Logarithmic Periodic Spacetime with Curvature Coupling: A Driver-Response Ansatz and Constant-Amplitude Oscillations in Black Hole Entropy

In quantum gravity phenomenology, incorporating discrete scale invariance (DSI) into general relativity often faces mathematical inconsistencies. Conventional perturbative schemes around flat spacetime not only struggle to embed into the field equation framework but also inevitably lead to the relative oscillation amplitude of the gravitational potential being suppressed by the background mass. To address this issue, this paper proposes a “driver-response” effective field theory framework. First, based on the algebraic structure of Fibonacci anyons in string-net topological order, the DSI factor is rigidly fixed to the square of the golden ratio, . Subsequently, a self-consistent metric ansatz is constructed that is strictly asymptotically flat in the temporal-radial sector, where the angular geometry is prescribed a priori by DSI (driver variable), while the radial and temporal geometries respond as response variables. Substituting this metric into the Einstein equations reveals that the effective source term required to maintain this self-consistent solution has an energy density proportional to the product of the background mass and the angular perturbation, , and exhibits an opposite-phase coupling to the angular perturbation. This can be interpreted as a phenomenological manifestation of a “topology-curvature nonlinear cross-coupling.” Within this framework, the weak-field limit naturally yields a logarithmic periodic Newtonian potential with constant relative amplitude. In the strong-field region, through first-order perturbative expansion of the horizon equation and area integration, the radial horizon shift and the angular metric perturbation are found to undergo a nonlinear partial cancellation. Consequently, the relative correction to the black hole entropy manifests as a logarithmic periodic oscillation with constant amplitude and a period locked by the irrational number . This feature distinguishes itself from conventional monotonic correction models and provides a clear, testable target for future observations such as LISA and the Event Horizon Telescope (EHT). Keywords: Discrete scale invariance; Effective field theory; Black hole entropy; Logarithmic periodic correction; Golden ratio 摘要 在量子引力唯象学中,将离散标度不变性(DSI)