Paper A: The Algebra of E₈₍₈₎, the Split Octonion, and the Origin of the Framework

First paper in the E₈₍₈₎ Split Form Unification series. Constructs the split octonion and derives the symmetry breaking cascade E₈ → E₇ × SU(2) from pure algebra, with no physical input. Establishes the Cayley-Dickson grading of the Weyl vector (Theorem 1), the exceptional Jordan algebra H₃⁰ and its 26-dimensional traceless arena, the zero-sum principle forcing n=3 as the minimum interaction vertex, and the 4704 Weyl invariant. Three independent confirmations of the cascade are given: the Kostant partition function, the McKay correspondence via the binary octahedral group 2O, and the cross-gauge interaction count. All results are computationally verified (NumPy/SageMath). No physical identification is made; that begins in Paper B.