Abstract
We study when different admissible projected realizations represent the same physical probability theory, and what obstructs their equivalence. Each member of a supplied family has a complex state sector, represented observable algebra, declared retained generator, and positive compatible metric. Strong operational equivalence intertwines these physical structures and preserves probabilities on equivalence classes of pointed preparation and measurement data. Necessary invariants include representation kernels, commutants, and the full spectral measures of physicalized generators. For a full matrix observable algebra, irreducible unital physical representations of the same abstract self-adjoint dynamical element yield a finite-dimensional sufficient equivalence criterion. Dimension-three examples exhibit distinct equivalent projections and an individually admissible sector excluded from their strong equivalence class by a spectral mismatch. Normalization, Gleason transport and matrix-algebra representation equivalence are standard ingredients used within this conditional framework. Projective probability retains explicit complex-dimension, additivity and preparation hypotheses. Neither projection classification nor metric uniqueness is established, and no probability measure on projection labels is supplied.