We formulate a weak-field geometric extension of the HγC framework, in which the effective gravitational response generated by the relational accumulation field is reinterpreted as an emergent metric perturbation rather than as a fundamental geometric structure. The accumulation field A(x) is understood as a coarse-grained macroscopic descriptor of retained relational organization, whose spatial gradients generate an effective gravitational bias. The aim of the present paper is to identify the minimal geometric layer required to express this bias in weak-field spacetime language. In this formulation, the standard nonrelativistic limit for massive particles is recovered, while the same weak-field geometry also provides a minimal basis for discussing null propagation and light bending. The construction is based on an isotropic weak-field completion of the accumulation potential and should be regarded as a controlled geometric reformulation rather than as a complete relativistic theory. This result clarifies how geometry can emerge as an effective large-scale encoding of accumulated relational structure and establishes the weak-field bridge needed to connect the HγC framework to lensing-related observables. Detailed relativistic consistency, quantitative lensing phenomenology, and nonequilibrium systems are left for future work.
Hans Van Cools (Tue,) studied this question.