Emergence of Quantum Mechanics from Relational Dynamics: Closing O3 in the EPPQ Framework

This work closes O3 (Emergence of Quantum Mechanics) in the EPPQ (Emergent Pre-Quantizable) framework by providing a mathematically controlled derivation of the Schrödinger equation from deterministic relational dynamics. Starting from the PMAH+ACR evolution established in O2, we construct an effective Markov chain on the space of metastable basins associated with low-energy metamodes. Under the separation of timescales and approximate detailed balance (both supported by prior analytical results and numerical simulations), the transition matrix M is shown to be a small perturbation of the identity. By symmetrization and spectral analysis, we prove that the logarithm of the transition operator is well-defined and yields an effective Hermitian generator. The dynamics decomposes into a skew-symmetric (unitary) component and a symmetric (dissipative) component. In the continuous-time limit, this leads to the Schrödinger equation: i ħₑff dψ/dt = Hₑff ψ where ψᵢ = √μᵢ arises from the coarse-grained probability distribution over basins, and ħₑff is identified (up to a dimensionless constant) with the fundamental time step τ_Θ. All approximations are explicitly controlled by the small parameter ε = 1/min τᵢ, and the emergence of unitary quantum evolution is shown to result from a controlled O (ε) breaking of reversibility induced by the historical asymmetry of the PMAH action. This work provides a rigorous bridge between deterministic relational dynamics and quantum mechanics, completing the dynamical foundation of the EPPQ program. --- Relation to the EPPQ program: - O2 (emergence of time and PMAH dynamics): https: //doi. org/10. 5281/zenodo. 19585980- SOC-Θ 1 (stationarity and avalanche mechanism): https: //doi. org/10. 5281/zenodo. 19596783- SOC-Θ 2 (criticality and scaling laws): https: //doi. org/10. 5281/zenodo. 19598463- λ-SOC (emergence of the inertia parameter): https: //doi. org/10. 5281/zenodo. 19599410 This paper should be read as the effective quantum layer emerging from the metastable structure established in the SOC-Θ series.