TWIST NUMBERS AND FOUR-DIMENSIONAL INTEGER J ALGEBRA

From i² = -1 to j + j² + j³ + j⁴ = -1Complex numbers adjoin i to the integers. Two dimensions. One relation: i² = -1. Twist numbers adjoin j to the integers. Four dimensions. One relation: j + j² + j³ + j⁴ = -1. Commutative, exact, integer.This paper defines the twist number ring J = Z[j], a four-dimensional commutative integer algebra generated by a single unit j satisfying the fifth cyclotomic relation. The ring is isomorphic to the classical cyclotomic integers Z[zeta_5], but is presented in a notation that foregrounds the defining property over the construction, and emphasizes integer coefficients over real embeddings.The paper develops: the closure relation and multiplication table; comparison with quaternions (commutative, one generator, integer coefficients); the golden ratio as a native algebraic element (phi = -(j² + j³)); the real twist subring Z[phi] with the golden closure phi² = phi + 1; the tower Z subset Z[phi] subset Z[j] with dimensions 1, 2, 4; matrix representations with all entries in {-1, 0, 1}; the distinguished unit J = 1 + j² with norm 1 and absolute value 1/phi; the decomposition M_J = I + D_J (identity plus phase motor); and the dual embedding diagram showing how the Galois action transforms the pentagon {5} into the pentagram {5/2}, with the product formula (1/phi) x phi = 1.The element J = 1 + j² is the starting point of the TWIST-J framework for discrete mathematical physics (twistj.com). The present document treats twist numbers as a standalone mathematical object.www.twistj.com