Emergent Geometry from Lyapunov Stability: Effective Metric Structure from the Second Variation of the RMB Stability Functional

This note develops a minimal and background-independent mechanism for emergent geometric description within the Space–Matter–Motion (RMB) theory.Starting from the RMB stability functional, it is shown that Lyapunov-stable attractors naturally induce an effective metric-like structure through the second variation. The resulting positive-definite quadratic form defines a stability metric on the configuration space of organized motion. The construction is pre-dynamical and does not assume a spacetime manifold, metric, or geometric postulates. Instead, geometric notions arise as secondary diagnostic structures encoding local stability properties of motion. Under fluctuations, the inverse stability operator admits a statistical interpretation in terms of perturbation correlations. This mechanism links directly to the RMB stability functional (Stability Principle Note) and to the master equation derived in the RMB Mathematical Core Note, providing a coherent bridge between stability, variational structure, and emergent geometry within the RMB framework. The work serves as a foundational note complementing the canonical RMB formulation and its subsequent dynamical and phenomenological applications.