Relational Closure, Diagram Hilbert Space, and Entropic Projection Toward a Unified Operator Origin of Gauge Symmetry and Emergent Spacetime

We develop a relational operator framework in which gauge symmetry and spacetime geometry emerge from algebraic relations among microscopic observables defined on a Diagram Hilbert Space (DHS). The DHS is a Hilbert space spanned by interaction diagrams representing relational configurations of microscopic degrees of freedom. Primitive observables satisfy closure relations forming Lie algebras. The minimal non-Abelian closure cell is shown to be isomorphic to su(2). Multiplicity of equivalent closure cells naturally produces internal symmetry groups SU(n), while local comparison of closure sectors induces gauge connections whose curvature reproduces Yang–Mills structure. Physical observables are encoded in a complex operator Z = M + iQ whose real and imaginary components represent mass–energy and charge sectors. Macroscopic physics arises through entropic coarse-graining within an Entropic Ontological Transfer Framework (EOTF). Correlations of relational observables generate an effective spacetime metric. Under thermodynamic assumptions for local horizon entropy, Einstein’s equations emerge as an equation of state, while stationarity of the free energy associated with the charge sector yields Maxwell’s equations. A finite-dimensional toy model demonstrates relational closure, sector multiplicity, gauge transport, and coarse-grained geometry. The framework suggests that gauge interactions and spacetime geometry may share a common relational operator origin.