HγC X: Projected Accumulation Geometry and Weak-Field Lensing in the HγC Framework

We extend the HγC framework to the interpretation of weak-field lensing observables. Previous work established a minimal geometric bridge by embedding the effective accumulation potential ΦA = αA into an emergent weak-field metric, thereby providing a common description of slow massive motion and leading-order null propagation. Building on that result, we argue that weak-field lensing in the HγC framework should not be interpreted simply as a direct tracer of the instantaneous baryonic distribution, but rather as a probe of projected accumulation geometry. The central reason is that the effective gravitational response of the accumulation sector is profile-dependent. It is governed not by local baryonic source terms in isolation, but by the developed spatial structure of the accumulation field after sourcing, smoothing, redistribution, and environmental modulation have acted on the system. Once this profile dependence is translated into the weak-field geometric language, the lensing signal becomes sensitive to the projected accumulation profile along the line of sight rather than to the local baryonic map alone. As a consequence, the lensing-relevant geometry may be broader, smoother, or spatially displaced relative to the dominant visible baryonic component. This interpretive shift becomes especially important in dynamically disturbed cluster mergers. We argue that such systems should be treated as nonequilibrium accumulation environments in which collisional gas, galaxy populations, and lensing-inferred geometry need not remain spatially coincident. Within this perspective, offset lensing peaks are not excluded in principle. In particular, the Bullet Cluster is discussed as a proof-of-principle case showing why a spatial offset between the dominant gas distribution and the lensing-relevant geometry does not by itself constitute a conceptual refutation of the HγC framework. The aim of the present paper is not to provide a full quantitative reconstruction of merger systems, nor a complete relativistic theory of lensing. Rather, it is to establish the weak-field interpretive basis on which lensing observables can be understood as probes of projected accumulation geometry. We conclude that nonequilibrium cluster lensing offers a natural phenomenological arena for testing persistence, redistribution, and finite relaxation in the accumulation sector.