Quantum dimensions in the universal branching rule for the symmetric square of the adjoint

For a simple complex Lie algebra 𝔤 with Vogel parameters (α, β, γ), let θ be the highest root, 𝔰𝔩₂^θ its 𝔰𝔩₂-subalgebra and 𝔤₀′ the centralizer of 𝔰𝔩₂^θ. A companion paper decomposes each Vogel piece Y₂(ν) of the symmetric square of the adjoint under 𝔰𝔩₂^θ ⊕ 𝔤₀′ by sixteen labels, rational functions on Vogel's plane that are added class by class. We lift this rule to quantum dimensions. Thirteen labels become products of q-numbers of linear forms, built from the point of 𝔤 and the shifted points of its two blocks. The other three are fixed by the quantum dimension of Y₂(ν); they are not such products, but their sum has a closed form. We prove, by a classification-based and computer-assisted argument, that for every simple 𝔤 other than 𝔰𝔩₂ and 𝔰𝔬₈ the quantum dimension of every class, in the principal specialization of 𝔤, is the sum of its q-labels; a block of Dynkin index k enters at q^(1/k). The ingredients are the splitting ρ_𝔤 − ρ_𝔤₀′ = (t − 1)θ/2 of Landsberg and Manivel, with t the dual Coxeter number, a root-string lemma and a closed product for the Cartan components V(λ + θ_X), a three-term decomposition of V ⊗ 𝔤′_X, V = 𝔤₁, for each block 𝔤′_X, and a proof, series by series, of Mkrtchyan's quantum dimension formulas for Y₂(β) and Y₂(γ) and of Westbury's formula for the adjoint. We record an exact check of 687 classes on 30 algebras, an independent recomputation on 13 algebras, and a prediction of the classes of A₁, A₂ and D₄ from the five exceptional algebras. The scripts in anc/ use exact arithmetic and reproduce every computation cited in the paper; each log ends with "checks passed: True".