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\title{
Variational Generation of Relational Structure\\
in the Relational Emergence Model
}
\author{
Yoshifumi Maruko\\
Independent Researcher\\
Yamagata, Japan
}
\date{2026}

\begin{document}

\maketitle

\begin{abstract}

Relational Quantum Mechanics (RQM) defines physical states relationally but does not specify
how relational structure itself becomes articulated.
The Relational Emergence Model (REM) introduces a generative layer beneath relational description.

We model relational articulation as a variational selection problem over Hilbert-space factorizations:

\[
\Phi = \Phi_S + \lambda \Phi_H
\]

combining informational differentiation and dynamical stability.

The parameter $\lambda$ acts as a control parameter governing the balance between
informational structure and dynamical coherence.

A three-qubit example demonstrates regime competition between informationally optimal
and dynamically stabilized factorizations, illustrating how relational articulation
may emerge through constrained optimization.

\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Relational Quantum Mechanics (RQM) proposes that physical states exist only relative
to other systems~\cite{rovelli}.
While this removes observer-independent states, it leaves open a prior question:
how relational structure itself becomes articulated.

The Relational Emergence Model (REM) introduces a generative layer beneath relational description.
Rather than assuming subsystem structure, REM models relational articulation as a
structured selection problem.
The key proposal:
relational structure is not assumed --- it is selected.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Variational Selection Principle}

Hilbert space admits multiple inequivalent tensor-product structures:

\begin{equation}
\mathcal H \cong \mathcal H_A \otimes \mathcal H_B.
\end{equation}

We consider a set of candidate factorizations:

\begin{equation}
\mathcal F = \{F_k\},
\end{equation}

which in the bipartite case is parametrized by the coset manifold
$\mathcal F \cong U(N)/(U(n_A)\otimes U(n_B))$,
i.e.\ all unitarily inequivalent tensor-product decompositions
of $\mathcal H$ with $\dim \mathcal H_A = n_A$ fixed.
In the three-qubit example ($N=8, n_A=2$) the continuous space has
real dimension $\dim_{\mathbb{R}}\mathcal F = 2(N^2-n_A^2-n_B^2) = 24$;
the three bipartitions $A|BC$, $B|AC$, $C|AB$ studied below are
representative \emph{local} factorizations whose boundaries align with
individual qubits, chosen for physical transparency rather than as an
exhaustive search.
These three are not guaranteed to be global optima of $\Phi$ over the full
continuous manifold; they serve as illustrative representatives.
A full optimization over $\mathcal F$ is left for future work.

We define a selection functional $\Phi(F)$ and the selection rule:

\begin{equation}
F^* = \arg\max_F \Phi(F).
\end{equation}

Lowest-order ansatz:

\begin{equation}
\Phi = \Phi_S + \lambda \Phi_H.
\end{equation}

Informational contribution:

\begin{equation}
\Phi_S = I(A:B).
\end{equation}

Dynamical contribution:

\begin{equation}
\Phi_H = - \langle H_{\partial F} \rangle,
\end{equation}

where $H_{\partial F}$ denotes the boundary interaction Hamiltonian
(see Section~\ref{sec:hdefinition}).

Equivalent minimization formulation:

\begin{equation}
\mathcal L = -\Phi_S + \lambda \mathcal C_H,
\quad
\mathcal C_H = \langle H_{\partial F} \rangle.
\end{equation}

Since $\mathcal C_H = -\Phi_H$, minimizing $\mathcal L$
is equivalent to maximizing $\Phi$.

\paragraph{Scalability.}
The factorization space $\mathcal F \cong U(N)/(U(n_A)\otimes U(n_B))$ grows
combinatorially with system size, posing a computational challenge for large $N$.
Practical strategies include restricting to locally-generated factorizations
(Lie-algebra gradient flow on $\mathcal F$), tensor-network parametrizations
of candidate states, and hierarchical coarse-graining in which the variational problem
is solved layer by layer.
The three-qubit model studied here is an existence proof;
scaling behavior is left as an open problem for future work.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Information-Theoretic Quantities}

Density operator:

\begin{equation}
\rho_{AB} = |\Psi\rangle \langle \Psi|.
\end{equation}

Reduced states:

\begin{align}
\rho_A &= \mathrm{Tr}_B \rho_{AB},\\
\rho_B &= \mathrm{Tr}_A \rho_{AB}.
\end{align}

Von Neumann entropy (log base 2):

\begin{equation}
S(\rho) = - \mathrm{Tr}(\rho \log_2 \rho).
\end{equation}

Mutual information:

\begin{equation}
I(A:B) = S(\rho_A) + S(\rho_B) - S(\rho_{AB}).
\end{equation}

For pure bipartitions ($S(\rho_{AB}) = 0$):

\begin{equation}
I(A:B) = 2\,S(\rho_A).
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{General Definition of $H_{\partial F}$}
\label{sec:hdefinition}

For a Hamiltonian $H = \sum_\alpha h_\alpha$ decomposed into elementary interaction terms,
the boundary Hamiltonian for factorization $F: \mathcal H_A \otimes \mathcal H_B$ is:

\begin{equation}
H_{\partial F} = \sum_{\alpha:\, \mathrm{supp}(h_\alpha) \not\subseteq A
                              \,\mathrm{and}\,
                              \mathrm{supp}(h_\alpha) \not\subseteq B} h_\alpha,
\end{equation}

i.e., the sum over all terms $h_\alpha$ whose support intersects both $A$ and $B$:

\begin{equation}
H = H_A + H_B + H_{\partial F},
\end{equation}

where $H_A$ ($H_B$) contains terms entirely within $A$ ($B$).
The Hilbert--Schmidt decomposition of any operator $X$ into subsystem components
is analogously defined in REM5~\cite{rem5}.

\paragraph{Relation to Carroll--Singh quantum mereology.}
Carroll and Singh~\cite{carroll} identify preferred tensor-product structure
by minimizing a dynamical coupling measure, selecting factorizations whose
subsystems evolve quasi-classically under the Hamiltonian.
REM is complementary but distinct in two respects.
First, REM introduces an explicit \emph{informational} objective $\Phi_S = I(A:B)$
that competes with the dynamical term; Carroll--Singh considers only
dynamical stability and has no analogue of $\lambda$.
Second, the competition between $\Phi_S$ and $\Phi_H$ produces a
\emph{regime structure} parametrized by $\lambda$: the REM framework
subsumes the purely dynamical selection ($\lambda \to \infty$) as a special
case, while also capturing the informationally-dominant regime ($\lambda \to 0$)
absent from Quantum Mereology.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Three-Qubit Numerical Example}

Consider the state:

\begin{equation}
|\Psi\rangle =
\sqrt{0.6}\,|000\rangle
+
\sqrt{0.3}\,|111\rangle
+
\sqrt{0.1}\,|101\rangle.
\end{equation}

\subsection*{Reduced density matrices}

\begin{align}
\rho_A
&=
\begin{pmatrix}
0.6 & 0\\
0 & 0.4
\end{pmatrix},
\quad
\rho_C
=
\begin{pmatrix}
0.6 & 0\\
0 & 0.4
\end{pmatrix},
\\[6pt]
\rho_B
&=
\begin{pmatrix}
0.7 & \sqrt{0.03}\\
\sqrt{0.03} & 0.3
\end{pmatrix}.
\end{align}

Eigenvalues and entropies:

\begin{align}
\mathrm{eig}(\rho_A) &= \{0.6,\ 0.4\},
&S(\rho_A) &= 0.9710 \text{ bits},\\
\mathrm{eig}(\rho_B) &= \{0.7646,\ 0.2354\},
&S(\rho_B) &= 0.7873 \text{ bits},\\
\mathrm{eig}(\rho_C) &= \{0.6,\ 0.4\},
&S(\rho_C) &= 0.9710 \text{ bits}.
\end{align}

Mutual informations (pure state: $I = 2S$; all values rounded to 4 decimal places):

\begin{align}
I(A:BC) &= 1.9419,\\
I(B:AC) &= 1.5747,\\
I(C:AB) &= 1.9419.
\end{align}

The B partition is informationally disfavored relative to A and C.

\subsection*{Hamiltonian and regime competition}

To demonstrate genuine regime competition, we choose the Hamiltonian:

\begin{equation}
H = Z_1 Z_3,
\end{equation}

a next-nearest-neighbor (non-adjacent qubit) coupling with expectation value:

\begin{equation}
\langle Z_1 Z_3 \rangle = 0.6(+1) + 0.3(+1) + 0.1(+1) = 1.0.
\end{equation}

Boundary contributions for each candidate factorization:

\begin{center}
\begin{tabular}{lll}
\toprule
Partition & $H_{\partial F}$ & $\Phi_H = -\langle H_{\partial F}\rangle$ \\
\midrule
$A|BC$ & $Z_1 Z_3$ & $-1.0$ \\
$B|AC$ & $0$ (both $Z_1, Z_3$ in $AC$) & $\phantom{-}0.0$ \\
$C|AB$ & $Z_1 Z_3$ & $-1.0$ \\
\bottomrule
\end{tabular}
\end{center}

The B$|$AC partition has \emph{zero} boundary coupling: both interacting qubits (1 and 3)
reside on the same side.
This makes B$|$AC dynamically stable despite its lower informational content.

\subsection*{Crossover parameter $\lambda^*$}

The combined functional $\Phi(F,\lambda) = \Phi_S(F) + \lambda\,\Phi_H(F)$
gives the following $\lambda$-dependent values:

\begin{center}
\begin{tabular}{ccccc}
\toprule
$\lambda$ & $\Phi(A|BC)$ & $\Phi(B|AC)$ & $\Phi(C|AB)$ & $F^*$ \\
\midrule
$0.0$ & $1.9419$ & $1.5747$ & $1.9419$ & $A|BC = C|AB$ \\
$0.2$ & $1.7419$ & $1.5747$ & $1.7419$ & $A|BC = C|AB$ \\
$0.3672$ & $1.5749$ & $1.5747$ & $1.5749$ & (crossover) \\
$0.5$ & $1.4419$ & $1.5747$ & $1.4419$ & $B|AC$ \\
$1.0$ & $0.9419$ & $1.5747$ & $0.9419$ & $B|AC$ \\
\bottomrule
\end{tabular}
\end{center}

The crossover occurs at:

\begin{equation}
\lambda^* =
\frac{\Phi_S(A|BC) - \Phi_S(B|AC)}{\Phi_H(B|AC) - \Phi_H(A|BC)}
=
\frac{1.9419 - 1.5747}{0 - (-1.0)}
\approx 0.3672.
\end{equation}

For $\lambda < \lambda^*$: the informationally optimal factorization ($A|BC$ or $C|AB$) is selected.
For $\lambda > \lambda^*$: the dynamically stable factorization ($B|AC$, with zero boundary coupling) is selected.
This constitutes a genuine regime competition with distinct $F^*$ on each side.

\begin{figure}[h]
\centering
\includegraphics[width=0.78\linewidth]{fig_regime_competition.pdf}
\caption{%
Regime competition in the three-qubit example ($H = Z_1 Z_3$).
The combined functional $\Phi(F,\lambda)$ is shown for the three candidate bipartitions.
For $\lambda < \lambda^* \approx 0.3672$ (blue region), the informationally optimal
factorization $A|BC$ (= $C|AB$, solid blue) is selected.
For $\lambda > \lambda^*$ (red region), the dynamically stable factorization $B|AC$
(dashed red, zero boundary coupling) is selected.
The crossover $\lambda^*$ is marked by the dotted vertical line.
Near $\lambda^*$, the functional values of competing factorizations become nearly degenerate,
suggesting a regime of structural fluctuation in which no single tensor-product
structure is strongly preferred---analogous to a critical point in a phase transition.
}
\label{fig:regime}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Physical Interpretation of $\lambda$}

The functional has a free-energy-like structure:

\begin{equation}
\Phi \sim S - \beta E,
\end{equation}

suggesting the correspondence:

\begin{equation}
\lambda \leftrightarrow \beta.
\end{equation}

In open-system settings (following Zurek~\cite{zurek}),
where environmental coupling induces decoherence, a natural timescale interpretation arises:

\begin{equation}
\lambda \sim \frac{\tau_D}{\tau_{dyn}},
\end{equation}

where $\tau_D$ is the decoherence timescale and $\tau_{dyn} \sim 1/\Delta E$ is the
dynamical timescale set by the energy gap.
In the isolated-system limit considered here, $\lambda$ acts as an external control parameter.
A minimal microscopic sketch of its origin proceeds as follows.
Coupling the system to a Markovian environment with Lindblad operators
$\{L_k\}$ of strength $g$ introduces a decoherence rate
$\Gamma \sim g^2 \langle H_{\partial F}^2\rangle / \hbar^2$
for off-diagonal elements of $\rho$ across the boundary.
The associated decoherence time $\tau_D \sim 1/\Gamma \sim \hbar^2/(g^2\langle H_{\partial F}^2\rangle)$
and the dynamical timescale $\tau_{dyn}\sim \hbar/\Delta E$ together give

\begin{equation}
\lambda \sim \frac{\tau_D}{\tau_{dyn}}
\sim \frac{\hbar\,\Delta E}{g^2\langle H_{\partial F}^2\rangle}.
\end{equation}

Strong environmental coupling ($g$ large) thus corresponds to small $\tau_D$, small $\lambda$,
and an informationally-dominated regime;
weak coupling ($g\to 0$) gives large $\lambda$ and dynamical stabilization.
A full derivation from a microscopic system--environment model is left for future work.

\smallskip
\noindent
\textit{Small $\lambda$}: information-dominated regime ($F^*$ maximizes mutual information).\\
\textit{Large $\lambda$}: dynamically stabilized regime ($F^*$ minimizes boundary coupling).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Relation to REM5}

REM5~\cite{rem5} introduces a quadratic dynamical cost:

\begin{equation}
\mathcal C_H = \mathrm{Tr}(\rho\, H_{\partial F}^2),
\end{equation}

which penalizes cross-boundary fluctuations rather than mean coupling strength.
The present work uses the lowest-order scalar expectation:

\begin{equation}
\Phi_H = -\langle H_{\partial F} \rangle
\end{equation}

as a minimal generative ansatz.
The two choices are appropriate in different physical regimes.
The linear form $\Phi_H$ is adequate when the boundary interaction is
weak and approximately uniform, so that the mean coupling captures the
leading-order dynamical cost.
The quadratic form $\mathcal C_H = \mathrm{Tr}(\rho H_{\partial F}^2)$ becomes
essential when boundary interactions are heterogeneous or fluctuating,
since it captures the variance of $H_{\partial F}$ under the state $\rho$
and thus distinguishes factorizations that have the same mean coupling
but differ in their susceptibility to local perturbations.
In the three-qubit example with $H = Z_1 Z_3$, the boundary term takes
only two values ($Z_1 Z_3$ or $0$), so mean and variance are perfectly
correlated and both criteria yield the same $F^*$;
the two formulations are expected to diverge for larger systems with
inhomogeneous couplings.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion}

REM provides a generative framework for relational structure.
Relational articulation is modeled as variational selection on factorization space.
The control parameter $\lambda$ governs the balance between informational differentiation
and dynamical stability.
The three-qubit example with $H = Z_1 Z_3$ demonstrates genuine regime competition:
at $\lambda^* \approx 0.3672$ the selected factorization switches from the informationally
optimal $A|BC$ (or $C|AB$) partition to the dynamically stable $B|AC$ partition.
The model establishes a mathematically tractable foundation for generative relational
quantum structure.

Several natural directions for future work follow directly.
First, in open quantum systems the parameter $\lambda$ need not be fixed:
as environmental coupling strength varies, $\lambda \sim \tau_D / \tau_{dyn}$
will itself evolve, driving the system dynamically across the regime boundary $\lambda^*$.
This suggests that decoherence-induced einselection~\cite{zurek} may be reinterpreted
as a $\lambda$-flow toward the dynamically stabilized regime---a picture testable
in quantum simulators by tuning system--environment coupling strength.
Second, replacing the linear dynamical cost $\Phi_H = -\langle H_{\partial F}\rangle$
with the quadratic form $\mathcal C_H = \mathrm{Tr}(\rho H_{\partial F}^2)$ of REM5~\cite{rem5}
penalizes boundary fluctuations rather than mean coupling, which may shift $\lambda^*$
and alter the sharpness of the structural crossover.
Systematic comparison of the two cost functions is deferred to future work.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{10}

\bibitem{rovelli}
C.~Rovelli,
Relational Quantum Mechanics,
\textit{Int.~J.~Theor.~Phys.}~\textbf{35}, 1637 (1996).

\bibitem{zurek}
W.~H.~Zurek,
Decoherence, einselection, and the quantum origins of the classical,
\textit{Rev.~Mod.~Phys.}~\textbf{75}, 715 (2003).

\bibitem{zanardi}
P.~Zanardi,
Virtual quantum subsystems,
\textit{Phys.~Rev.~Lett.}~\textbf{87}, 077902 (2001).

\bibitem{carroll}
S.~M.~Carroll and A.~Singh,
Quantum mereology: Factorizing Hilbert space into subsystems with quasi-classical dynamics,
\textit{Phys.~Rev.~A}~\textbf{103}, 022213 (2021).

\bibitem{rem5}
Y.~Maruko,
Relational Emergence Model (REM~5.12):
Variational Selection of Tensor-Product Structure in Quantum Systems,
Independent preprint (2026).

\end{thebibliography}

\end{document}