Paper D: The Dance - Interactions, Coupling Constants, and the Unified Action from the Jordan Triple Product

The fourth paper in a series deriving Standard Model physics from the split real form of E₈, this paper identifies the interaction vertex of the framework and constructs an action that unifies it with gravity. The vertex is shown to be the Jordan triple product on the split octonions 𝕆_s, the unique Lorentz-covariant trilinear operation on this algebra (up to overall normalization). Its structure constants are all exactly ±1, so the interaction strength is fixed by arithmetic rather than tuned. Pauli exclusion and Bose enhancement follow from the triple's slot structure via Artin's theorem, and Feynman diagrams are forced by the non-associativity of the 4-point function. Principal results: The grand unified coupling α_GUT = 2/49 is derived from the Jordan Casimir eigenvalue √7, with zero free parameters. A Z₂-grading by the center Z(E₇) ≅ ℤ₂ classifies every E₇ representation as symplectic or orthogonal. Proton stability and dark-matter stability follow as consequences of representation type, exact at all tensor orders. The three visible sectors (gauge 133, matter 1463, Higgs 1539) form a closed, self-generating algebra under the tensor product, separated from the dark sector (912) by sealed creation channels. A complete matter Lagrangian is constructed on the D₄ root lattice from the algebra alone, and standard QED is recovered at tree level with zero imported ingredients. The Scalar Contraction Unified Theorem establishes that gravity, quantum measurement, and particle interaction share a common algebraic origin in the Freudenthal cubic, specialized at three depths. Opening the octonionic link variables from the quaternionic subalgebra to the full split octonion yields a unified action containing both MacDowell-Mansouri gravity and gauge forces in a single line.