Reconstructability of Inverse Problems under Symmetry: Separating Structural, Effective, and Physical Upper Bounds

Abstract: In our previous work we introduced the Reconstruction Dimension, an upper bound on the reconstructable degrees of freedom determined solely by the symmetry. In actual physical systems, however, the state quantities are generated through physical processes $P_o$ induced by the cause $O$. The state space is therefore often degenerate, and the upper bound determined by the representation-theoretic structure is not reached. This calls for a distinction between the upper bound determined solely by symmetry and the one realized after the physical process. We therefore introduce a three-level structure of upper bounds: the Structural, Effective, and Physical Reconstruction Dimensions (SRD, ERD, PRD), determined respectively by the representation-theoretic structure of the intermediate space, by the concrete design of the reconstruction map, and by the degeneracy of the state space under the physical process $P_o$. For each irreducible component they form the hierarchy $\mathrm{SRD}\ge\mathrm{ERD}\ge\mathrm{PRD}$, so that the two gaps identify at which level reconstructability is lost. The framework is illustrated through two contrasting systems: the orientation dynamics of flake-like particles, where the cause is a velocity gradinet, and the two-body problem, where the cause itself is defined through the symmetry. This paper provides a concrete framework for reconstruction in physical inverse problems.