Entropy-calibrated hyperbolic curvature

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>=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.