Thermodynamic Projection of a Complex Operator: Emergent Einstein–Maxwell Dynamics from a Diagram Hilbert Space

We present an operator–entropic framework in which classical spacetime geometry and a gauge field arise as macroscopic variables associated with a coarse-grained description of microscopic degrees of freedom. The microscopic theory is formulated on an abstract Diagram Hilbert Space and is summarized, at the level relevant for macroscopic sourcing, by two Hermitian operators and , combined into a single complex operator . A macroscopic state is obtained by maximum-entropy inference under a chosen set of coarse constraints, yielding a Gibbs density matrix that depends parametrically on a macroscopic metric and a macroscopic potential . The macroscopic stress tensor and current are defined as functional derivatives of the Gibbs free energy with respect to and , respectively. With standard local horizon-thermodynamic input, Einstein’s equation emerges as a local equation of state in the sense of Jacobson. Maxwell’s equations emerge from stationarity of the free energy with respect to , with the homogeneous identities following kinematically from . As an explicit macroscopic example, a collisionless hidden ensemble (interpreted as a sector of -eigenmodes) yields an isothermal halo profile with , producing asymptotically flat galactic rotation curves. Implications for an effective cosmological constant and for black-hole entropy are discussed within the same coarse-graining viewpoint.