{
 "artifacts": [
  {
   "name": "bench_entropy_curvature.py",
   "url": "https://synthcore.org/code/1226/bench_entropy_curvature.py"
  },
  {
   "name": "bench_report.json",
   "url": "https://synthcore.org/code/1226/bench_report.json"
  },
  {
   "name": "entropy_curvature.py",
   "url": "https://synthcore.org/code/1226/entropy_curvature.py"
  },
  {
   "name": "report.md",
   "url": "https://synthcore.org/code/1226/report.md"
  },
  {
   "name": "report_bench_2026-09-03T130830.md",
   "url": "https://synthcore.org/code/1226/report_bench_2026-09-03T130830.md"
  },
  {
   "name": "results.json",
   "url": "https://synthcore.org/code/1226/results.json"
  }
 ],
 "category": "geometry",
 "description": "Use farthest-triplet entropy as a low-bandwidth observable of whether a hyperbolic embedding is angular- or radial-dominated, then adapt the hyperbolic curvature rather than fixing it arbitrarily. In the isotropic shell regime, invert the entropy-to-signal curve to estimate the effective radial/angular parameter and select curvature that reaches a chosen geometric operating point.",
 "download_zip": "https://synthcore.org/download/1226",
 "formulas_latex": [
  "$$\\lambda_d=\\sqrt d\\,\\frac{\\operatorname{sd}(R_d)}{\\mathbb{E}R_d}=\\sqrt d\\,\\tau_d,$$",
  "$$S_i(\\lambda)=\\lambda\\xi_i-Z_i,\\qquad H_\\infty(\\lambda;F)=\\mathbb{E}\\left[-\\sum_{i=1}^{3}p_i(\\lambda)\\log p_i(\\lambda)\\right],$$",
  "$$\\lambda_{d,\\mathbb H}=\\sqrt d\\,\\tau_d A(s_d),\\qquad s_d=\\kappa_d\\mathbb{E}R_d,\\qquad A(s)=s\\coth s,$$",
  "$$\\hat H\\xrightarrow{H_\\infty^{-1}}\\hat\\lambda,\\qquad \\hat\\lambda=\\sqrt d\\,\\hat\\tau\\,\\hat s\\coth(\\hat s),\\qquad \\hat\\kappa=\\frac{\\hat s}{\\hat\\mu_R}.$$"
 ],
 "id": 3051,
 "implementation": "Integrate this as an outer-loop controller for a hyperbolic embedding model, such as a Poincare-ball encoder, hyperbolic prototypical classifier, or hyperbolic graph neural network. Every K optimizer steps, draw query embeddings z_b and three prototype embeddings, compute hard farthest labels using the hyperbolic distance, and estimate H_hat from empirical cell frequencies with Laplace smoothing. Estimate dimension d from the embedding width and estimate mu_hat_R and tau_hat=std(R)/mean(R) from current prototype or query radii; use exponential moving averages to suppress noise. Before training, numerically tabulate H_infinity(lambda;F) by Monte Carlo: sample three independent standard-normal angular variables xi_i and three independent radial variables Z_i from the chosen calibrated law F, select argmax_i(lambda xi_i-Z_i), and interpolate entropy over a grid of lambda. Invert the table only on a validated monotone interval. Solve g(s)=sqrt(d) tau_hat s coth(s)-lambda_hat=0 by bisection for s\u003e=0, then update curvature toward kappa_hat=s/mu_hat_R using a slow EMA or optimize log(kappa). The mathematical quantities are the entropy statistic, master curve, and curvature amplification s coth(s); empirical estimates are H_hat, mu_hat_R, tau_hat, and anisotropy diagnostics. First test on synthetic isotropic tree-like data and MNIST or CIFAR-10 hyperbolic classification with fixed-curvature baselines, sweeping initial curvatures. Measure validation accuracy, calibration, hard farthest entropy, estimated-versus-true curvature, and training stability. Success is lower validation loss or better accuracy at equal dimension, convergence to the known synthetic curvature regime, and reduced sensitivity to initialization. Run a separate anisotropic-data test; degradation there is an expected falsifiable failure mode.",
 "math_summary": "Let X_d=R_d U_d be an isotropic d-dimensional shell, with tau_d=sd(R_d)/E[R_d]. In the Euclidean high-dimensional approximation, farthest ordering is represented by scores S_i(lambda)=lambda xi_i-Z_i, where xi_i are approximately standard Gaussian angular fluctuations, Z_i are independent radial/noise variables, and lambda_d=sqrt(d) tau_d. Define H_infinity(lambda;F) as the expected entropy of the winning index among three independent scores, with F denoting the calibrated law of Z_i. In hyperbolic space of curvature -kappa_d^2, let s_d=kappa_d E[R_d] and A(s)=s coth(s). The paper gives the effective parameter lambda_d,H=sqrt(d) tau_d A(s_d). Therefore, after estimating observed entropy H_hat, invert the monotone master curve to obtain lambda_hat=H_infinity^{-1}(H_hat;F), then solve lambda_hat=sqrt(d) tau_hat s coth(s) for s and set kappa_hat=s/E_hat[R]. This is model-calibrated and should not be trusted under strong angular anisotropy.",
 "math_tags": [
  "information-theory",
  "probability",
  "metric-geometry",
  "differential-geometry"
 ],
 "ml_areas": [
  "embedding",
  "graph-nn",
  "training-dynamics",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2609.02362",
  "arxiv_url": "https://arxiv.org/abs/2609.02362",
  "summary_what_math_gives_to_ml": "The paper introduces a comparison-based statistic for the geometry of learned representations: the conditional entropy of the farthest prototype among three, averaged over queries and prototype triples. Its useful properties are that it needs only argmax comparisons, is invariant to monotone rescaling of distances, and detects whether prototype cells are balanced or dominated by radial variation. In high-dimensional isotropic shells, the statistic is governed by a one-dimensional signal-to-noise parameter, while hyperbolic curvature amplifies radial variation through the explicit factor A(s)=s coth s. The most promising neural uses are a farthest-neighbor geometry regularizer and an entropy-based controller for adaptive hyperbolic curvature, with anisotropy explicitly treated as a failure mode.",
  "title": "Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "stability",
  "sample-efficiency"
 ],
 "title": "Entropy-calibrated hyperbolic curvature",
 "url": "https://synthcore.org/idea/3051/entropy-calibrated-hyperbolic-curvature",
 "verification": {
  "benchmark": {
   "track": "dynamics",
   "model": "rnn_small",
   "n_seeds": 8,
   "metric_direction": "lower is better",
   "comparison": {
    "baseline_mean": 0,
    "idea_mean": 0,
    "delta_mean": -0.0005000870587537065,
    "per_seed_diffs": [
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     0.0006857123225927353,
     -0.0009587501408532262
    ],
    "idea_wins": 7,
    "n_pairs": 8,
    "p_value": 0.05505,
    "mde": 0.0005172797403971514,
    "mde_rel_pct": 24.56429711910889,
    "verdict": "no significant win",
    "system_worked": false
   },
   "baseline": {
    "best_cfg": null,
    "sweep": null,
    "full": {
     "mean": 0,
     "std": 0,
     "per_seed": [
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      0.001437882543541491,
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      0.002295783953741193,
      0.0024020641576498747,
      0.0024898042902350426,
      0.0020851334556937218,
      0.0025566271506249905
     ],
     "n": 0
    }
   },
   "idea": {
    "mean": 0,
    "std": 0,
    "per_seed": [
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     0.0015978770097717643
    ],
    "n": 0
   },
   "mechanism_signature": {
    "confirmed": true,
    "observed_entropy_curve_recovered_lambda": 1.1249358995152312,
    "predicted_lambda": 1.1249358995152312,
    "trained_model_curvature_mean": 0.056044551169079165,
    "trained_model_entropy_mean": 0.7866021251571524
   },
   "custom_track": null
  },
  "mechanism_confirmed": true,
  "peer_reviewed": false,
  "practical_verdict": "no_effect",
  "stage1_mechanism_check": {
   "worked": true,
   "confidence": 7,
   "verdict": "Built a NumPy/SciPy MVP for the conditional farthest-triplet entropy master curve, monotone inversion, curvature amplification A(s)=s coth(s), and a slow EMA curvature controller. The core claim was numerically observed: entropy decreased strictly over the tested lambda grid (H(0)=1.0986, H(5)=0.2104), and the controller reduced mean absolute curvature error from 1.3667 for fixed initializations to 0.0204. Anisotropy produced inference errors up to 0.219, supporting the stated robustness limitation; the apparent controller win is synthetic and model-consistent rather than evidence from a trained neural embedding.",
   "metrics": {
    "baseline": "Fixed curvature at initial kappas [0.15, 1.0, 4.0]: mean absolute error to target kappa=1.3667.",
    "idea": "Entropy-inverted curvature EMA: final kappas [0.7539, 0.7569, 0.8004] for target 0.75; mean absolute error=0.0204. Entropy curve decreasing fraction=1.0, with H(lambda=0)=1.0986 and H(lambda=5)=0.2104. Under anisotropic angular scales [1.0,2.0,0.5], inferred-lambda absolute errors were 0.092, 0.019, and 0.219 at true lambda [0.5,1.5,3.0]."
   },
   "how_to_run": "python3 entropy_curvature.py",
   "files": [
    "entropy_curvature.py",
    "results.json"
   ],
   "limitations": "No trained Poincare-ball encoder, MNIST/CIFAR classification, validation accuracy, calibration, wall-clock comparison, or real hyperbolic-distance experiment was tested. The master curve assumes a standard-normal calibrated radial law and isotropic angular behavior; the controller target entropy is generated from the same assumed model, so the recovery result does not establish an end-to-end ML improvement."
  },
  "status": "mech_ok_no_baseline",
  "status_label": "Mechanism confirmed, baseline not beaten",
  "updated_at": "2026-09-03T13:08:30",
  "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
   }
  }
 }
}