The Relational Emergence Model (REM) is a generative framework for the articulation of relational structure in quantum theory. While Relational Quantum Mechanics (RQM) defines physical states relative to other systems, it does not specify how subsystem structure itself becomes determined. REM addresses this gap by modeling subsystem differentiation as a variational selection process over candidate tensor-product factorizations of Hilbert space. The model introduces a three-layer architecture: Layer 0 — a pre-relational Hilbert space without a privileged tensor-product structureLayer 1 — differentiation as relational articulationLayer 2 — decoherence as stabilization of articulated structure Differentiation is formulated as an optimization problem on a factorization manifold, F ≅ U(N)/(U(n_A) ⊗ U(n_B)), with selection governed by a variational functional Φ = Φ_S + λ Φ_H, combining informational articulation (mutual information) and dynamical stability (boundary Hamiltonian contribution). The parameter λ controls the balance between informational correlation and dynamical coherence, admitting interpretations analogous to inverse temperature, timescale ratio, or renormalization-scale dependence. A three-qubit toy model demonstrates nontrivial factorization selection and a crossover between informationally favored and dynamically stabilized regimes. The framework connects relational quantum mechanics, decoherence theory, information geometry, subsystem structure in gauge theory, and variational optimization on quotient manifolds. The present work is not proposed as a complete physical theory but as a mathematically structured foundational proposal for studying how relational structure itself may become selected, stabilized, and rendered physically meaningful. The REM framework aims to clarify the generative layer underlying relational description and provides a basis for further development including continuous factorization flow, tensor-network formulations, categorical structure, and operational implementations.
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
Quantum theory does not uniquely determine subsystem structure. This paper proposes a variational principle on factorization space to select preferred tensor-product structure. The optimization objective balances informational articulation (mutual information across subsystem boundaries) and dynamical separability (interaction coupling across the partition). The framework is formulated on the coset manifold F ≅ U(N)/(U(n_A)⊗U(n_B)) and provides a generative extension of relational quantum mechanics (Rovelli 1996), in which subsystem structure itself is treated as a variational degree of freedom rather than a primitive assumption. A three-qubit illustration demonstrates regime competition with crossover at λ* ≈ 0.28, consistent with results in the REM series (REM2–REM4). This paper is the fifth in the Relational Emergence Model (REM) series.
REM_lambda Version 4 Version 4 presents a corrected numerical implementation of the variational selection framework used in the Relational Emergence Model (REM). The model evaluates candidate Hilbert-space factorisations through competition between informational differentiation and dynamical boundary cost.
Φ(F)=ΦS(F)−λCH(F) The informational term is
ΦS(F)=I(A:B) and the sign-definite quadratic dynamical cost is
CH(F)=Tr(ρH∂F2)=⟨H∂F2⟩ Numerical correction in Version 4 Version 4 corrects two implementation issues in the numerical analysis accompanying REM_lambda Version 3. The discrete-cut implementation previously calculated
⟨H∂F⟩2 instead of the intended quadratic boundary-fluctuation cost
⟨H∂F2⟩ The previous continuous-factorisation routine transformed the state and the full Hamiltonian together, making the dynamical cost invariant under the candidate unitary transformation. Consequently, the Version 3 optimiser effectively maximised mutual information rather than the complete REM functional. Version 4 corrects both issues and recomputes all affected numerical results. Three-qubit XY-chain benchmark The numerical benchmark uses the asymmetric three-qubit XY-chain Hamiltonian
H=J12(X1X2+Y1Y2)+J23(X2X3+Y2Y3)+h∑iZi The parameters are
J12=1.5,J23=0.6,h=0.2 The ground state is obtained through exact diagonalisation. For the two site-contiguous candidate partitions, the corrected dynamical cost is evaluated as
CH=⟨H∂F2⟩ At
λ=0.2 the best site-contiguous partition achieves
Φ=0.560 The crossover between the information-dominated and dynamics-dominated contiguous partitions remains
λ∗≈0.165 Continuous factorisation optimisation For each candidate unitary
U, the rotated Hamiltonian is
HU=UHU† It is decomposed into local and interaction components:
HU=HA(U)+HB(U)+H∂(U) The local components are obtained through partial-trace projection with an identity-component correction. The continuous optimisation evaluates the complete REM functional:
Φ(U)=I(A:B)Uψ−λ⟨ψU∣H∂(U)2∣ψU⟩ A search over a four-parameter generator submanifold recovers
Φ=1.05 This exceeds the best site-contiguous result:
Φ=0.560 The corrected numerical improvement is therefore
0.560→1.05 The previously reported Version 3 result
0.68→1.32 is withdrawn because it was obtained using an incorrect objective function. Factorisation-space dimension The full factorisation space is represented by
U(8)/(U(4)⊗U(2)) The tensor-product representation of
U(4)×U(2) has a one-dimensional kernel. The effective local-unitary subgroup therefore has dimension
16+4−1=19 Since
dimU(8)=64 the quotient manifold has real dimension
64−19=45 The present numerical implementation explores a four-parameter generator submanifold of this 45-dimensional quotient manifold. The recovered factorisation is a higher-scoring solution within the explored submanifold. It is not claimed to be the global optimum over the full factorisation manifold. Finite-size scaling Finite-size calculations were performed for
N=3,4,6,8 under weak-center, linear, and step coupling profiles. All finite-size values were recomputed using the corrected quadratic dynamical cost
CH=⟨H∂F2⟩ Validation and reproducibility The accompanying reproducibility package includes: exact diagonalisation for asymmetric XY chains; evaluation of site-contiguous candidate cuts; factorisation-dependent reconstruction of the interaction Hamiltonian; continuous optimisation on a four-parameter generator submanifold; finite-size scaling calculations; deterministic benchmark values; 16 automated tests; GitHub Actions continuous integration for Python 3.12 and 3.13. The regression tests verify: exact agreement with
⟨H∂F2⟩; the distinction between
⟨H∂F2⟩ and
⟨H∂F⟩2; factorisation dependence of the reconstructed boundary cost; consistency at the identity transformation; reproduction of corrected benchmark values; successful finite-size and output-generation workflows. Scope Version 4 supersedes the numerical results reported in Version 3. The theoretical REM functional and the use of the sign-definite quadratic boundary-fluctuation cost remain unchanged. Source and reproducibility GitHub repository: https://github.com/malko73/rem Version 4 DOI: https://doi.org/10.5281/zenodo.21427776
Calculus of Element Generated Sets for Quantum Mechanics - Emergent Hilbert Sectors from Generative Relational Structure This monograph develops a structural embedding of quantum mechanics within the framework of the Calculus of Element Generated Sets (CEGS) constituing a CEGS4QM, extending earlier exploratory work on quantum interference utilising CEGS into a unified generative formulation. Rather than treating Hilbert space, probability, tensor-product decomposition, or unitary structure as primitive ingredients of reality, the framework investigates how such structures may emerge from a deeper recursive accessibility organization governed by relational transport and constraint stabilization. Within this construction, quantum states appear as stabilized accessibility configurations generated through conjugate completion and compatibility-preserving transport structure. Ordinary quantum mechanics is recovered as a locally stabilized Hilbert-compatible regime embedded within a broader recursive hierarchy of generative accessibility sectors. Entanglement is reformulated not as a relation between fundamentally separate subsystems, but as a synchronization-preserving projection structure arising from unified generative accessibility organization. Compatibility structure, relational transport, and recursive conjugate completion together generate higher-order accessibility layers beyond ordinary Hilbert representations while preserving standard quantum-mechanical behavior within stabilized observational domains. The framework does not modify empirical quantum mechanics. Instead, it proposes that the Hilbert-space formalism itself may represent a locally stabilized observational structure emerging from a more general recursive generative substrate. The work describes a generalized structural framework based on a set based calculus (CEGS) for investigating emergent quantum representation, recursive accessibility geometry, and compatibility-preserving relational transport beyond primitive Hilbert ontology. Related: Following related articles may add significantly for its understanding: de Leeuw, N. (2026). CEGS4QM TSO & Hilbert Addendum: Bound and Free spinors. Zenodo. https://doi.org/10.5281/zenodo.20703671 de Leeuw, N. (2026). Identity Manifolds and Representation Spaces for Divergent Series - A Geometric Analytical Framework for Recursive Reconstruction of Infinite divergent Series . Zenodo. https://doi.org/10.5281/zenodo.20707284
We study subsystem-structure selection in small interacting quantum systems. We compare two criteria: (i) maximization of cross-boundary correlations and (ii) minimization of dynamical coupling across subsystem boundaries. Using exact diagonalization of asymmetric spin chains, we show that these criteria can favor different tensor factorizations of Hilbert space. We derive the crossover parameter λ* at which the preferred factorization switches. The existence of this regime demonstrates that subsystem structure need not be uniquely determined by dynamical stability alone. This paper is part of the Relational Emergence Model (REM) series (REM4 of 5).