Beyond Dual Flatness: Curvature Emergence via Anisotropic Metric Perturbations

Dually flat manifolds occupy a privileged position in information geometry, enabling closed-form geodesics, canonical divergences, and the celebrated Pythagorean theorem for Bregman divergences. Yet the structural conditions that preserve or destroy this flatness under metric perturbation remain incompletely characterized. We introduce a dichotomy framework distinguishing two classes of perturbation: isotropic perturbations, which uniformly scale the base divergence, and anisotropic point-dependent perturbations, which introduce directional selectivity modulated by position in parameter space. Our main result establishes that this dichotomy precisely characterizes the stability of the Pythagorean theorem: isotropic perturbations preserve the theorem for arbitrary perturbation magnitude, while anisotropic point-dependent perturbations violate it for all non-degenerate triples. The key mechanism is that point-dependent modulation induces non-vanishing derivatives in the Christoffel symbols, which generate curvature in the dual connections. We provide explicit formulas for the connection perturbation and the leading-order curvature tensor. Numerical experiments on 2D Gaussian (ε_max = 9.75) and 4-category categorical (ε_max = 0.435) manifolds confirm the theoretical predictions with machine-precision agreement. These results provide, within the studied perturbation family, a minimal sufficient condition for curvature emergence under the non-alignment condition (dφ ∧ α ≠ 0) and offer practical guidance for the design of geometrically faithful divergence modifications.