Curvature and noise—a geometric framework for information stability

Classical measures of information stability—such as variance, entropy, and channel capacity—quantify uncertainty but do not explicitly describe the structural response of information under deformation. We propose a geometric formulation in which information channels are characterized by three invariant quantities: curvature, strain, and compressibility. Their ratio defines a scalar stability parameter governing equilibrium and instability within an informational manifold. Noise is interpreted as a curvature–strain imbalance that induces geometric flow and entropy production. A symbolic potential is introduced to characterize stability boundaries, yielding a convexity criterion that delineates reversible and irreversible information flow. When applied to Gaussian and binary channel models, the construction recovers qualitative stability features, such as monotonic degradation and characteristic transition regimes, that are consistent with classical signal-to-noise and redundancy trade-offs. The construction is not a substitute for probabilistic information theory. It is a reduced geometric diagnostic that tracks how stability degrades under deformation, and it is intended to be read alongside standard channel models. Assumptions and limitations are stated explicitly.