We investigate the classical three-body problem from the perspective of definability of solution concepts within dynamical systems theory. Although the equations of motion generate well-defined trajectories, the notion of an orbital solution requires invariant identity conditions that persist under time evolution. We formalize this requirement in terms of representation-independent equivalence relations commuting with the dynamical flow. While such structures exist in the integrable two-body problem, we show that they generically fail in three-body dynamics due to the absence of dynamically stable identity criteria. This failure is independent of analytical solvability, numerical approximability, or chaotic sensitivity. The argument is formulated as a structural no-go result: orbital identity cannot be defined in a representation-independent manner for generic three-body systems. The analysis reframes the classical three-body problem as exhibiting a fundamental limitation in the admissibility of orbit-based solution concepts rather than a merely technical obstruction.
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