A-4 Emergent Quantum-Like Features in Black Hole Ringdown from Foundational Scale Structure

This work examines the quantum-like features recently reported in black hole ringdown observations and provides a minimal structural framework for describing these features without introducing new fields or modifying general relativity. The approach is based on a foundational scale structure in which the metric is not treated as a primary quantity; instead, the local time scale T(x) and spatial scale L(x) serve as the fundamental variables. Within this framework, small variations in the gravitational potential δΦ are reparameterized in terms of fluctuations of a scale-structure parameter X(x). The parameter X(x) does not represent a new physical field but acts as an indicator characterizing the relation between T(x) and L(x). This minimal reparameterization allows the quantum-like amplitude and phase fluctuations observed in ringdown signals to be consistently associated with small changes in the underlying scale structure. The purpose of this work is strictly complementary: it does not propose a new theory of gravity, does not interpret the physical origin of the scale structure, and does not attempt to fully explain observational data. Instead, it provides the smallest structural extension necessary to accommodate the reported features while remaining fully compatible with both general relativity and quantum theory. This paper aims to offer a mathematically transparent and observationally consistent foundation for future theoretical and observational studies of black hole ringdown phenomena.