Repeated tensor products describe increasingly complicated combinations of a basic symmetry representation. One can count how often a chosen representation appears, and collect those counts into a power series. Is that series algebraic, meaning that it satisfies a polynomial equation? This candidate gives a uniform answer for complex orthogonal groups. Fix any nonzero, finite-dimensional algebraic representation. Its ordinary multiplicity series is algebraic exactly in dimensions one, two and three. In every higher dimension it is not algebraic, although it still satisfies a linear differential equation with polynomial coefficients. The important consequence is that choosing a different genuine representation cannot restore algebraicity above dimension three. The qualification genuine matters: formal subtraction can cancel the obstruction. The paper gives a dimension-four illustration. The proof combines explicit low-dimensional formulas with character integrals and the growth of their coefficients. Invariant series for symmetric determinantal algebras appear as a special case. Several ingredients are classical. In particular, the scalar asymptotic constant is already in Regev, and the dimension-three invariant formula comes from Almkvist, Dicks and Formanek. The contribution presented here is the unified classification and its fixed-representation consequences, not a claim to have invented those inputs. The package contains thirteen pages of written mathematics, source comparisons, internal reviews and exact finite diagnostics. Those computations do not prove the universal result. This is an unrefereed Evidence Press candidate. Historical priority and external specialist validation are not established. The full paper and evidence are linked. This is an AI-generated voice summary, not additional mathematical evidence.