A universal branching rule for the symmetric square of the adjoint under highest-root breaking

For a simple complex Lie algebra 𝔤 with Vogel parameters (α, β, γ), the symmetric square of the adjoint decomposes as S²𝔤 = ℂ ⊕ Y₂(α) ⊕ Y₂(β) ⊕ Y₂(γ). Let θ be the highest root, 𝔰𝔩₂^θ its 𝔰𝔩₂-subalgebra, 𝔤₀′ its centralizer, and H = 𝔰𝔩₂^θ ⊕ 𝔤₀′ (the quaternionic symmetric pair). We decompose each Y₂(ν) under H by a single rule on Vogel's plane. An isotypic component is labeled by ν, the 𝔰𝔩₂^θ-spin and the eigenvalue of the Casimir of 𝔤₀′. Sixteen labels suffice, each with a Casimir value linear in the parameters and a dimension rational in them; the dimension of a class is the sum of its labels, added as rational functions and then evaluated. We prove the rule for every simple 𝔤 other than 𝔰𝔩₂ and 𝔰𝔬₈, assuming Vogel's parametrization of the Casimir spectrum on S² for 𝔤 and for the simple algebras in its chain of blocks; the proof also uses an index condition and the location of the point β = γ, both read off the classification, and dim V(2θ) = 300 for 𝔰𝔬₈. Vogel's dimension formula for the pieces is not assumed; it follows by induction on the rank. The key steps are a Casimir identity on the odd part of S²𝔤, which reduces spin 1/2 to a Vandermonde moment problem, and a three-moment computation at spin 0. Corollaries include the analogous rule for Λ²𝔤, which needs the parametrization for 𝔤 alone, and the number of H-invariants in each piece. We record verification on 31 algebras, a computer-assisted integrality theorem on six lines of the plane with a conjecture for the whole plane, and an identity of level-one central charges that holds on the whole plane.The scripts in anc/ use exact rational arithmetic and reproduce every computation in the paper, including the computer-assisted proof of the integrality theorem (anc/ubr_int.py).