HγC XII: Nonequilibrium Evolution of Projected Accumulation Geometry in the HγC Framework

We formulate the next dynamical step in the HγC lensing programme by developing a minimal nonequilibrium description of projected accumulation geometry in disturbed systems. Earlier work established that weak-field lensing should be interpreted not as a direct tracer of the instantaneous baryonic distribution, but as a probe of projected accumulation geometry. The natural next question is therefore how this projected geometry evolves in time when collisional gas, galaxy populations, and lensing-relevant structure do not remain pointwise coincident. In the present paper, we introduce a minimal phenomenological description of time-dependent accumulation-profile evolution and use it to clarify how projected accumulation geometry can acquire persistence, lag, and offset morphology in merger environments. The aim is not to provide a precision reconstruction of observed cluster collisions, a hydrodynamic treatment of gas passage, or a complete relativistic theory of lensing. Rather, it is to establish the simplest dynamical framework in which nonequilibrium lensing structures can be interpreted as natural consequences of finite profile relaxation within the HγC programme. The central conceptual point is that the geometry relevant to weak-field lensing need not track the visible baryonic configuration instantaneously. Instead, it reflects a developed profile with its own response timescale and smoothing properties. Within this perspective, an offset between collisional gas and lensing-inferred geometric structure is not, by itself, an automatic contradiction, but a diagnostic manifestation of nonequilibrium profile dynamics. We conclude that disturbed cluster mergers should be treated not merely as binary tests of alignment, but as structured probes of profile persistence, redistribution, lag, and finite relaxation in the accumulation sector. In this sense, the present paper supplies the nonequilibrium dynamical closure of projected accumulation geometry in the HγC framework.