Metastability and Timescale Separation in the EPPQ Framework: A Rigorous Derivation of Emergent Quantum Mechanics (O3 — Part III)

This work provides a rigorous mathematical foundation for metastability and timescale separation in the Emergent Pre-Quantizable (EPPQ) framework. Building on previous results (O3 Parts I and II), where quantum mechanics was derived under structural assumptions, this paper removes those assumptions by proving that: - Fixed points of the deterministic relational dynamics correspond to strict local minima of the interface functional. - Each minimum defines a basin of attraction with well-defined Lyapunov structure. - The discrete Hessian at each metamode is positive definite and admits a spectral gap. - Intra-basin relaxation is controlled by the Hessian gap, while inter-basin transitions follow a large deviation principle. - A strong separation of timescales (τᵣel ≪ τₑsc) emerges naturally from the underlying dynamics. These results complete the axiomatic derivation of emergent quantum mechanics within the EPPQ program, showing that the effective Markovian dynamics and Hilbert space structure arise from purely relational and deterministic principles. This paper corresponds to O3 — Part III of the EPPQ research program. Related works: - O3 Part I: https: //doi. org/10. 5281/zenodo. 19613313- O3 Part II: https: //doi. org/10. 5281/zenodo. 19622307- SOC-Θ 1: https: //doi. org/10. 5281/zenodo. 19596783- SOC-Θ 2: https: //doi. org/10. 5281/zenodo. 19598463- Lambda SOC: https: //doi. org/10. 5281/zenodo. 19599410