Operator Entropy and the Emergent Unification of Gravity and Gauge Interactions

Gravitation and gauge interactions are commonly treated as distinct manifestations of nature, yet mounting theoretical evidence suggests a shared informational origin. We present a framework in which spacetime, matter and gauge fields emerge from an entropy-maximizing projection of a complex operator Z = M + iQ acting in a Diagram Hilbert Space (DHS) spanned by diagrammatic entities such as Feynman graphs, Wilson loops and knot invariants. The real and imaginary components of the projection reproduce Einstein’s and Maxwell’s equations, while higher-order correlations generate small, testable deviations. Within this formalism we derive the Bekenstein–Hawking entropy prefactor from diagram counting, reproduce the QCD one-loop β-function, obtain an operator-based renormalization-group flow, explain galactic rotation curves through entropic equilibrium of hidden eigenstates, and predict cross-sector couplings between the mass and charge operators. The approach provides a renormalizable, information-theoretic route to the unification of gravity and gauge interactions.