Emergent Quadratic Stability and Recursive Z₄ Phase Structure as the Structural Origin of Quantum Propagators

We show that a discrete recursive classification framework naturally gives rise to quadratic stability, a Z₄ cyclic phase structure, and a Hilbert-like algebra, yielding Born-type probability rules and stationary path propagation without assuming Hilbert space a priori. The framework derives phase, frequency, energy, and wavelength as emergent invariants of identity-preserving recursive cycles. In the large-depth limit, recursive path statistics produce a Gaussian kernel consistent with short-time quantum propagators. This provides a minimal combinatorial and structural origin for key elements of quantum mechanics.