Symmetry-Resolved Phase Transitions of Electromagnetic Degrees of Freedom Under RIS Control

The theory of physical degrees of freedom (DoF) developed by Franceschetti–Migliore–Minero (FMM) establishes a fundamental phase transition in the singular-value spectrum of electromagnetic radiation operators under maximal rotational symmetry. In this work, we revisit this result from a symmetry-explicit operator-theoretic perspective and extend it to scenarios with reduced and controllable symmetries, with particular emphasis on reconfigurable intelligent surfaces (RISs). We model the radiation process as a compact operator acting between admissible source and observation spaces and characterize its symmetry through group equivariance. This formulation enables a systematic decomposition of the operator into irreducible representation sectors associated with the effective symmetry group, defined as the intersection of symmetries supported jointly by the source architecture, RIS geometry and programmability, receiver configuration, and propagation environment. We show that the FMM phase transition persists within each symmetry sector and that the total DoF budget is redistributed across sectors according to symmetry constraints. A key outcome of this analysis is the distinction between physical and effective degrees of freedom. While breaking the maximal SO(2) symmetry does not increase the total number of electromagnetic DoF dictated by physics, symmetry reduction modifies their allocation across sectors, potentially lifting degeneracies and increasing the number of degrees of freedom that can be effectively addressed by a given excitation, RIS control, and measurement architecture, even when the total number of physical DoF remains fixed by fundamental limits. This clarifies the role of controlled symmetry breaking as a design mechanism rather than a means to surpass fundamental limits. The proposed framework bridges electromagnetic operator theory, representation theory, and RIS-enabled system design, providing both rigorous symmetry-resolved DoF accounting and actionable insights for excitation, surface programmability, and measurement strategies under practical architectural constraints.