Selection Principles for the Differentiation Map in the Relational Emergence Model

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.