CURVSPREAD: CRITICAL CONFINEMENT FOR HYPERBOLIC GRAPH SELF-SUPERVISION

Dispersion prevents collapse in self-supervised learning, but hyperbolic space has no finite radius at which spreading naturally stops. We show that whether spreading stops depends on the reference measure of the spreading law. A wrapped law, specified in tangent coordinates and pushed through the exponential map as in tangent-space isotropy objectives, always inherits a finite radial scale from the tangent law. An intrinsic law, specified against hyperbolic volume under linear radial confinement, has a finite equilibrium exactly when the confinement coefficient exceeds the volume-growth rate $(d-1)\sqrt{\kappa }$. The ratio of the two is thus a single critical coordinate across dimensions and curvatures, with the boundary at one. We introduce CurvSpread, which adds the exponential-map Jacobian to a local-spacing entropy surrogate and calibrates confinement to this rate, so that the confinement strength is set by the geometry rather than tuned per dataset. We prove that its finite-sample objective, before clipping, has exactly this threshold for every sample size. Across 25 dimension–curvature settings, the operational midpoints of the trained-encoder response are centered on the predicted boundary (median 1.00), while the sharpness of that response varies with geometry. One calibration, selected on three development graphs, is competitive on ten node-classification benchmarks.