We present a unified operator framework in which spacetime geometry, gauge dynamics, and matter spectra emerge from entropy-maximizing projections in a Diagram Hilbert Space built from microscopic interaction histories. A single complex operator Z=M+iQ combines mass–energy and charge degrees of freedom. Statistical extremization of the Gibbs–von Neumann entropy under constraints on ⟨M⟩ and ⟨Q⟩ yields Einstein’s equations from the real sector and Maxwell’s equations from the imaginary one. The operator spectrum inherits its structure from QCD partition functions, while macroscopic projections reproduce gravitational and electromagnetic fields as thermodynamic equations of state. Hidden eigenstates of Z form self-gravitating isothermal halos consistent with galactic rotation curves, and residual entropic energy density produces an effective cosmological constant. The same formalism accounts for black-hole entropy as projection saturation and identifies gravitons as collective excitations of the mass operator. This entropy-projected operator framework offers a renormalizable and testable route toward quantum gravity directly linked to the Standard Model.