The octic invariant of E₈ and the exceptional Wolf direction

The Casimir degrees of E₈ are {2, 8, 12, 14, 18, 20, 24, 30}, so beyond the quadratic Casimir its first primitive invariant is the octic p₈(Φ̂) = Tr(ad_Φ̂⁸) = Σ_α ⟨α, Φ̂⟩⁸. Starting from E₈ alone, we ask which adjoint symmetry breaking this first invariant selects. On the Cartan unit sphere p₈ ranges over [57/2, 39], with maximum at the highest-root direction (centralizer E₇ × U(1)) and minimum at Spin(14); we prove this by linear-programming bounds resting on the spherical 7-design property of the roots. The extremal values and the maximizer's uniqueness are theorems of Borodachov; new here are uniqueness at the minimum and the Morse structure. The restricted Hessian is, by Schur's lemma, block-scalar, obeys Tr H = 7(480 − 16 p₈), and singles out the highest-root stratum as the unique negative-definite one (Morse index 7), with eigenvalue −144 = −8 h∨(E₇); this direction is the minimal nilpotent orbit and the quaternion-Kähler (Wolf) direction of E₈. The algebra then stops at one place: the sign ν of the degree-eight coupling is a Wilson coefficient, and both signs give a stable phase (ν < 0 selects E₇, ν > 0 the Spin(14)/SU(8) side). We map both branches and fix ν < 0 as an explicit structural choice; on that branch the vacuum is E₈/(E₇ × U(1)), whose quaternion-Kähler base is the exceptional Wolf space EIX = E₈/(E₇ × SU(2)), and its spectrum is 114 massless Goldstone modes, one radial mode and a single level of 133 modes at −144ν, the Morse eigenvalue carried onto the whole unbroken e₇ by Schur's lemma. The extremal and Morse results are unconditional: E₈ produces the fork and ranks it, and EIX is the quaternion-Kähler base of the vacuum on the branch we choose.