The Relational Emergence Model (REM) extends Relational Quantum Mechanics (RQM) by treating relational structure as *generated* rather than assumed. This companion paper addresses the central open question left in the main REM framework: what selects the differentiation map 𝒟 that articulates a specific tensor-product decomposition out of the pre-relational Hilbert space (Layer 0 → Layer 1). We introduce a variational formulation over candidate factorizations ℱ, with functionals based on von Neumann entropy maximization (Φ_S), mutual-information maximization (Φ_I), and energetic stability (Φ_H). A combined functional Φ = Φ_S + λ Φ_H allows continuous interpolation between purely informational and dynamically stabilized regimes. A concrete 3-qubit numerical example (with entropy landscape visualization) demonstrates that informational extremization induces nontrivial bipartition selection. The framework is further embedded in three mathematical structures: information geometry on factorization space, categorical quantum mechanics (Frobenius algebra selection), and gauge-theoretic subsystem construction (boundary degrees of freedom analogy). Connections to recent integrated-information approaches in relational quantum dynamics (Zaghi 2025) are discussed as natural extensions. This work complements the main REM paper by supplying the missing selection mechanism, transforming REM from a conceptual proposal into a mathematically structured generative model of relational quantum reality. Related publication: "Relational Emergence Model: A Generative Extension of Relational Quantum Mechanics" (Maruko 2026, Zenodo DOI: [https://doi.org/10.5281/zenodo.19123314]) This is the companion paper providing the selection mechanism for the differentiation map 𝒟 introduced in the main REM framework.
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
Short Description This paper investigates whether the relational turn in quantum theory and category-theoretic physics is sufficient to constitute a genuinely generative ontology. It distinguishes state change, probabilistic actualization, and compositional generation from the stronger transformation of the possibility architecture itself. The paper proposes Generative-Resonant Ontology, E = Phi R, organized by the recursive cycle Phi_t -> R_t -> S_t -> Phi_(t+1), in which actuality may participate in changing what can become possible next. Extended Description Modern physics has progressively displaced the classical image of reality as a collection of independently determined states. Classical mechanics formalized physical existence through states and trajectories; quantum mechanics placed structured possibility at the center of physical description; Relational Quantum Mechanics made physical values dependent upon interactions between systems; and categorical approaches to quantum theory foregrounded morphisms, processes, and composition. These developments constitute a profound movement from substance-centered toward relational and process-centered ontology. This paper argues, however, that relationality is not yet equivalent to generativity. A theory may describe relational actualization, compositional novelty, particle creation, or even highly complex emergent organization while still operating within a sufficiently pre-specified architecture of admissible states, entities, relations, and transformations. To make this distinction precise, the paper differentiates four meanings of generation: State change:S_t -> S_(t+1) Actualization of prior possibility:Phi_given -> R -> S Compositional generation:f, g -> g o f Possibility-space generation:Phi_t -> R_t -> S_t -> Phi_(t+1) The first three are classified as Weak Generativity, whereas Strong Generativity occurs when relationally constituted actuality participates in transforming the effective structure of subsequent possibility. This distinction motivates Generative-Resonant Ontology: E = Phi R expressed recursively as: Phi_t -> R_t -> S_t -> Phi_(t+1) Here, Phi is not a Hilbert space, wave function, probability distribution, or collection of possible states. It denotes structured, historically transformable openness. R denotes relational participation, and S denotes provisional crystallization. The decisive innovation is the final transition: S_t -> Phi_(t+1) Actuality is therefore not merely the endpoint of possibility actualization but may become an enabling condition for a transformed future possibility regime. The paper tests this distinction against quantum field theory, particle creation, phase transitions, spontaneous symmetry breaking, quasiparticles, and emergence. It distinguishes epistemic, descriptive, and ontological novelty and argues that emergent effective regimes constitute the strongest current candidates for strong generativity at the effective level, while avoiding the stronger claim that fundamental physical possibility spaces are known to evolve. Finally, the paper formulates a mathematical research program centered on: Omega_t --(R_t, S_t)--> Omega_(t+1) and, categorically: C_t -> C_(t+1) while recognizing the meta-rule problem that arises when the rule governing structural change is itself pre-specified. The resulting proposal is an ontological extension of the relational turn rather than a replacement for quantum theory. The central question is no longer merely which possibility becomes actual, but how actuality may transform what can become possible next. Highlights Distinguishes Relationality from Generativity. Introduces four meanings of generation:State Change, Actualization, Composition, and Possibility-Space Generation. Defines the first three as Weak Generativity and the fourth as Strong Generativity. Establishes the central distinction: Quantum Uncertainty != Generative Openness Proposes Generative-Resonant Ontology: E = Phi R Introduces the recursive structure: Phi_t -> R_t -> S_t -> Phi_(t+1) Identifies: S -> Phi' as the decisive transition from relational actualization to generative reopening. Distinguishes Epistemic Novelty, Descriptive Novelty, and Ontological Novelty. Treats phase transitions, spontaneous symmetry breaking, and emergent effective degrees of freedom as a boundary zone toward strong effective generativity. Replaces the simple opposition between closed and open systems with: Pre-Specified Relational SpaceversusSelf-Transforming Possibility Space Formulates a future mathematical research program around: Omega_t --(R_t, S_t)--> Omega_(t+1) and: C_t -> C_(t+1) Explicitly distinguishes physical formalism from ontological interpretation. Brief Summary of Each Part Part I. Framing the Problem: Is Quantum Theory Truly a Theory of Generation?Establishes the central problem of the paper. The transition from classical states to quantum possibility, relational events, and categorical processes does not automatically establish strong generativity. The key issue is whether relation merely actualizes possibility or can generate the structure of future possibility. Part II. Classical Physics: Existence as StateReconstructs classical physics as a state-centered ontology in which: S_t -> S_(t+1) occurs within a sufficiently specified state space Omega. The central distinction is between movement through possibility and transformation of the possibility architecture itself. Part III. Quantum Mechanics: From State to Possibility and RelationShows how superposition, measurement, interaction, and entanglement move physical ontology beyond classical determinate states toward structured possibility and relational actualization. The basic heuristic structure is: Phi_given -> R -> S while quantum possibility is distinguished from generative Phi. Part IV. Relational Quantum Mechanics: The Ontologization of RExamines RQM as a major ontological shift from relations between already determinate facts toward relations as constitutive of physical facts. It identifies the first generativity gap: R -> S does not imply S -> Phi' Part V. Categorical Quantum Mechanics: From Objects to Processes and CompositionAnalyzes the categorical move from objects toward morphisms, processes, and composition: f, g -> g o f It identifies the second generativity gap: Compositional novelty does not imply generation of relational grammar. Part VI. Refining the Meaning of “Closure”Rejects a simple characterization of category theory as “closed.” It introduces the distinction between Pre-Specified Relational Space and Self-Transforming Possibility Space and separates operational openness from generative openness. Part VII. Four Meanings of GenerationDistinguishes: State ChangeS_t -> S_(t+1) ActualizationPhi_given -> R -> S Compositionf, g -> g o f Possibility-Space GenerationPhi_t -> R_t -> S_t -> Phi_(t+1) The first three constitute Weak Generativity; the fourth constitutes Strong Generativity. Part VIII. Quantum Field Theory, Emergence, and Phase Transitions as CounterargumentsTests the distinction against particle creation, spontaneous symmetry breaking, phase transitions, and emergent degrees of freedom. It distinguishes: 0 -> 1 -> 2 from: Omega_t -> Omega_(t+1) and treats emergence as a serious boundary case. Part IX. Is Emergence Phi'?Distinguishes Epistemic Novelty, Descriptive Novelty, and Ontological Novelty. Emergence qualifies as a candidate for Phi' when newly stabilized structures transform what can subsequently exist, relate, or become dynamically accessible. Part X. Generative-Resonant OntologyPresents the theory fully through: E = Phi R and: Phi_t -> R_t -> S_t -> Phi_(t+1) S is reinterpreted as provisional crystallization rather than final actuality. Part XI. Phi and Quantum Possibility: Why Generative Openness Is Not Quantum UncertaintyEstablishes that: Phi != {S_1, S_2, ..., S_n} Phi != H Phi != |psi> Phi != P(S) Phi != Quantum Uncertainty The key distinction is between: “Which possibility becomes actual?” and: “What becomes possible because this actuality occurred?” Part XII. How Can Possibility Space Itself Be Formalized?Converts the ontology into a mathematical research program: Omega_t --(R_t, S_t)--> Omega_(t+1) and: C_t -> C_(t+1) It also identifies the meta-rule problem: if a transformation rule is itself entirely pre-specified, generativity may merely have been displaced to a higher formal level. Part XIII. The Boundary Between Physical Claims and Ontological ClaimsClarifies that Generative-Resonant Ontology does not modify quantum mechanics or claim new quantum predictions. Its current status is an ontological interpretation and formal research hypothesis constrained by, but not entailed by, existing physics. Part XIV. An Integrated Model: Three Ontological Stages in the Development of Physical ThoughtIntegrates the paper into three conceptual stages: Being as State:S_t -> S_(t+1) Being as Relational Actualization:Phi_given -> R -> S Being as Generative Reopening:Phi_t -> R_t -> S_t -> Phi_(t+1) These are treated as nested ontological expansions rather than simple historical replacements. Part XV. Conclusion: From a Relational World to a Generative WorldConcludes with the transition: R -> S to: R -> S -> Phi' and ultimately: Phi -> R -> S -> Phi' Reality is interpreted as a history in which relationally constituted actuality may participate in transforming the horizon of future possibility. Academic Contributions 1. Distinguishing Relationality from Generativity The paper introduces the fundamental distinction: Relationality != Generativity Relationality concerns how actuality depends upon or is constituted through relations. Generativity concerns whether relational actuality transforms the structure of subsequent possibility. 2. A Fourfold Taxonomy of Generation The paper distinguishes: State Ch
Coherence First develops a relational reformulation of Radial Coherential Dynamics (RCD) centered on a prior question: what physically selects the factorization of a quantum system into subsystems? Because entanglement is defined only relative to a tensor-product structure or a partition of observable algebras, programs that derive classicality, locality, or geometry from entanglement already presuppose a cut. The paper proposes the Coherential Selection Principle (CSP): the physical factorization is the one that maximizes the persistence of coherent information within its parts while minimizing coherent persistence across their boundaries. In this framework, subsystems are dynamically persistent regions of a relational quantum structure, and the coherential field C(x,t)C(x,t) is reinterpreted as a possible continuous, effective description of a deeper network of selected relations. A numerical proof of concept is presented for an open three-qubit system. Keeping the Hamiltonian fixed while rotating the environmentally monitored algebra shifts the factorization favored by a provisional functional. The effect survives a generic non-commuting control, remains stable under sampling, and follows a continuous five-point rotation of the monitored algebra. The result establishes that open dynamics can transport and recover a recognizable structure of parts that is not fixed by the Hamiltonian alone. The full CSP, the emergence of geometric locality, and the proposed connection to cosmological relaxation remain the next stages of the program. Version 0.2. Preprint; not peer reviewed. This record contains the PDF, LaTeX source, numerical scripts, and reproducibility materials used in the closed- and open-system studies.