Complex Thermodynamic Projection on Diagram Hilbert Space Local Maximum-Entropy Coarse-Graining, Response Functionals, and Effective Einstein–Maxwell Dynamics

We develop a thermodynamic coarse-graining framework in which classical geometry and gauge dynamics are recovered as effective macroscopic descriptions of a microscopic diagrammatic ensemble. Microscopic degrees of freedom are encoded in a separable diagram Hilbert space 𝐻 = ℓ2(𝐺) with diagram basis, equipped with a observable algebra 𝒜 ⊂ 𝐵(𝐻!). We assume self-adjoint operators 𝑀 and 𝑄 representing mass–energy and charge, and we fix the relativistic identification 𝐻 ≡ 𝐸 ≡ 𝑀𝑐2 so that 𝑀 = 𝐻/𝑐2 (hence 𝐻 = 𝑀 in natural units). Macrostates are defined by constrained maximization of von Neumann entropy, yielding global and local generalized Gibbs states with Lagrange-multiplier fields 𝛽(𝑥) and 𝛾(𝑥). Locality is introduced constructively via coarse partitions of diagram space and associated completely positive, trace-preserving coarse-graining maps; coarse local densities are defined as conditional expectations of 𝑀 and 𝑄 onto a commutative subalgebra generated by coarse projectors. We define a generating functional 𝑊[𝑔, 𝐴] = ln 𝑍[𝑔, 𝐴] from a controlled family of coarse-grained operators 𝑀[𝑔, 𝐴], and we define emergent stress-energy and current as functional derivatives with respect to the macroscopic sources 𝑔&' and 𝐴(. Under standard local horizon-thermodynamic postulates (Unruh temperature and Clausius relation), Einstein’s equation is recovered as a local equation of state, while Maxwell’s equations follow from stationarity of an effective thermodynamic potential containing the leading gauge-invariant field-cost term. Mixed mass–charge response is quantified by 𝜒 = ∂+ ∂ 𝑊 reducing to Cov(𝑀,𝑄) in the commuting case, with commutatorcontrolled Kubo terms when [𝑀,𝑄] ≠ 0. We provide a finite-ensemble toy model with explicit coarse partition, susceptibilities, and a computable mixing parameter. Assumptions, limitations, and open problems are stated explicitly.