Two Exceptions Among the Exceptional: Derivative Layers of Big Algebras and Yakimova’s Conjecture

Hausel's big algebra is a commutative algebra of polynomial operators attached to an irreducible representation of a complex simple Lie algebra. We study its filtration by invariant derivatives and settle Yakimova's exceptional generating-set conjecture: homogeneous generating systems satisfying the recursive symmetrisation condition exist in G₂, E₆ and E₇, and fail to exist in F₄ and E₈. Every invariant satisfying that condition is good in every representation, meaning that all its derivatives belong to the corresponding big algebra. Exact Cartan recursion and direct contraction certify the exceptional formulas and their potentials; commutator ranks determine the corresponding adjoint-good spaces. Under Hausel's Satake identification, the first layer realizes ordinary cohomology inside the intersection cohomology of an affine Schubert variety. The second adjoint layer gives a root-sum tensor for the almost-dual product and, in simply-laced types, Saito's metric. The weighted tensors form flat pencils in every rank of types B and C, with constant contravariant connection coefficients and a complete Jordan stratification. An explicit flat pencil is also obtained in type F₄. Derivative depth is determined in rank two and equals the Coxeter number minus one for regular highest weights in the classical types, G₂, E₆ and E₇. A conditional mixed Hodge–Riemann theorem specifies the pairing and cone hypotheses required for the principal specialization.