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 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 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.
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
Deterministic Quantum Mechanics (DQM) is a research program investigating whether the probabilistic content of quantum mechanics can be understood as an emergent statistical description of an underlying deterministic physical theory. This paper consolidates the program’s cumulative results into a single account.The central proposal is that a complete microscopic state evolves deterministically, that the quantum state is a coarse-grained ensemble descriptor rather than the complete physical state, and that measurement outcomes are selected by which deterministic basin of an enlarged phase space a microscopic trajectory occupies. Under this proposal, Born probabilities are basin measures rather than fundamental stochastic laws.For a qubit, this program is realized exactly: a hidden variable λ on the unit sphere, a preparation-dependent density μ_r(λ) = [r·λ]₊/π, and the deterministic response A(n,λ) = sgn(n·λ) reproduce the qubit Born rule P(±|r,n) = (1 ± r·n)/2 exactly, together with sequential-measurement statistics and the associated expectation values. This preparation measure is itself shown to be the deterministic pushforward of a uniformly distributed microscopic preparation coordinate, removing any appeal to fundamental preparation-level stochasticity.Measurement is then developed as deterministic basin selection within a finite-time Hamiltonian architecture, with a finite-time stability theorem governing when microscopic perturbations cannot dislodge a trajectory from its basin. Entangled correlations are reproduced by an autonomous, rotationally covariant Bell Hamiltonian containing dynamical setting registers and a conserved-momentum phase shear; this mechanism reproduces the singlet correlation E(a,b) = −a·b and the CHSH value 2√2 in its ideal limit, while preserving exactly uniform local marginals at finite setting inertia and yielding an explicit, quantitatively bounded finite-inertia correction. A finite Hamiltonian bath (Model-C) supplies a deterministic amplification and record-retention mechanism; its analytic structure — including an exact completed-square cancellation of the static bath backreaction and an exact finite-bath spectral equation — is derived independently of, and is shown to survive, a documented numerical implementation failure (the C14 program), which is retained in the record as a diagnosed, non-Hamiltonian right-hand side rather than as evidence against the underlying construction.Extending the program beyond the qubit exposes a genuine mathematical obstruction: no deterministic basin defined directly on the qutrit pure-state manifold ℂP² can reproduce an arbitrary rank-one Born probability, because positivity forces any exact basin onto a measure-zero point. This is proved rigorously. The resolution is not the abandonment of determinism but the enlargement of the hidden state beyond the quantum ray itself, to a genuine mechanical phase space such as the cotangent bundle T*SU(3), on which the quantum state re-emerges as a coarse-grained moment and on which contextual, Kochen–Specker-compatible measurement dynamics can in principle be realized.Throughout, the paper enforces a strict evidentiary hierarchy — mathematical construction, analytic theorem, numerical validation, and experimental confirmation are never conflated — and closes with an explicit account of what has been proved, what remains open, and what would be required to render the program experimentally falsifiable.
Within CPQ theory, we propose that Planck's constant can be understood as the minimal action quantum of the discrete matrix, given in scale by the product of the minimal excitation energy and the minimal time step, \hbar \sim E_{min} \cdot t_P. This identification, together with resonant boundary conditions in the localising \Delta V well, leads naturally to the relation E = \hbar\omega, the uncertainty principle as the Nyquist sampling limit of a discrete medium, and low-energy Schrödinger dynamics. The article builds on the derivation of the de Broglie wave from the gradient coupling [2] and structurally closes the chain: discreteness of the matrix → minimal action → resonance → E = \hbar\omega → uncertainty principle → Schrödinger. Porschová, A. (2026). CPQ Theory — A Unified Framework from Spacetime Quanta to Cosmology (Books I–III). Zenodo. https://doi.org/10.5281/zenodo.18686137 Porschová, A. (2026). CPQ Field Theory — Applications (Books 01–03). Zenodo. https://doi.org/10.5281/zenodo.19100819 Porschová, A. (2026). The Matrix — Parameters and Dynamics: Foundation of CPQ Theory (3.1). Zenodo. https://doi.org/10.5281/zenodo.19431867