Operator Dressing Calculus III: Universal Phi Continuum Closure

Operator Dressing Calculus III develops the continuum-closure side of CVM operator dressing. It formulates the universal dressing field \Phi as a compensator that changes the native conformal representative while preserving the dressed metric and dressed matter variables. The paper derives the associated dressing constraint, places it alongside the Hamiltonian and diffeomorphism constraints, and proves strong canonical closure for the compensator branch when the dressed gravitational-matter constraint algebra closes. It also defines the conditions for admitting scalar-field branches, constructing physical observables, and recovering the standard Schwarzschild result in the regular vacuum branch.