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 (τ_rel ≪ τ_esc) 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
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 (τ_rel ≪ τ_esc) 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
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 ħ_eff dψ/dt = H_eff ψ where ψ_i = √μ_i arises from the coarse-grained probability distribution over basins, and ħ_eff 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 τ_i, 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.
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: π_i M_{ij} = π_j M_{ji} + 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.
This work (O3 — Part II) completes the derivation of the quantum mechanical formalism within the Emergent Pre-Quantizable (EPPQ) framework by establishing the necessary emergence of the complex Hilbert space structure from purely relational dynamics. Building upon O3 — Part I, where the Schrödinger equation and the Born rule were derived conditionally from coarse-grained Markov dynamics on metastable basins, this paper addresses the remaining foundational gap: the origin of complex amplitudes and quantum phases. We demonstrate that a discrete U(1) gauge structure naturally arises from the contextual layer of the EPPQ vacuum, formalized as a principal bundle over the vacuum graph. The associated holonomies (Wilson lines) induce geometric phases along microscopic trajectories. By analyzing the interference of these trajectories, we prove that: • Transition amplitudes must take values in the complex field ℂ • The Born rule P = |ψ|² is the unique probability measure compatible with superposition and gauge invariance • The effective Hamiltonian emerges from the skew-symmetric component of the coarse-grained Markov generator and is directly determined by contextual holonomies Furthermore, we show that the same structure yields, in a specific spectral sector, the Yukawa matrices of the Standard Model, including nontrivial complex phases responsible for CP violation. The results establish that the complex structure of quantum mechanics is not an independent postulate, but a necessary consequence of relational holonomy and trajectory interference. This provides the missing link between the geometric constructions of O1/O4 and the emergent quantum dynamics of O3. This work is part of a broader program (EPPQ) aimed at deriving spacetime, quantum theory, and particle physics from a minimal relational ontology.
The EPPQ (Emergent Pre-Quantizable) framework seeks to derive spacetime andquantum mechanics from a purely relational substrate governed by a primordial distinctionα and the Principle of Absolute Historicity (PAH). While previous work (O1) established aconditional reconstruction of Riemannian geometry, the open problem O2—the emergenceof a causal, unidirectional temporal dynamics—has remained unresolved. The PAH, inits original form, is a static constraint; it prohibits ahistorical configurations but doesnot furnish a rule for selecting the actual historical trajectory among many admissibleones. This ambiguity is a manifestation of the Axiom of Choice in the relational setting:without a dynamical principle, the system’s evolution is either arbitrary or requires external,non-relational oracles. In this paper, we establish a rigorous local dynamics for O2 by elevating the PAH toa variational principle: the Principle of Minimal Historical Action (PMAH). We define adiscrete action functional that penalizes both local inconsistencies and the “forgetting” ofhistorical depth. We then introduce the Axiom of Relational Causality (ACR), which selectsthe physical, Markovian solution among the many non-causal extrema of the action. Underempirically motivated structural hypotheses (energy gap, monotonicity of the historicalcost), we prove a local well-posedness theorem: in the vicinity of a low-energy metamode,the PMAH+ACR dynamics is deterministic, uniquely defined, and converges exponentiallyto a fixed point via a contraction mapping. This result transforms the PAH from a passiverestriction into an active, time-generating engine. It resolves the local Axiom of Choiceambiguity and establishes the arrow of time as a consequence of historical dissipation. Thetheorem provides the rigorous dynamical foundation required for the emergence of quantummechanics (O3) and, ultimately, for closing the geometric problem O1.
Relational Quantum Mechanics (RQM) proposes that physical states exist only relative to other systems, but does not explain how such relational structures arise. This paper introduces the Relational Emergence Model (REM), which addresses this foundational gap by modeling the generation of relational structure as an explicit physical process. The central contribution is the differentiation map 𝒟 — not a unitary operator, but a structural bipartition-selecting map that produces explicit system–observer correlations from a pre-relational Hilbert space (Layer 0). Environmental decoherence 𝓔 then acts as a CPTP map on the articulated structure, stabilizing effectively classical facts (Layer 2). This three-layer architecture (pre-relational potential → relational articulation → phenomenal stabilization) provides a mathematically structured account of how relational reality becomes determinate. REM is compared with RQM, QBism, and Many-Worlds interpretations. Unlike these approaches, REM does not presuppose that relational structures are already formed — it models the generative phase that precedes perspectival consistency. The framework also provides a natural analysis of Wigner's Friend scenarios as multi-level differentiation processes. A structural correspondence (not equivalence) with Buddhist concepts of dependent origination (pratītyasamutpāda) and emptiness (śūnyatā) is identified: Layer 0 corresponds to the absence of intrinsic nature prior to differentiation; Layer 1 corresponds to the arising of relational structure through differentiation events. --- This is Paper 1 of the 6-paper Relational Emergence Model (REM) series: - REM1 (this paper) — Foundational framework: three-layer architecture and differentiation map- REM2 — Selection principles for the differentiation map (variational formulation) https://doi.org/10.5281/zenodo.19155166- REM3 — Generative differentiation via variational selection on factorization manifold https://doi.org/10.5281/zenodo.19157709- REM4 — Competing informational and dynamical criteria for subsystem structure https://doi.org/10.5281/zenodo.19637584- REM5 — Variational selection of tensor-product structure in quantum systems https://doi.org/10.5281/zenodo.19637651- REM_lambda — Variational generation of relational structure (physical interpretation of λ) https://doi.org/10.5281/zenodo.19641344
This work introduces a minimal variational framework in which localized energy basins and their fixed points form the foundation for emergent quantum dynamics. Rather than imposing quantization axiomatically, discrete spectra arise naturally from geometric regularity, topological invariants, and adjacency-driven transport within a continuous relational configuration space. Key contributions include: Emergent Schrödinger Dynamics: Oscillatory modes near basin fixed points admit a complex structure, leading to a Schrödinger-type evolution derived from a quadratic variational principle rather than canonical quantization. Topology-Induced Quantization: Integer-valued invariants such as winding and linking numbers enforce discrete excitation spectra without probabilistic postulates. Hydrogenic Consistency: The framework reproduces the hydrogen spectrum with emergent parameters, matching experimental wavelengths to within 0.1 nm. Conceptual Foundations: Spacetime is treated as an emergent descriptor of basin adjacency and deformation, not as a pre-imposed manifold. This approach provides a conservative and mathematically rigorous route to quantum-like behavior grounded in geometry and topology, while remaining compatible with classical correspondence in the coarse-grained limit. Extensions to relativistic regimes, gauge structures, and gravitational coupling are deferred to future work. Keywords: Emergent quantum dynamics, fixed-point geometry, basin framework, topology-induced quantization, Schrödinger evolution, variational principles.