Meaning
A finite physical realization is an actual member of the admissible realization fiber.
A realization-complete proposal is physically realized when its fiber is nonempty. Choosing x_D in that fiber supplies a concrete finite realization, with the physical requirements already satisfied on its declared carrier.
Example (illustrative)
Suppose two candidates implement the same proposal. If the first loses a required sector but the second passes every physical clause, the second witnesses nonemptiness of the fiber. The first candidate’s failure remains useful evidence about that attempt.
Scope and limits
One finite realization does not establish a compatible family across refinements, a continuum theory, or a relativistic mass gap.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 4.3.