Complex Phase Structure Emerging from Nontrivial Quadratic Closure: A Foundational Connection to Finite-Order Exchange Systems and Discrete Born-Type Weights

This preprint separates two mathematically distinct quadratic structures in a closed exchange model. The algebraic closure form Q_0(x)=sum_k x_k^2 contains no conjugation, whereas the positive readout Q_+(x)=sum_k x_k^*x_k is introduced only at the observation stage. For a nontrivial solution of Q_0=0 over a commutative extension K containing the real field, the solution vector must lie outside R^N. In the minimal two-component case x^2+y^2=0 with x nonzero, the ratio y/x is a square root of -1 and generates the minimal subfield R(i), isomorphic to C, without using complex conjugation. With additional assumptions of continuity, reversibility, positive-readout preservation, exchange symmetry, finite order, and squared-amplitude readout, the previously derived discrete channel weights cos^2(pi m/n) and sin^2(pi m/n) are recovered. The paper also proves that the existing kernel U=P_s+zeta P_a does not generally preserve the Q_0=0 closure cone: for x_0=(1,i)^T, Q_0(Ux_0)=i(1-zeta^2). Thus the dynamical connection between algebraic quadratic closure and the finite-order exchange kernel remains an explicit open problem, labeled C2. The work does not derive the full Born frequency rule; it identifies the nonreal direction forced by nontrivial quadratic closure and classifies the additional constructions required to reach discrete Born-type weights.