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 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 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 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 (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.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
This work presents a rigorous, first-principles audit of unitarity in quantum mechanics, treating unitarity not as a foundational axiom but as a hypothesis whose validity depends on explicit physical and mathematical assumptions. The analysis is conducted entirely within the operational framework of open quantum systems, using completely positive trace-preserving (CPTP) maps to describe the dynamics of reduced subsystems. By systematically isolating and auditing every prerequisite required for unitary evolution—including perfect isolation, Hilbert-space factorization, linearity, the Born rule, and time parameterization—the manuscript demonstrates that unitarity is neither fundamental nor universal, but instead an emergent, scale-dependent consistency condition. The paper distinguishes clearly between mathematical bookkeeping identities and physically meaningful conservation laws, showing that probability conservation does not require unitarity, and that reversibility and energy conservation are not guaranteed by unitary evolution alone. Explicit, mathematically consistent non-unitary evolution models are constructed, including Lindblad dynamics, non-Hermitian Hamiltonians, stochastic Schrödinger equations, non-Markovian evolution, and quantum Darwinism. The work further identifies the precise regimes in which unitary dynamics emerges as an effective approximation—such as weak coupling, short timescales, or limited observational resolution—and delineates the conditions under which it fails. Empirical evidence from atomic spectroscopy, quantum information processing, decoherence experiments, and measurement theory is used to anchor the analysis without assuming fundamental unitarity. The central conclusion is unambiguous: no physically accessible open or effective subsystem undergoes strictly unitary evolution. Unitarity is shown to be an emergent, conditional, and scale-dependent property rather than a fundamental law of nature. This manuscript is intended as a foundational clarification of the status of unitarity in quantum theory and is suitable for researchers in quantum foundations, open quantum systems, quantum information, and quantum gravity. FUSION OS INNOVATIONS