Tensor Hierarchy of Curvature Feedback — From Scalar β to Rank-4 and Beyond

Curvature feedback describes the intrinsic ability of a geometric system to restore coherence after disturbance. Earlier works (R81–R82) treated this process with a scalar feedback constant β governing exponential relaxation of curvature energy. The present study generalizes β into a hierarchy of tensors, each rank representing a higher level of geometric self-awareness. The scalar β defines isotropic decay; the vector βᵢ introduces directional flux; the rank-2 tensor βᵢⱼ describes coupled-axis elasticity; the rank-3 tensor βᵢⱼₖ governs nonlinear cross-coupling and handedness; and the rank-4 tensor βᵢⱼₖₗ encapsulates curvature-of-curvature feedback—the self-healing of spacetime itself. Across roughly thirty orders of magnitude in spatial scale, from protons to galaxies, the characteristic feedback rate follows a single power law β ∝ R⁻¹·¹, implying that all matter and spacetime belong to one continuous self-correcting continuum. Each tensor rank corresponds to known physical behavior: isotropic nuclear stability, magnetic and thermal anisotropy, nonlinear field mixing, and gravitational re-coherence. Finite βᵢⱼₖₗ ensures that curvature singularities cannot form; collapse halts when feedback saturates. Forces and fields emerge as visible expressions of this tensorial self-correction, uniting electromagnetism, nuclear binding, and gravitation under one geometric principle. The resulting framework portrays the universe not as a passive arena but as an active geometry that remembers and repairs itself. Reality is curvature feedback made visible.