Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification

Abstract

We introduce farthest-cell triplet entropy, the conditional Shannon entropy of the farthest-prototype label given three random prototypes. For independent queries and prototypes, its estimator records only the farthest label, not coordinates or numerical distances. The statistic is bounded by $\log 3$, is invariant under common strictly increasing transformations of the dissimilarities, and has an exact mutual-information interpretation. In high-dimensional isotropic radial models $X_d=R_dU_d$, the Euclidean ordering reduces to scores $λ_dξ_{i,d}-Z_i$, where $λ_d=\sqrt d\,\operatorname{sd}(R_d)/\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\log 3$, $H_{\infty}(λ;F)$, and $0$. In hyperbolic space of curvature $-κ_d^2$, the same master curve appears at $λ_{d,\mathbb H}=\sqrt d\,τ_d A(s_d)$, where $τ_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$, $s_d=κ_d\mathbb{E} R_d$, and $A(s)=s\coth s$. With a calibrated radial law and $τ_d$, and a monotone operating interval, entropy inversion identifies the scale-invariant target $s_d^2$; absolute curvature requires an external length unit. CPU simulations give Euclidean and hyperbolic master-curve RMSEs of $0.0164$ and $0.0209$. Inversion from observed synthetic latent coordinates has a median relative error in $κ$ of $6.6\%$, while angular anisotropy increases this error to $68.9\%$. Thus the entropy statistic is comparison-based, whereas curvature recovery remains model-calibrated, is not graph-only, and is not robust to anisotropy.