When the body computes: The information geometry of cheap control

A body can sustain useful motion that a controller would otherwise have to produce. But when does exploiting that motion make control cheaper, and can an information-based reward encourage it? We develop an account of cheap control that separates what a policy encodes from what it physically imposes. A common divergence geometry connects two complementary views of embodied dynamics: action influence through empowerment, and state propagation through morphological computation. Rewarding the latter induces policy compression gated by the body, targeting precisely the controller information that the body's own dynamics render redundant. A causal decomposition extends the result to feedback. We then separate informational from physical cheapness. Physical savings depend on an alignment between what the body sustains, what the task permits, and what control costs. Experiments recover the selective compression mechanism across linear and nonlinear control. The resulting theory explains how embodiment simplifies control, and when information savings become physical ones.