {
    "baseline": "12-step denoising MSE 0.4081426; 4-step training loss 0.2311514",
    "idea": "12-step denoising MSE 0.4640646; 4-step training loss 0.2209387; energy nonincreasing fraction 1.0; gradient norm 1.1651 -\u003e 0.4993; cumulative path length 2.3558; Euler boundary observed gamma=1.00, predicted gamma=1.00; contraction prediction matched exactly"
   },
   "how_to_run": "python3 experiment.py",
   "files": [
    "experiment.py",
    "results.json"
   ],
   "limitations": "The learned-flow comparison used a tiny synthetic fixed-point denoising task rather than MNIST/CIFAR, only one seed, and a fixed step size rather than a full eta/Lhat sweep for the learned neural energy. Łojasiewicz convergence, compact sublevel-set assumptions, wall-clock speed, and large-scale accuracy were not tested."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-01T19:14:32",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}