Unitary Field

We present a compact mathematical formulation of the Unitary Field, defined as a triadic relation among domain, structure, and execution. The normalized scalar core D D= 1, R= , ε= R−D, ε together with the explicit choice of the positive branch, determines R= Φ, ε= Φ−1 , and the self-similar discrete scale Xn = Φn. The fundamental relation lifts scale- covariantly to ∆Xn = εXn = Xn−1, Xn+1 = Xn + Xn−1. Preserving the endpoint typing of D,ε, followed by the declared structural policy of terminal groupoid completion, yields the pair groupoid on two objects. Its linearization is M2; inversion induces a canonical anti-involution; the recurrence selects the generators J and K; and bilateral operator closure reaches End(A). Four derived operators satisfy Clifford relations of signature (3,1) and generate the full endomorphism algebra. The logical status of assumptions, definitions, the relational policy, canonical constructions, internal theorems, and interpretations is kept explicit throughout. In particular, an internal quadratic scalarization of an operator subspace does not require an external observation functional as a mathematical primitive.