{
 "artifacts": [
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1085/bench_report.json"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1085/report.md"
  },
  {
   "name": "report_bench_2026-09-02T000929.md",
   "url": "https://synthcore.org/code/1085/report_bench_2026-09-02T000929.md"
  },
  {
   "name": "stage2_bench.py",
   "url": "https://synthcore.org/code/1085/stage2_bench.py"
  },
  {
   "name": "topological_fluctuation.py",
   "url": "https://synthcore.org/code/1085/topological_fluctuation.py"
  }
 ],
 "category": "architecture",
 "description": "Replace a deterministic graph propagation layer by a stable stochastic linearized latent dynamics whose frequency-resolved covariance matrix defines spectral bands. Train or initialize the graph operator so that a selected covariance band has a nonzero Chern number and remains separated by a measurable spectral gap, producing representations that are robust to local perturbations and can support boundary-localized responses.",
 "download_zip": "https://synthcore.org/download/1085",
 "formulas_latex": [
  "$$\\dot z_k(t)=A(k)z_k(t)+\\xi_k(t),\\qquad \\operatorname{Re}\\lambda_j(A(k))\u003c0$$",
  "$$S(\\omega,k)=(-i\\omega I-A(k))^{-1}D(k)(i\\omega I-A(k)^\\dagger)^{-1}$$",
  "$$C_n(\\omega)=\\frac{1}{2\\pi}\\int_{\\mathrm{BZ}} i\\left(\\langle\\partial_{k_x}u_n|\\partial_{k_y}u_n\\rangle-\\langle\\partial_{k_y}u_n|\\partial_{k_x}u_n\\rangle\\right)d^2k$$",
  "$$g_n(\\omega)=\\min_k\\left[s_n(\\omega,k)-s_{n+1}(\\omega,k)\\right],\\qquad g_n=0\\ \\text{is the predicted transition boundary}$$"
 ],
 "id": 2743,
 "implementation": "1. Integration point: implement a latent state-space graph layer on an \\(L_x\\times L_y\\) periodic grid or mesh. Store a small channel-space drift operator \\(A(k)\\), obtained by Fourier-transforming a local graph or convolution stencil. Add antisymmetric chiral couplings between channels, while constraining the symmetric part to be negative definite, for example \\(A(k)=-(B(k)B(k)^\\dagger+\\epsilon I)+H(k)\\), where \\(H(k)\\) is skew-Hermitian. Inject Gaussian noise with covariance \\(D(k)\\), simulate short trajectories, and pass filtered latent states to the predictor. 2. Pseudocode: update \\(z_{t+\\Delta t}=z_t+\\Delta t A z_t+\\sqrt{\\Delta t}L\\varepsilon_t\\), with \\(LL^\\dagger=D\\); estimate \\(S(\\omega,k)\\) by averaging FFT periodograms; eigendecompose each \\(S(\\omega,k)\\); compute the Fukui lattice Chern number from normalized neighboring eigenvector overlaps; and add \\(\\lambda_g\\max(0,g_0-g_n)^2\\) to the task loss. Initially use the Chern number as a monitor rather than differentiating through its integer rounding, and optimize a smooth Berry-curvature or gap surrogate. 3. Computed from the mechanism: the resolvent spectrum, covariance eigenbands, band gap, Berry links, and Chern estimate. Estimated empirically: finite-sample spectra, useful frequency windows, and downstream accuracy. 4. First cheap experiment: use a two-channel graph convolution on a \\(16\\times16\\) periodic grid and train node classification on synthetic data whose labels depend on long-range wave propagation. Compare a deterministic GNN, a noisy GNN, and this covariance-band layer while sweeping chirality \\(q\\), noise strength \\(\\sigma^2\\), and observation frequency \\(\\omega\\). The quantitative prediction is that the measured Chern number remains constant while \\(g_n(\\omega)\u003e0\\), changes only near a frequency or noise value where \\(g_n\\) approaches zero, and produces stronger boundary localization after opening the torus into a strip. Locate the transition by the minimum covariance gap and require agreement within 20 percent; quantify localization using inverse participation ratio or boundary-to-bulk energy ratio.",
 "math_summary": "Let \\(z_k(t)\\in\\mathbb{C}^d\\) be the Fourier mode of a latent field on a two-dimensional periodic graph, \\(A(k)\\) its learned drift matrix, and \\(\\xi_k(t)\\) zero-mean noise with covariance \\(D(k)\\). The stable stochastic dynamics are \\(\\dot z_k=A(k)z_k+\\xi_k\\), with \\(\\operatorname{Re}\\lambda_j(A(k))\u003c0\\). At angular frequency \\(\\omega\\), the fluctuation spectrum is \\(S(\\omega,k)=G(\\omega,k)D(k)G(\\omega,k)^\\dagger\\), where \\(G(\\omega,k)=(-i\\omega I-A(k))^{-1}\\). Let \\(u_n(\\omega,k)\\) be a normalized eigenvector of covariance band \\(n\\), satisfying \\(S u_n=s_n u_n\\). Its Chern number is \\(C_n=(2\\pi)^{-1}\\int_{\\mathrm{BZ}}\\Omega_n(k)\\,d^2k\\), with Berry curvature \\(\\Omega_n=i(\\langle\\partial_{k_x}u_n|\\partial_{k_y}u_n\\rangle-\\langle\\partial_{k_y}u_n|\\partial_{k_x}u_n\\rangle)\\). The topology can change only when the covariance-band gap \\(g_n=\\min_k|s_n-s_{n+1}|\\) closes.",
 "math_tags": [
  "statistical-mechanics",
  "stochastic-processes",
  "spectral-theory",
  "differential-geometry",
  "dynamical-systems"
 ],
 "ml_areas": [
  "graph-nn",
  "ssm",
  "regularization",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.26055",
  "arxiv_url": "https://arxiv.org/abs/2608.26055",
  "summary_what_math_gives_to_ml": "The paper offers a nonstandard mechanism in which topology is carried by a frequency-resolved fluctuation or displacement covariance spectrum, even when the deterministic dynamics are topologically trivial. Its transferable asset is a computable covariance-band construction: eigenvectors of the stochastic response matrix can acquire nonzero Chern numbers, undergo gap-closing transitions as noise chirality or observation frequency changes, and generate boundary-localized fluctuation modes. A neural implementation should use stochastic latent dynamics on a two-dimensional periodic graph, estimate its frequency-resolved covariance bands, and either regularize or exploit their Chern structure for robust bulk representations and boundary-sensitive inference.",
  "title": "Topology of Fluctuation Bands in Chiral Active Matter",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 7,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Topological Fluctuation Graph Layer",
 "url": "https://synthcore.org/idea/2743/topological-fluctuation-graph-layer",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0.001892383472295478,
    "idea_mean": 0.0024836512457113713,
    "delta_mean": 0.0005912677734158933,
    "per_seed_diffs": [
     0.0001953578321263194,
     0.0009697097120806575,
     0.00078917620703578,
     0.0010487011168152094,
     0.0002084987936541438,
     0.00029010442085564137,
     0.0010423766216263175,
     0.00018621748313307762
    ],
    "idea_wins": 0,
    "n_pairs": 8,
    "p_value": 0.0081,
    "mde": 0.000339375496055173,
    "mde_rel_pct": 17.933759252478964,
    "verdict": "idea worse (significant)",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 18,
     "lr": 0.006,
     "weight_decay": 0
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 18,
       "lr": 0.0015,
       "weight_decay": 0
      },
      "mean": 0.009193785954266787
     },
     {
      "cfg": {
       "epochs": 18,
       "lr": 0.003,
       "weight_decay": 0
      },
      "mean": 0.0050034308223985136
     },
     {
      "cfg": {
       "epochs": 18,
       "lr": 0.006,
       "weight_decay": 0
      },
      "mean": 0.002262153459014371
     }
    ],
    "full": {
     "mean": 0.001892383472295478,
     "std": 0.0005824768770598943,
     "per_seed": [
      0.0016680180560797453,
      0.0018032685620710254,
      0.002757444279268384,
      0.0028198829386383295,
      0.0014833298046141863,
      0.0013202878180891275,
      0.0011803944362327456,
      0.0021064418833702803
     ],
     "n": 8
    }
   },
   "idea": {
    "mean": 0.0024836512457113713,
    "std": 0.0007913986756607849,
    "per_seed": [
     0.0018633758882060647,
     0.002772978274151683,
     0.003546620486304164,
     0.003868584055453539,
     0.00169182859826833,
     0.001610392238944769,
     0.002222771057859063,
     0.002292659366503358
    ],
    "n": 8
   },
   "mechanism_signature": {
    "confirmed": false,
    "observed_response_ratio": 1.0069190264772103,
    "predicted_response_ratio": "\u003c= 1.0",
    "prediction": "dissipative chiral latent dynamics should remain stable and reduce local perturbation response",
    "trained_model_observation": {
     "baseline": {
      "output_rms": 0.1667313128709793,
      "perturbation": 0.01,
      "response_rms": 0.00008569472265662625
     },
     "idea": {
      "output_rms": 0.16530634462833405,
      "perturbation": 0.01,
      "response_rms": 0.00008628764771856368
     }
    }
   },
   "custom_track": null
  },
  "mechanism_confirmed": false,
  "peer_reviewed": false,
  "practical_verdict": "harms",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 7,
   "verdict": "Built a readable toy topological fluctuation graph layer in topological_fluctuation.py, including stable chiral drift, resolvent covariance spectra, Fukui Chern estimation, Euler stability checking, covariance gaps, and open-strip localization. The mechanism manifested: the measured Euler threshold was 0.14752 versus the predicted 0.14752, Chern phases were 0,+1,+1,-1,-1,0 across the mass sweep, and chirality increased edge concentration from edge ratio/IPR 0.368/0.088 at q=0 to 1.000/1.000 at q=1. This is a promising toy verification, not evidence of downstream ML accuracy or a trained graph-layer win.",
   "metrics": {
    "baseline": "Nonchiral q=0 strip: edge ratio 0.368 and IPR 0.088.",
    "idea": "Chiral q=1 strip: edge ratio 1.000 and IPR 1.000; Chern sweep gave 0,+1,+1,-1,-1,0; stability-radius crossing was dt≈0.14752."
   },
   "how_to_run": "python3 topological_fluctuation.py",
   "files": [
    "topological_fluctuation.py"
   ],
   "limitations": "No neural-network training, synthetic node-classification task, finite-sample stochastic trajectory/periodogram estimation, learned operator optimization, perturbation robustness, or systematic frequency/noise sweep was performed. The strip diagnostic is a small Hamiltonian proxy for boundary localization, and finite momentum grids make exact transition-gap verification sensitive to whether gap-closing momenta are sampled."
  },
  "status": "failed_benchmark",
  "status_label": "Failed on benchmark",
  "updated_at": "2026-09-02T00:09:29",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}