Mathematical Foundations of the Order Parameter Spacetime Theory

This paper establishes rigorous mathematical foundations for the Order Parameter Spacetime Theory. The core content comprises four parts. First, we give a strict differential-geometric definition of the complex metric — defining it as a map from sections of the tangent bundle on a real manifold to the complexified tangent bundle, whose real part corresponds to the reversible inertial response and the imaginary part to irreversible dissipation. The complex geodesic equation is rigorously derived from a variational principle of the complex metric. Second, we classify the non-Riemannian geometric structures in the theory — proving that the activation of the information dissipation tensor induces torsion and non-metricity; torsion describes the degree of geometric twisting when the information dissipation channel opens, and non-metricity describes the non-integrability of the metric length caused by information loss. In the limit of vanishing information dissipation tensor, both torsion and non-metricity vanish, recovering standard Riemannian geometry. Third, we establish a topological classification of Type-B traversable singularities — using the degeneracy of the metric determinant and the asymptotic behavior of sectional curvatures, we give a precise definition of Type-B singularities, classify them into three fundamental topological types (traversable, horizon-enveloped, and degenerate), and prove a constraint relation between the finiteness of the relaxation time and the curvature divergence. Fourth, we derive the information–Einstein equations from the complete variation of the unified action, proving that when the information dissipation tensor goes to zero, the equations reduce exactly to the standard Einstein field equations; when the curvature goes to zero, the equations reduce exactly to the equation of motion for the order-parameter field in flat spacetime. The mathematical framework built here introduces no new physical hypotheses; rather, it unifies the geometric concepts already used throughout the series of papers into an axiomatic mathematical system, providing the final mathematical support for the logical closure of the Order Parameter Spacetime Theory.