Approximate Reversibility and the Emergence of Unitarity in the EPPQ Framework (O3 — Part IV)

This article constitutes Part IV of the O3 program within the Emergent Pre-Quantizable (EPPQ) framework, completing the axiomatic derivation of quantum mechanics from a deterministic relational substrate. The EPPQ program is based on a minimal relational ontology defined by a primordial distinction (α), the Principle of Absolute Historicity (PAH), and a local consistency operator Θ governing the evolution of configurations. Previous parts of O3 established that low-energy dynamics organize into metastable basins with strong timescale separation, and that—under an assumption of approximate reversibility—the Schrödinger equation, the Born rule, and the complex Hilbert space structure emerge. The present article removes this final assumption and proves it as a theorem. Specifically, we show that the effective Markov chain describing transitions between metastable basins is Nearly Completely Decomposable (NCD) with parameter ε ≪ 1, and satisfies the approximate detailed balance condition: πᵢ M₈₉ = πⱼ M₉₈ + O (ε²) The proof combines: - Simon–Ando theory of NCD Markov chains- Large deviation estimates of inter-basin transitions- Controlled asymmetry of the PMAH (Principle of Minimal Historical Action) Key results include: • Rigorous identification of the EPPQ effective dynamics as an NCD Markov system • Derivation of approximate reversibility with explicit O (ε²) bounds • Justification of the symmetrization S = D M D⁻¹ • Emergence of a Hermitian effective Hamiltonian • Proof that unitarity is exact in the thermodynamic limit • A deterministic mechanism for decoherence via the dissipative sector This result closes the final structural gap in O3 (Parts I–IV), establishing that unitary quantum mechanics is not fundamental, but emerges from deterministic relational dynamics. Additionally, the work identifies a residual irreversibility sector (O (ε²) ) that may encode non-abelian gauge structures, suggesting a pathway toward the emergence of the Standard Model gauge sector in O4. The EPPQ framework thus provides a unified program in which: - spacetime, - quantum mechanics, - and gauge interactions emerge from a common relational foundation.