Dimensional Behavior in Recursive Geometry: Stick Theory and the Collapse of Structure into Flow

Stick Theory is a recursive geometrical framework that models emergent dimensional behavior through the structural collapse of connected linear elements. Unlike string theory, which relies on vibrational abstraction in higher-dimensional space, Stick Theory derives motion, fluidity, and field-like behavior from recursive saturation of rigid structure. This paper introduces the Faulkner Structural Motion Equation (FSME) and the Dimensional Flow Threshold (DFT), providing a complete mathematical foundation for modeling motion as the derivative of recursive identity collapse. The theory is supported by mechanical analogs such as the Hoberman sphere, offering a testable and visually grounded alternative to smooth-field physics.