Geometric Response From Coherence Cost

This paper presents a structural derivation of the stress–energy tensor and the Einstein field equations from minimal conditions of coherent persistence. Starting from the requirement that any persistent system must be able to compare the outcome of a transformation back to a reference (coherent return), incomplete closure generates a quantitative coherence cost that must be locally accounted for. This cost may be stored, transported, and redistributed. The analysis shows that consistent accounting of coherence cost forces the emergence of energy, momentum, and stress as distinct aspects of a single bookkeeping structure, identified as the stress–energy tensor. Its local conservation follows directly from return and does not rely on spacetime symmetries, action principles, or variational methods. Because the source is symmetric and locally conserved, coherent closure further requires a geometric response that is itself divergence-free. The Einstein tensor provides the minimal such response, yielding the Einstein field equations as a structural consistency condition between conserved coherence cost and geometry. The result explains why all forms of energy gravitate and why matter, radiation, pressure, and stress contribute on equal footing. Stress–energy and gravitational geometry thus emerge as necessities of coherent persistence rather than as postulates of gravitational theory.