{
 "artifacts": [
  {
   "name": "README.md",
   "url": "https://synthcore.org/code/1221/README.md"
  },
  {
   "name": "bench_experiment.py",
   "url": "https://synthcore.org/code/1221/bench_experiment.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1221/bench_report.json"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1221/report.md"
  },
  {
   "name": "report_bench_2026-09-03T124505.md",
   "url": "https://synthcore.org/code/1221/report_bench_2026-09-03T124505.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1221/results.json"
  },
  {
   "name": "run_experiment.py",
   "url": "https://synthcore.org/code/1221/run_experiment.py"
  }
 ],
 "category": "architecture",
 "description": "Use several phase-locked states as distinct attractors of one recurrent network and shape their basin asymmetry through the phase-delay composite spectrum. This creates a controllable associative-memory architecture in which a desired memory receives a larger basin without adding a separate classifier or explicit nearest-neighbor lookup.",
 "download_zip": "https://synthcore.org/download/1221",
 "formulas_latex": [
  "$$C^{(q)}_{ij}=A_{ij}\\cos(\\psi_j^{(q)}-\\psi_i^{(q)}-\\alpha_{ij}),\\qquad L^{(q)}=\\operatorname{diag}(C^{(q)}\\mathbf 1)-C^{(q)}$$",
  "$$r_q=K\\min_{k\\ne\\mathrm{gauge}}\\operatorname{Re}\\lambda_k\\!\\left(L^{(q)}\\right),\\qquad \\|\\delta\\theta(t)\\|\\approx e^{-r_qt}\\|\\delta\\theta(0)\\|$$",
  "$$\\mathcal L_{\\mathrm{basin}}=\\sum_{q=1}^{M}[\\gamma_q-r_q]_+^2+\\lambda_b\\sum_{q=1}^{M}(\\widehat B_q-b_q)^2$$"
 ],
 "id": 3036,
 "implementation": "1. Integration point: implement an associative-memory neural ODE or recurrent layer with one shared graph \\(A\\), shared delay matrix \\(\\alpha\\), and \\(M\\) prototype phase patterns \\(\\psi^{(q)}\\). Decode the attractor index from the final phase state with a cosine-similarity head. 2. Pseudocode: initialize prototypes from class-conditioned latent embeddings; for every prototype compute \\(C^{(q)}\\), \\(L^{(q)}\\), and the smallest non-gauge real eigenvalue \\(r_q\\); simulate short trajectories from noisy versions of each prototype; identify the converged attractor; estimate \\(\\widehat B_q=N_q/N\\) from uniformly sampled initial states; optimize task loss plus \\(\\mathcal L_{\\mathrm{basin}}\\). To enlarge one target basin, increase its requested \\(\\gamma_q\\) or target \\(b_q\\), while retaining positive margins for other memories. 3. Computed from the paper's mechanism: phase-delay composite spectra and their relationship to multistability and basin organization. Estimated empirically: global basin fractions, because the exact spectrum-to-volume mapping is network-dependent and should not be assumed universal. 4. First cheap experiment: use an 8-dimensional phase attractor memory with four prototype patterns and noisy-MNIST class embeddings. Compare unconstrained training with spectral basin allocation using 10,000 random initial states per model. The predicted signature is monotonic: increasing \\(r_q\\) should increase the measured basin fraction \\(\\widehat B_q\\), and attractors with larger \\(r_q\\) should show perturbation-recovery slopes near \\(-r_q\\) on a semilog plot. Falsify the mechanism if basin fraction fails to correlate positively with \\(r_q\\) across at least 20 independently initialized graphs.",
 "math_summary": "Let \\(q\\in\\{1,\\ldots,M\\}\\) index target phase-locked patterns \\(\\psi^{(q)}\\). For each pattern define \\(C^{(q)}_{ij}=A_{ij}\\cos(\\psi_j^{(q)}-\\psi_i^{(q)}-\\alpha_{ij})\\) and \\(L^{(q)}=\\operatorname{diag}(C^{(q)}\\mathbf 1)-C^{(q)}\\). The local contraction rate of attractor \\(q\\) is \\(r_q=K\\min_{k\\ne\\mathrm{gauge}}\\operatorname{Re}\\lambda_k(L^{(q)})\\), and its slowest local mode has time constant approximately \\(1/r_q\\). The paper's transferable mechanism is that this same composite spectrum organizes basin sizes, so spectral shaping can be used as a basin-control proxy. Define desired margins \\(\\gamma_q\\) and basin targets \\(b_q\\), and optimize \\(\\mathcal L=\\sum_q[\\gamma_q-r_q]_+^2+\\lambda_b\\sum_q(\\widehat B_q-b_q)^2\\), where \\(\\widehat B_q\\) is the measured fraction of initial states converging to attractor \\(q\\).",
 "math_tags": [
  "dynamical-systems",
  "spectral-theory",
  "linear-algebra",
  "bifurcations"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics",
  "embedding"
 ],
 "paper": {
  "arxiv_id": "2609.02047",
  "arxiv_url": "https://arxiv.org/abs/2609.02047",
  "summary_what_math_gives_to_ml": "The paper offers a nonstandard spectral mechanism in which a composite matrix combining network connectivity and heterogeneous phase delays predicts both linear stability and basin sizes of phase-locked states. This is transferable to phase-based recurrent or neural-ODE architectures by treating learned interactions as a graph with trainable phase offsets and constraining the spectrum of the corresponding cosine-weighted interaction matrix. The most direct implementation is a spectral-margin regularizer and initialization procedure for attracting latent memories or multiple dynamical modes. Its predictions are falsifiable: divergence should occur near the predicted eigenvalue boundary, while basin frequencies should change systematically when the composite spectrum is reshaped.",
  "title": "Phase-delays shape multistability and basin sizes in Kuramoto networks: analytical estimates from network structure",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 7,
  "novelty": 8,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Spectral Basin Allocation for Multimodal Neural Memories",
 "url": "https://synthcore.org/idea/3036/spectral-basin-allocation-for-multimodal-neural-memories",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0,
    "idea_mean": 0,
    "delta_mean": 0.0000705174,
    "per_seed_diffs": [],
    "idea_wins": 1,
    "n_pairs": 8,
    "p_value": 0.1228,
    "mde": 0,
    "mde_rel_pct": null,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": {
     "epochs": 10,
     "lr": 0.006,
     "weight_decay": 0.0001
    },
    "sweep": [
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.001,
       "weight_decay": 0
      },
      "mean": 0.0220816971
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.001,
       "weight_decay": 0.0001
      },
      "mean": 0.0219710653
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.003,
       "weight_decay": 0
      },
      "mean": 0.0036966768
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.003,
       "weight_decay": 0.0001
      },
      "mean": 0.0036578036
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.006,
       "weight_decay": 0
      },
      "mean": 0.0028201772
     },
     {
      "cfg": {
       "epochs": 10,
       "lr": 0.006,
       "weight_decay": 0.0001
      },
      "mean": 0.0027613395
     }
    ],
    "full": {
     "mean": 0,
     "std": 0,
     "per_seed": null,
     "n": 0
    }
   },
   "idea": {
    "mean": 0,
    "std": 0,
    "per_seed": [],
    "n": 0
   },
   "mechanism_signature": {
    "confirmed": true,
    "phase_gap_change": -0.1730552,
    "rate_change": 0.1695761
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "no_effect",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 7,
   "verdict": "Built a reproducible 12-oscillator delayed-Kuramoto associative-memory MVP with composite-Laplacian spectral rates, phase-locked prototypes, direct integration, and empirical basin classification. The local contraction claim was clearly verified: predicted rate 0.5070508 matched the Jacobian rate 0.5070508, and the measured semilog decay slope was -0.50834. Coupling reallocation increased the target q=1 basin from 0.0777 to 0.1677, demonstrating controllable basin asymmetry, but its local contraction rate decreased from 0.3660 to 0.2588, so the stronger universal rate-to-basin claim was not supported.",
   "metrics": {
    "baseline": "weights [0.5, 0.5]; q=1 rate 0.3660; q=1 basin 0.0777; q=0 basin 0.9223",
    "idea": "weights [0.9, 0.1]; q=1 rate 0.2588; q=1 basin 0.1677; q=0 basin 0.8323; math relative rate error 1.97e-15"
   },
   "how_to_run": "python3 run_experiment.py",
   "files": [
    "run_experiment.py",
    "results.json",
    "README.md"
   ],
   "limitations": "This MVP uses manually selected coupling weights rather than optimizing the proposed basin-aware objective, synthetic phase-gradient prototypes rather than MNIST embeddings, and no learned neural layer. The 20-graph correlation check is weak because only two attractors had positive rates in the sampled systems; no 10,000-sample or GPU experiment was performed."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-03T12:45:05",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": true,
    "tested": true
   },
   "practical_benchmark": {
    "beats_baseline": false,
    "tested": true,
    "verdict": "no_effect"
   },
   "toy_mechanism_gate": {
    "confirmed": true,
    "tested": true
   }
  }
 }
}