Abstract 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)引入广义相对论常面临数学不自洽的困境:传统的平直时空微扰方案不仅难以嵌入场方程框架,且必然导致引力势的相对振荡振幅被背景质量压低。为探讨这一问题,本文提出了一种“驱动-响应”有效场论框架。首先,基于弦网拓扑序中斐波那契任意子的代数结构,将DSI因子刚性锁定为黄金比例的平方 。随后,构建了一个时-径向扇区严格渐近平直的自治度规拟设,其中角向几何由DSI先验给定(驱动变量),而径向与时间几何作为响应变量。将该度规代入爱因斯坦方程反推发现,维持该自洽解所需的有效源项能量密度正比于背景质量与角向扰动的乘积, ,且与角向扰动呈反相耦合,这可被诠释为“拓扑-曲率非线性交叉耦合”的唯象体现。基于此框架,弱场极限自然给出具有常数相对振幅的对数周期牛顿势。在强场区,通过对视界方程进行一阶微扰展开与面积积分,发现径向视界偏移与角向度规扰动发生非线性部分抵消,最终使黑洞熵的相对修正表现为一个振幅恒定、周期由无理数锁定的对数周期振荡。该特征区别于常规的单调修正模型,并为LISA与EHT等未来观测提供了明确的可检验靶标。 关键词: 离散标度不变性; 有效场论; 黑洞熵; 对数周期修正; 黄金比例
zhengda 李 (Fri,) studied this question.