GIFT 3.3 - Supplement S1: Mathematical Foundations

This supplement presents the mathematical architecture underlying GIFT. Part I develops the E₈ exceptional Lie algebra with the exceptional chain identity. Part II introduces G₂ holonomy manifolds, including the correct characterization of the g₂ subalgebra as the kernel of the Lie derivative map. Part III presents a K₇ topological blueprint via the twisted connected sum framework, deriving Betti numbers and cohomological invariants from asymptotically cylindrical building blocks (conditional on existence of appropriate smooth G₂ manifold). Part IV establishes the algebraic reference form determining det(g) = 65/32, with Joyce's theorem guaranteeing existence of a torsion-free metric. PINN validation achieves a torsion scaling law ∇φ(L) = 8.46 × 10⁻⁴/L² and spectral fingerprint [1, 10, 9, 30] at 5.8σ significance. All algebraic results are formally verified in Lean 4.