Exact Belief Compression from Wreath-Product Symmetry: An AR-Inspired Pocket Cube Study

Augmented reality and robotics tasks often need to be solved under visual obstruction and time constraints. Cameras that only show a single view require an agent to plan a solution against anything it cannot see. We show that in a class of such tasks, the group structure of the hidden space makes tracking the exact belief cheap, without any learning or approximation. In particular, when the state space can be written as a wreath product $C_k \wr S_N$ with an orientation-parity constraint and the camera reveals a face subset, then simple per-position counting bounds every fibre of the observation map independently of the group order. Our study uses the $2{\times}2{\times}2$ Pocket Cube as a toy model. When observed from a single angle, the three visible faces collapse the posterior over $3.7$ million states to at most six candidates, so the exact Bayesian update becomes a constant-time lookup. Planning a solution inherits the same reduction in per-step cost. In this model system, the reachable belief graph stays small enough for exact value iteration and we certify the partially observable Markov decision process optimum. We show that beyond the toy model, the same approach can be used on the Pyraminx puzzle, and for the real-world task of circuit-board inspection.