Foundations of an E₈ group field theory: action uniqueness, vacuum selection, and a four-dimensional algebraic substrate

We formulate a local quantum field theory for an adjoint-valued scalar field Φ: E₈ → 𝔢₈ on the compact real form of the exceptional Lie group. The framework is based on four standard meta-principles of local Lagrangian QFT and four E₈-specific postulates. We trace its consequences in four steps: (i) Separate E₈L × E₈R × E₈Ad invariance, the Casimir-degree spectrum of 𝔢₈, Ostrogradski exclusion, and stability collapse the IR action to two leading and nine sub-leading Wilson coefficients with a bounded remainder. (ii) On the open half-line c₂ < 0 the action admits a symmetry-breaking vacuum on the round 247-sphere of 𝔢₈, stratified by Levi sub-root-systems. (iii) Two algebraic filters together with one structural-geometric input select EIX, the unique compact quaternion-Kähler symmetric space of E₈ (EIX = E₈ / (E₇ × SU (2) ) ). (iv) Suter's rank–antichain identity yields a four-dimensional abelian sector 𝔞 ⊂ 𝔪EIX with dim_ℝ 𝔞 = 4 and P_μ, P_ν = 0, and the lower homotopy of EIX vanishes, ruling out Kibble-type defects from the phase transition. The promotion to a smooth Lorentzian four-manifold is recorded as a hypothesis: dimension and abelian closure follow from (iv) ; Lorentzian signature and the global Poincaré subgroup are closed at leading bosonic Gaussian (and at a Dₛtab-interior point) by Osterwalder–Schrader reconstruction; metric reconstruction is closed at leading + BV-BRST sub-leading Sakharov order via a Camporesi–Higuchi spectral-zeta computation on EIX, with structural coefficient 𝒱ᵢnd^ (EIX) = 144 to within a ≤ 3. 7% finite-part bound. We record explicitly the structural selections and the residual open problems. The construction is parallel to the non-compact E₈ programs of Lisi (2007), Manogue–Dray–Wilson (2022), and Wilson (2025), and structurally compatible with division-algebra approaches such as Furey (2018). It is not addressed by the Distler–Garibaldi no-go theorem (2010). Verification scripts are available at https: //github. com/lukasbednarik/E8-GFT.