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.
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.
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.
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.
We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.
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
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.
Quantum subsystem structure is typically specified independently of the workload, yet the decomposition that best preserves locality under a Hamiltonian need not be the one that best supports repeated computation. We develop workload–mereology quantum computing, a framework in which generalized tensor-product structure is selected by an explicit competition between dynamical naturalness and computational demand. Physical cost is quantified using the Gaussian scrambling criterion of operational quantum mereology, while task cost measures workload interactions cut by a candidate bipartition. After dimensionless normalization, their competition is captured by Jλ(F) = Ŝ(F) + λĈₜ(F). We prove that a factorization is uniquely optimal for some workload pressure precisely when its cost pair is an exposed point of the lower convex hull, and that a genuine compromise remains optimal over a nonempty interval precisely when it forms a strict intermediate lower-hull vertex. A four-qubit construction yields the transition threshold in closed form. Exhaustive reproducible simulations show that compromise factorizations are not exceptional: at n = 10 and λ = 1, they occur in 61.33% of 300 commuting Pauli-ZZ instances for uniform and lognormal couplings (60.67% for normal couplings) and in 59.00% of generic noncommuting XX+YY+ZZ instances. These results identify a task-dependent regime of quantum subsystem organization in which workload does more than act on a predetermined decomposition: it becomes a variational ingredient in determining which decomposition is preferred.