Action of the Euclidean versus projective group on an agent's internal space in curiosity driven exploration

According to the Projective Consciousness Model (PCM), in human spatial awareness, 3-dimensional projective geometry structures information integration and action planning through perspective taking within an internal representation space. The way different perspectives are related to and transform a world model defines a specific perception and imagination scheme. In mathematics, such a collection of transformations corresponds to a 'group', whose 'actions' characterize the geometry of a space. Imbuing world models with a group structure may capture different agents' spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such a geometric structure impacts agents' behaviors, comparing how the Euclidean versus projective groups act on epistemic value in active inference, drive curiosity, and exploration. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task. The projective group's nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward objects with uncertain locations due to limited sampling. The Euclidean group had no effect on epistemic value: no action was better than the initial idle state. In structuring a priori an agent's internal representation, we show how geometry can play a key role in information integration and action planning. Our results add further support to the PCM.