HγC IX: Emergent Geometry from Relational Accumulation in the Weak-Field Regime

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…