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, from four standard meta-principles of local Lagrangian QFT and four 𝔢₈-specific postulates, and trace its consequences in four steps. (i) Separate E₈^L × E₈^R × E₈^Ad 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 Bose–Einstein-type symmetry-breaking vacuum on the round 247-sphere of 𝔢₈, stratified by Levi sub-root-systems. (iii) Two algebraic filters (cubic-anomaly safety; existence of an SU(2) substrate for a Skyrme-type soliton) together with one structural-geometric input (compact quaternion-Kähler structure) select the unique adjoint orbit EIX = E₈ / (E₇ × SU(2)). (iv) Suter's rank–antichain identity yields a four-dimensional abelian sector 𝔞 ⊂ 𝔪_EIX 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 𝒟_stab-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 𝒱_ind^(EIX) = 432/3 = 144 to within a ≤ 3.7% finite-part bound. We record explicitly the structural selections (compact real form; single-copy adjoint-valued field; truncation (n,k) ≤ (4,4); BEC branch; Cas₂ vacuum truncation; Wolf-space input F3) and the residual open problems. The construction is parallel to the non-compact E₈ programs of Lisi (2007), Manogue–Dray–Wilson (2022), Wilson (2025), and Furey (2018), and is not addressed by the Distler–Garibaldi (2010) no-go theorem. Standard-Model embedding, particle content, and the Wilsonian calibration of Newton's constant are deferred. Verification scripts: https://github.com/lukasbednarik/E8-GFT