GIFT 3.2 - Supplement S1: Mathematical Foundations

This supplement presents the mathematical architecture underlying GIFT. Part I develops E8 exceptional Lie algebra with the Exceptional Chain theorem. Part II introduces G2 holonomy manifolds. Part III establishes K7 manifold construction via twisted connected sum, building compact G2 manifolds by gluing asymptotically cylindrical building blocks. Part IV establishes the algebraic reference form determining det(g) = 65/32; Joyce's theorem guarantees a torsion-free metric exists within this framework. All results are formally verified in Lean 4.